Trilayer Hubbard Model

• Trilayer Hubbard model is a strongly correlated electron framework defined on three coupled layers with intralayer hopping, interlayer hybridization, and local Coulomb repulsion.
• It encompasses diverse realizations such as ABC graphene, multilayer cuprates, and moiré systems, each exhibiting unique band structures and interaction-driven phases.
• The model highlights phenomena like Mott transitions, layer-selective magnetism, and unconventional superconductivity emerging from varied lattice symmetries and electron correlations.

Searching arXiv for recent and foundational papers on trilayer Hubbard models across graphene, cuprates, and related multilayer systems. The trilayer Hubbard model denotes a family of strongly correlated electron models defined on three coupled layers, with intralayer kinetic energy, interlayer hybridization, and local Coulomb repulsion as the minimal ingredients. In practice, the term covers several inequivalent realizations: three coupled honeycomb sheets for rhombohedral graphene, three square-lattice layers for multilayer cuprates, triangular moiré lattices in ABC-trilayer-graphene/hexagonal-boron-nitride and alternating twisted trilayer WSe$_2$, as well as multi-orbital or extended-Hubbard variants relevant to nickelates and exact pairing constructions. Across these settings, the central questions are the same: how interlayer tunneling reshapes the low-energy bands, when a Mott state appears, how magnetic correlations are distributed across the three layers, and under what conditions superconductivity or exact paired eigenstates emerge (Dai et al., 2020, Liu et al., 9 Jul 2025, Zhang et al., 2018, Liu et al., 1 Jul 2026, Chen et al., 9 Aug 2025).

1. Model classes and Hamiltonian structure

Representative trilayer Hubbard formulations differ primarily by lattice geometry, orbital content, and whether the three layers are equivalent. In ABC graphene trilayers, the model is a single-band Hubbard Hamiltonian on three coupled honeycomb sheets with nearest-neighbor intralayer hopping $t$, vertical interlayer hopping $t_\perp$, chemical potential $\mu$, and on-site repulsion $U$. In multilayer cuprates, the standard formulation uses three square-lattice layers, two equivalent outer layers (OL) and one inner layer (IL), with nearest- and next-nearest-neighbor hoppings $t$ and $t'$, interlayer hopping $t_\perp$, and layer-dependent chemical potentials $\mu_m$. In moiré realizations such as ABC-TLG/hBN and twisted trilayer WSe$_2$, the low-energy problem is often projected to a triangular lattice with an effective hopping $t$0 and a Hubbard $t$1, while gate or displacement fields tune the bandwidth and layer polarization. Extended and multi-orbital variants add interlayer density interactions $t$2 or Kanamori terms with Hund’s coupling $t$3 (Dai et al., 2020, Liu et al., 9 Jul 2025, Chen et al., 2019, Jeong et al., 9 Jun 2026, Liu et al., 1 Jul 2026, Chen et al., 9 Aug 2025).

A concise way to compare these realizations is to separate them by lattice and interaction content.

Realization	Degrees of freedom	Characteristic terms
ABC graphene trilayer	Three honeycomb layers	$t$4, $t$5, $t$6, optional layer potentials
Multilayer cuprate trilayer	OL–IL–OL square lattices	$t$7, $t$8, $t$9, $t_\perp$0, $t_\perp$1, $t_\perp$2
Moiré trilayers	Triangular lattice Wannier band	$t_\perp$3, $t_\perp$4, displacement-field tuning
Extended or multi-orbital trilayers	Layer/orbital-resolved fermions	$t_\perp$5, Kanamori $t_\perp$6, cross-layer couplings

For the honeycomb ABC case, the Hamiltonian is written explicitly as

$t_\perp$7

with $t_\perp$8 eV and $t_\perp$9 (Dai et al., 2020). For the square-lattice cuprate trilayer,

$\mu$0

with $\mu$1, $\mu$2, and $\mu$3 in the cited DCA study (Liu et al., 9 Jul 2025).

