Two-Band Hubbard Model

• The two-band Hubbard model is a multiorbital extension that incorporates distinct electron orbitals per site, enabling studies of strong correlations and quantum phase transitions.
• It captures complex phenomena such as simultaneous and orbital-selective Mott transitions, unconventional superconductivity, and diverse magnetic and orbital orders.
• The model’s rich phase diagram encompasses correlated metallic states, Mott insulators, excitonic condensations, and nonequilibrium dynamics seen in engineered optical lattices.

The two-band Hubbard model generalizes the canonical single-band Hubbard model by introducing multiple electronic orbitals or bands per site, thereby incorporating rich spin, charge, and orbital degrees of freedom fundamental to correlation phenomena in complex materials. The model captures key physics of multiorbital transition metal oxides, iron-based superconductors, and engineered optical lattice systems, and encompasses a diverse landscape of strongly correlated ground states, quantum phase transitions, and nonequilibrium dynamics.

1. Model Formulation and Essential Parameters

The prototypical two-band Hubbard model is defined on a lattice, with electrons carrying a spin-½ degree of freedom and occupying two local orbital or band states (labeled generically by $a=1,2$). The standard Hamiltonian (in Kanamori parametrization at half-filling, Bethe lattice) reads: $$\begin{aligned} H &= H_{\rm kin} + H_{\rm int}, \ H_{\rm kin} &= -\sum_{\langle ij\rangle}\sum_{a=1}^2 \sum_{\sigma} t_a\bigl(c^\dagger_{i a\sigma}c_{j a\sigma} + \textrm{h.c.}\bigr) \ H_{\rm int} &= U\sum_{i,a} n_{i a\uparrow} n_{i a\downarrow} + U_2\sum_{i,\sigma,\sigma'} n_{i 1\sigma} n_{i 2\sigma'} + \textrm{(Hund, pair-hopping)} \end{aligned}$$ where $c^\dagger_{i a\sigma}$ creates an electron on site $i$, orbital $a$, with spin $\sigma$, and $n_{i a\sigma} = c^\dagger_{i a\sigma}c_{i a\sigma}$. $t_a$ sets the orbital-dependent bandwidth, $U$ ($U_2$) are intra-band (inter-band) repulsions; Hund's exchange $J$ and pair-hopping terms complete the Kanamori interaction. Typical cases of interest set $U_2=U$ and $J=0$ for SU(4)-symmetric models, or explore the role of finite $J$ in realistic settings (Gusmão et al., 2 Apr 2025).

Key control parameters include filling $n$, intra- and inter-orbital interactions, Hund’s exchange, crystal-field splitting, and the relative bandwidth ratio $t_2/t_1$.

2. Correlated Metallic, Mott, and Orbital-Selective Physics

The two-band Hubbard model exhibits complex phase behavior as a function of correlation strength $U$ and Hund's coupling $J$. At half-filling ($n=2$ per site) with equal $U_2=U$ and $J=0$, the system undergoes a single, simultaneous Mott transition in both bands irrespective of bandwidth asymmetry, with both orbital quasiparticle weights $Z_{1,2} \to 0$ at the same $U_c$ (Gusmão et al., 2 Apr 2025). The central metallic phase is characterized by a standard three-peak structure in the wide band, and a narrow quasiparticle peak with pseudogap features in the narrow band, but the absence of an orbital-selective Mott transition (OSMT).

For $J>0$, the interorbital holon-doublon coupling is weakened, enabling true OSMT behavior in strongly asymmetric bandwidths: the narrower band localizes (opens a gap) at lower $U_{c,2}$, while the wider band remains metallic until a higher threshold is crossed (Gusmão et al., 2 Apr 2025). The narrow-band pseudogap and central peak observed for $J=0$ serve as precursors to OSMT under finite $J$.

The nature of the Mott transition is controlled by band filling and exchange: at half-filling and small $J$, the transition is first order, but goes continuous for larger $J$ (Franco et al., 2018). Hund’s coupling reduces the critical $U_c$ for the Mott transition at half-filling and increases it at quarter-filling ($n=1$), reflecting the Janus effect of $J$ (Maurya et al., 2021).

3. Magnetic and Orbital Ordered States

When unrestricted to the paramagnetic sector, the two-band Hubbard model admits a variety of magnetic and orbital-ordered phases:

• Antiferromagnetism at half-filling: Perfect nesting yields a Slater AF insulator for $U>0$ (small $U$), crossing over to a Heisenberg AF Mott insulator at strong interaction (Schickling, 2013, Maurya et al., 2021). Hund's coupling only weakly influences the AF order, as the ordered moment is determined primarily by $U$ (Schickling, 2013).
• Ferromagnetism and orbital order at quarter-filling: At $n=1$ and finite $J$, the ground state is a fully spin-polarized (ferromagnetic) insulator with staggered (antiferro-)orbital order, separated from the paramagnetic metal by a first-order transition as $U$ increases (Maurya et al., 2021, Franco et al., 2018). This phase is stabilized by double-exchange and is entirely absent in the single-band case.
• Supersolid and spin-state orders: Near the excitonic regime and spin-state transitions (as in perovskite cobaltates), coexistence of staggered orbital polarization ("spin-state order", SSO) and excitonic condensation (EC) yields a "supersolid" region, with further differentiation of AFM order on the HS sublattice (Niyazi et al., 2020).

