--- title: Trilayer Hubbard Model url: https://www.emergentmind.com/topics/trilayer-hubbard-model type: topic --- # Trilayer Hubbard Model 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 [2009.14647, 2507.06614, 1809.05110, 2607.01038, 2508.06802]. ## 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 \(t_{\rm eff}=W/9\) and a Hubbard \(U\), while gate or displacement fields tune the bandwidth and layer polarization. Extended and multi-orbital variants add interlayer density interactions \(V_\perp\) or Kanamori terms with Hund’s coupling \(J_H\) [2009.14647, 2507.06614, 1901.04621, 2606.11495, 2607.01038, 2508.06802]. 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 | \(t\), \(t_\perp\), \(U\), optional layer potentials | | Multilayer cuprate trilayer | OL–IL–OL square lattices | \(t\), \(t'\), \(t_\perp\), \(U\), \(\mu_{OL}\), \(\mu_{IL}\) | | Moiré trilayers | Triangular lattice Wannier band | \(t_{\rm eff}=W/9\), \(U\), displacement-field tuning | | Extended or multi-orbital trilayers | Layer/orbital-resolved fermions | \(V_\perp\), Kanamori \(U,U',J_H\), cross-layer couplings | For the honeycomb ABC case, the Hamiltonian is written explicitly as \[ \begin{aligned} H &= -\,t\sum_{\ell=1}^3\sum_{\langle ij\rangle,\sigma} \bigl(a_{i\ell\sigma}^\dagger b_{j\ell\sigma}^{\vphantom\dagger} + \mathrm{h.c.}\bigr) \;-\;t_\perp\sum_{i,\sigma}\bigl(b_{i,1,\sigma}^\dagger a_{i,2,\sigma}^{\vphantom\dagger} + b_{i,2,\sigma}^\dagger a_{i,3,\sigma}^{\vphantom\dagger} + \mathrm{h.c.}\bigr) \\ &\quad-\;\mu\sum_{\ell=1}^3\sum_{i,\sigma}\bigl(n^a_{i\ell\sigma}+n^b_{i\ell\sigma}\bigr) \;+\;U\sum_{\ell=1}^3\sum_{i}\bigl(n^a_{i\ell\uparrow}n^a_{i\ell\downarrow} +n^b_{i\ell\uparrow}n^b_{i\ell\downarrow}\bigr)\,, \end{aligned} \] with \(t\approx2.7\) eV and \(t_\perp\approx0.138\,t\) [2009.14647]. For the square-lattice cuprate trilayer, \[ \begin{aligned} \hat H &= -\,\sum_{\langle ij\rangle,\sigma,m}\;t\,\bigl(c_{i\sigma m}^\dagger\,c_{j\sigma m} + \mathrm{H.c.}\bigr) \;-\;\sum_{\langle\langle ij\rangle\rangle,\sigma,m}\;t'\,(c_{i\sigma m}^\dagger\,c_{j\sigma m} + \mathrm{H.c.}) \\ &\quad - \;t_\perp\sum_{i,\sigma}\Bigl(c_{i\sigma,2}^\dagger\,c_{i\sigma,1} +c_{i\sigma,2}^\dagger\,c_{i\sigma,3}+\mathrm{H.c.}\Bigr) +\;U\sum_{i,m}\;n_{i m\uparrow}\,n_{i m\downarrow} \;-\;\sum_{i,m,\sigma}\mu_{m}\;n_{i m\sigma}\,, \end{aligned} \] with \(t=1\), \(t'=-0.15\,t\), and \(U=7\,t\) in the cited DCA study [2507.06614]. ## 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 \(U=0\), there are six bands; two low-energy bands touch at each \(K\) point with a cubic dispersion \(E\propto k^3\), 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 \(U\), which predisposes the system to interaction-driven ordering [2009.14647]. In square-lattice trilayers with large \(t_\perp\), diagonalization produces bonding, nonbonding, and antibonding bands, \[ \varepsilon_{B}(\mathbf k)=t\gamma_{\mathbf k}-\sqrt{2}\,t_{\perp},\quad \varepsilon_{NB}(\mathbf k)=t\gamma_{\mathbf k},\quad \varepsilon_{AB}(\mathbf k)=t\gamma_{\mathbf k}+\sqrt{2}\,t_{\perp}, \] with \(\gamma_{\mathbf k}=2(\cos k_x+\cos k_y)\). Near \(n=2/3\), 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 [2606.24126]. In moiré triangular-lattice realizations, the principal control parameter is the ratio \(U/t\) or \(U/W\), and this ratio is strongly displacement-field dependent. In ABC-TLG/hBN, spectroscopy and continuum modeling give \(U \approx 25\) meV, moiré wavelength \(l_M\approx15\) nm, and \(W\) evolving from \(\approx25\) meV at \(\Delta\approx0\) to \(\approx12\) meV as \(|\Delta|\to80\) meV, so that \(U/t_{\rm eff}\) rises from \(\sim9\) to \(\sim19\) using \(W=9t_{\rm eff}\) [2202.12330]. The transport study of the same platform quotes a similar tuning window, \(W(D=0)\approx25\) meV, \(W(D=-0.6\,{\rm V/nm})\approx10\) meV, and \(U/t\) evolving from \(\sim9\) to \(\sim22\) [1901.04621]. Alternating twisted trilayer WSe\(_2\) realizes a strongly correlated triangular-lattice Hubbard regime by mirror-enhanced confinement in the middle layer. The cited work reports \(W\approx10\) meV at \(\theta=3.87^\circ\), \(t\approx W/9\simeq1.1\) meV, \(U\approx20\)–\(30\) meV, and \(U/t\sim20\)–\(30\) [2606.11495]. 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 [2501.18491]. ## 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 \(L=6,9,12,15,18\) lattices finds a metal-to-Mott transition at \(\mu=0\), with a consistent critical coupling \(U_c\approx4\,t\) for \(t_\perp=0.138\,t\). The transition is diagnosed by the dc conductivity \[ \sigma_{dc}(T)=\tfrac{\beta^2}{\pi}\,\Lambda_{xx}(0,\beta/2), \] the Fermi-level density of states \(N(0)\simeq\beta\,G(r=0,\tau=\beta/2)\), and the antiferromagnetic structure factor \[ S_{\rm AFM}=\frac1{N_s}\Bigl\langle\bigl[\sum_{\ell,i}(S_{i\ell,A}^z-S_{i\ell,B}^z)\bigr]^2\Bigr\rangle. \] For \(U\gtrsim4\,t\), the thermodynamic-limit extrapolation yields an AFM-ordered Mott insulator [2009.14647]. 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 \(r/a\gtrsim2.5\) at large \(\varepsilon\) [2204.06222]. 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 \(U_c^{\rm TTG}\approx0.6\,t\), 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 \(\Delta_V\), and even a small \(\Delta_{\rm hBN}\sim5\) meV pushes \(U_c\) upward substantially [2501.18491]. 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 \(\nu=1/4\) and \(\nu=1/2\) below \(T\approx20\) K, and FTIR-photocurrent spectroscopy resolves a broad absorption peak at \(\hbar\omega_{\rm Mott}\approx18\) meV at \(\nu=-1/2\), together with peaks near \(\hbar\omega_{\rm corr}\approx22\) meV at \(\nu=+1/4\) and \(\nu=+1/2\) [1901.04621, 2202.12330]. In alternating twisted trilayer WSe\(_2\), a robust correlated insulator at \(v=1\) persists for \(-300\,{\rm mV\,nm^{-1}}