2. Band structure, tuning parameters, and effective scales

The noninteracting structure of the trilayer Hubbard model is highly realization-dependent, and this dependence largely determines the correlated phase diagram. In ABC-stacked honeycomb trilayers at $\mu$4, there are six bands; two low-energy bands touch at each $\mu$5 point with a cubic dispersion $\mu$6, the density of states is enhanced near charge neutrality, and van Hove singularities appear just above and below the neutrality point. When hBN superlattice effects are included, narrow moiré subbands can have bandwidth comparable to $\mu$7, which predisposes the system to interaction-driven ordering (Dai et al., 2020).

In square-lattice trilayers with large $\mu$8, diagonalization produces bonding, nonbonding, and antibonding bands,

$\mu$9

with $U$0. Near $U$1, the bonding and nonbonding bands as a whole are close to half filling, which is the regime emphasized in the FLEX study of electron-hole asymmetry (Yamada et al., 23 Jun 2026).

In moiré triangular-lattice realizations, the principal control parameter is the ratio $U$2 or $U$3, and this ratio is strongly displacement-field dependent. In ABC-TLG/hBN, spectroscopy and continuum modeling give $U$4 meV, moiré wavelength $U$5 nm, and $U$6 evolving from $U$7 meV at $U$8 to $U$9 meV as $t$0 meV, so that $t$1 rises from $t$2 to $t$3 using $t$4 (Yang et al., 2022). The transport study of the same platform quotes a similar tuning window, $t$5 meV, $t$6 meV, and $t$7 evolving from $t$8 to $t$9 (Chen et al., 2019).

Alternating twisted trilayer WSe$t'$0 realizes a strongly correlated triangular-lattice Hubbard regime by mirror-enhanced confinement in the middle layer. The cited work reports $t'$1 meV at $t'$2, $t'$3 meV, $t'$4–$t'$5 meV, and $t'$6–$t'$7 (Jeong et al., 9 Jun 2026). In mirror-symmetric magic-angle twisted trilayer graphene, the atomistic mean-field Hubbard description yields characteristic flat bands together with a Dirac cone at charge neutrality; the flat bands have zero Chern numbers, but the multiband Berry-curvature distribution is non-trivial over the moiré Brillouin zone (Rodrigues et al., 30 Jan 2025).

3. Mott transitions and magnetic ordering

At half filling, the trilayer Hubbard model commonly supports interaction-driven insulating and magnetic states, but the critical coupling and the detailed order depend on geometry. In the ABC graphene trilayer on the honeycomb lattice, finite-temperature DQMC on $t'$8 lattices finds a metal-to-Mott transition at $t'$9, with a consistent critical coupling $t_\perp$0 for $t_\perp$1. The transition is diagnosed by the dc conductivity

$t_\perp$2

the Fermi-level density of states $t_\perp$3, and the antiferromagnetic structure factor

$t_\perp$4

For $t_\perp$5, the thermodynamic-limit extrapolation yields an AFM-ordered Mott insulator (Dai et al., 2020).

In the constrained-phase QMC study of rhombohedral trilayer graphene under a perpendicular electric field, antiferromagnetism is suppressed by the field, especially in the long-range part, but the dominant magnetic fluctuations remain antiferromagnetic. The same work reports that intralayer nearest-neighbor spin correlations remain negative, while long-range spin correlations become essentially zero for $t_\perp$6 at large $t_\perp$7 (Dai et al., 2022).

In the atomistic twisted trilayer graphene Hubbard model, turning on the on-site interaction at charge neutrality produces a metallic-to-antiferromagnetic phase transition at $t_\perp$8, with the AF gap opening in the flat bands while the Dirac cone remains ungapped. The AF region shrinks when mirror symmetry is broken by $t_\perp$9, and even a small $\mu_m$0 meV pushes $\mu_m$1 upward substantially (Rodrigues et al., 30 Jan 2025).