4. Superconductivity and Fluctuation-Driven Pairing

The two-band Hubbard model sustains a range of superconducting instabilities, notably:

• Repulsive $s$-wave pairing: In the absence of Hund's coupling ($J=0$) and for $U^{\prime} > U > 0$, enhanced on-site charge fluctuations enable a fully symmetric $s$-wave superconducting dome adjacent to the PM and metallic boundaries, even in the purely repulsive regime. The correlation-driven mechanism differs fundamentally from BCS and is controlled by the proximity to a first-order metal-paired-Mott transition (Koga et al., 2014, Steiner et al., 2016).
• Hund's-coupling induced spin-triplet pairing: For $J > 0$, "Hund metal" regimes with spin-freezing near integer filling yield $s$-wave, spin-triplet orbital-singlet superconductivity, maximized at the spin-freezing crossover. For $J < 0$ (realized e.g. via effective Jahn-Teller phonons), an "orbital-freezing" regime promotes intra-orbital, spin-singlet $s$-wave pairing, relevant for fullerides (Steiner et al., 2016).
• Competing triplet and singlet pairings: D-wave singlet superconductivity is possible only very close to half-filling and is suppressed by finite $J$, while triplet pairing is robust for a range of fillings and interaction strengths (Franco et al., 2018).

Notably, in models with explicit interband pairing processes or two-particle hybridization, effective interactions can be engineered for exact $\eta$-pairing states, with direct connection to potential mechanisms of high-$T_c$ superconductivity in hydride materials (Karnaukhov, 6 Jan 2026).

5. Excitonic Instabilities and Spin-State Physics

The two-band Hubbard model provides a minimal model for spin-triplet excitonic condensation and spin-state transitions:

• Excitonic condensation at spin-state crossing: Near the crossover between low-spin and high-spin atomic states, competition between high-spin/low-spin (checkerboard) order and triplet excitonic (orbital-off-diagonal) order is resolved in favor of the latter for sufficiently symmetric bandwidths or low crystal-field splitting. The excitonic instability maps in strong coupling to a condensation of spinful hard-core bosons and in weak coupling to a BCS-like pairing of electron-hole pairs (Kunes et al., 2013).
• Complex excitonic phase diagrams: DMFT calculations reveal multiple symmetry-allowed excitonic phases (polar, ferromagnetic, elliptic), with transitions controlled by temperature, chemical potential, and band asymmetry; multi-critical points enable magnetization switching via small gate voltages (Kunes, 2014).
• Supersolid and roton-like softening: Approaching the transition from an excitonic condensate to a spin-state ordered phase, dynamical susceptibilities exhibit roton-like softening in the excitonic sector and the emergence of Goldstone and flat Higgs-like modes; phase coexistence yields supersolid behavior (Niyazi et al., 2020).

Mean-field studies on quasiperiodic (Penrose) lattices show the excitonic instability becomes dramatically enhanced when the Fermi energy aligns with macroscopically degenerate "confined" or "string" states, demonstrating sensitivity of condensate patterns to lattice topology and local coordination (Inayoshi et al., 2020).

6. Nonequilibrium Dynamics, Surface, and Multiband Extensions

• Interaction Quenches and Non-Equilibrium Regimes: Time-dependent Gutzwiller/slave-boson approaches have revealed that, unlike the single-band case, the two-band model under a sudden interaction quench exhibits a broad regime of long-lived fluctuations with chaotic dynamics between metallic and insulating character (extended dynamical Mott transition), underpinned by orbital and interband oscillatory modes. Hund's coupling, even when finite, sustains singlet multiplet occupations deep within the nonequilibrium Mott regime (Behrmann et al., 2013).
• Surface Correlations and Charge Transfer: In multilayer geometries, the interplay of reduced delocalization and surface-to-bulk charge transfer strongly modifies the local quasiparticle weight. At quarter-filling, charge depletion can nearly compensate for correlation enhancement at the surface; only at sufficiently large $U$ does enhanced surface correlation reappear, highlighting potential subtleties for interpreting surface-sensitive experiments (Nourafkan et al., 2011).
• Extensions to Bose-Hubbard and Cavity QED Realizations: For bosons, the two-band Bose-Hubbard model with interband coupling realizes complex many-body Landau-Zener physics, chaotic thermalization, and interaction-induced collapse and revival in the presence of static fields (Parra-Murillo et al., 2015, Plötz et al., 2010). In cavity-mediated systems, driving a two-band Hubbard model out of equilibrium via strong light-matter coupling and detuned laser drive gives rise to nontrivial Floquet-renormalized interactions, excitonic enhancement, and broadened cavity-induced attractive interactions, opening pathways to engineered nonequilibrium phases (Wang et al., 2023).