Moiré triangular-lattice realizations exhibit experimentally identified correlated insulating states rather than a single universal half-filled Mott point. In ABC-TLG/hBN, resistivity peaks appear at $\mu_m$2 and $\mu_m$3 below $\mu_m$4 K, and FTIR-photocurrent spectroscopy resolves a broad absorption peak at $\mu_m$5 meV at $\mu_m$6, together with peaks near $\mu_m$7 meV at $\mu_m$8 and $\mu_m$9 (Chen et al., 2019, Yang et al., 2022). In alternating twisted trilayer WSe$_2$0, a robust correlated insulator at $_2$1 persists for $_2$2 with activation gaps $_2$3–$_2$4 meV, and a weaker intermediate insulator with $_2$5 meV appears between superconductivity and the half-filled insulating state; the cited work states that this may correspond to a non-magnetic or spin-liquid Mott phase (Jeong et al., 9 Jun 2026).

4. Superconductivity and pairing channels

A principal result across trilayer Hubbard studies is that superconductivity is typically unconventional and strongly constrained by lattice symmetry, layer differentiation, or orbital structure. In the honeycomb ABC trilayer Hubbard model, doping away from half filling produces a superconducting instability whose dominant channel is chiral $_2$6, rather than extended $_2$7 or $_2$8. The nearest-neighbor form factor is

$_2$9

and at $t$00 with $t$01, the pairing susceptibility $t$02 grows most rapidly on cooling while $t$03 approaches $t$04. The effective $t$05 pairing interaction strengthens with increasing $t$06 and is suppressed as $t$07 increases up to $t$08 (Dai et al., 2020).

Under a perpendicular electric field in rhombohedral trilayer graphene, the constrained-phase QMC study similarly finds correlation-driven $t$09 superconductivity, but in that formulation the largest interaction-induced vertex occurs in the next-nearest-neighbor $t$10 channel. The NNN $t$11 vertex remains positive for $t$12 and grows strongly with increasing $t$13 (Dai et al., 2022).

In square-lattice multilayer cuprates, the leading instability is $t$14-wave. The DCA+CT-AUX study solves the lattice-level Bethe-Salpeter eigenproblem and identifies $t$15 from $t$16. Its central result is that imbalanced doping with $t$17 enhances the global $t$18 above the single-layer benchmark; at $t$19, the maximum occurs near $t$20, where $t$21, while $t$22 (Liu et al., 9 Jul 2025).

The square-lattice FLEX study near one-third filling identifies a different pairing phenomenology. For $t$23, the leading superconducting eigenvalue $t$24 is symmetric about $t$25 and the gap has an $t$26 form, changing sign between bonding and nonbonding pockets. For $t$27, superconductivity is more favored on the hole-doped side $t$28, and the gap becomes essentially $t$29-independent, indicative of local interlayer/intra-cell pairing (Yamada et al., 23 Jun 2026).

Strongly correlated triangular-lattice moiré trilayers also exhibit superconductivity near correlated insulators. In alternating twisted trilayer WSe$t$30, superconductivity appears adjacent to the half-filled insulator and again near quarter filling. The half-filling dome SC1 has $t$31 mK, $t$32 mT, and $t$33 nm, while the quarter-filling feature SC2 has $t$34 mK and $t$35 mT. The cited work states that direct phase-sensitive measurements are not yet reported, but that the close analogy to triangular Hubbard models suggests spin-singlet $t$36-wave or chiral $t$37 superconductivity near half filling, and either $t$38- or $t$39-wave near quarter filling (Jeong et al., 9 Jun 2026).

5. Layer differentiation, asymmetry, and nonstandard pairing mechanisms

A distinctive feature of trilayer, as opposed to bilayer or single-layer, Hubbard physics is genuine layer differentiation. In the cuprate trilayer DCA study, the realistic regime is $t$40. There the IL develops a pseudogap below $t$41, while the OL remains metallic down to the lowest accessible temperatures. Layer-resolved Bethe-Salpeter eigenvectors show that only the IL component has a robust $t$42-wave form, and the paper concludes that the IL itself can drive $t$43-wave superconductivity while the OLs only serve as the charge reservoir; balanced doping $t$44 is detrimental to superconductivity (Liu et al., 9 Jul 2025).

The FLEX study near one-third filling identifies a different sort of differentiation, namely unequal many-body renormalization of the bonding, nonbonding, and antibonding bands. At $t$45, the quasiparticle weights remain close, with all $t$46. At $t$47, the nonbonding and antibonding bands are more strongly renormalized, with $t$48, $t$49, and $t$50. The paper attributes the resulting electron-hole asymmetry of superconductivity to this asymmetric renormalization (Yamada et al., 23 Jun 2026).

In the realistic two-orbital trilayer Hubbard model for La$t$51Ni$t$52O$t$53, DMRG resolves orbital-selective magnetic correlations: the $t$54 orbital exhibits both interlayer and cross-layer AFM correlations, while the $t$55 orbital shows exclusively cross-layer AFM correlations. The only quasi-long-range superconducting channel is the cross-layer pairing in the $t$56 orbital, with $t$57, while all other pairing channels decay exponentially. The cited work further states that Hund’s rule coupling is essential for forming the superconducting order (Chen et al., 9 Aug 2025).

An algebraically exact extension appears in the trilayer extended Hubbard model with interlayer density interaction $t$58. There, interlayer triplet-pairing operators $t$59 satisfy a restricted spectrum-generating algebra in the trilayer case, allowing exact condensate eigenstates

$t$60

with energy

$t$61

The pair correlator exhibits ODLRO,

$t$62

and when $t$63, these triplet-pair states coexist with the usual on-site $t$64-pairing sector (Liu et al., 1 Jul 2026).

6. Experimental realizations, scope, and open interpretive issues

The trilayer Hubbard model is not a single universal Hamiltonian with a single universal phase diagram. The literature instead supports several experimentally motivated subclasses. ABC-TLG/hBN on the topologically trivial side $t$65 admits a Wannier-based triangular-lattice description with four spin-valley flavors, on-site $t$66 meV, non-negligible nearest-neighbor repulsion, and displacement-field-controlled bandwidth. The same platform is proposed as a setting to study bandwidth- and doping-controlled Mott transitions, including the possibility of a continuous Mott transition into a quantum spin liquid insulator (Zhang et al., 2018).

A related misconception is that interlayer coupling alone determines the superconducting outcome. The available studies indicate a more conditional picture. In honeycomb ABC trilayers, increasing $t$67 suppresses the $t$68 instability, although it does not eliminate it over the range studied (Dai et al., 2020). In square-lattice trilayers, by contrast, the enhancement of superconductivity is tied to layer imbalance and differentiation rather than to a uniform increase of interlayer coherence; the underdoped IL is the pairing engine and the overdoped OLs provide metallic support (Liu et al., 9 Jul 2025). In the nickelate trilayer, the relevant mechanism is again different: cross-layer AFM inherited from the $t$69 sector and transferred by Hund’s coupling into the $t$70 sector (Chen et al., 9 Aug 2025).

Moiré platforms underscore the breadth of the concept. ABC-TLG/hBN displays Mott insulating states and superconducting domes in a gate-tunable triangular Hubbard regime (Chen et al., 2019). Alternating twisted trilayer WSe$t$71 realizes a strongly correlated triangular-lattice Hubbard system with superconductivity near both quarter and half filling, together with correlated insulators, strange-metal transport above $t$72, and bad-metal behavior at higher temperature (Jeong et al., 9 Jun 2026). Mirror-symmetric twisted trilayer graphene adds a distinct angle: even when the relevant flat bands have zero Chern number, the Hubbard description can host a tunable hidden quantum geometry through a non-trivial multiband Berry-curvature distribution (Rodrigues et al., 30 Jan 2025).

Taken together, these results define the trilayer Hubbard model less as a single textbook Hamiltonian than as a multilayer organizing principle. Its essential content is the competition between intralayer Mottness, interlayer hybridization, symmetry-imposed form factors, and layer- or orbital-selective many-body renormalization. This suggests that the most robust conclusions are structural rather than universal: trilayer geometry can stabilize AFM Mott states at half filling, generate genuinely differentiated layers or bands, and support unconventional superconductivity whose dominant channel depends sharply on lattice symmetry, filling, and the way the three layers communicate.