Extended Hubbard Models

• Extended Hubbard models are generalizations of the standard Hubbard Hamiltonian that incorporate nonlocal density interactions, multi-orbital effects, and anisotropic hopping.
• They offer insights into complex phases such as charge-density waves, superconductivity, and exotic magnetic orders in both electron and boson systems.
• Advanced computational and analytical techniques like DQMC, DMFT, and VMC are employed to map phase diagrams and critical points in these models.

The extended Hubbard model is a generalization of the canonical Hubbard Hamiltonian that incorporates nonlocal density–density interactions, multi-orbital structure, longer-range hopping, and further extensions capturing the physics of real correlated materials. This model is central to contemporary theory and quantum simulation of strongly correlated electrons and bosons, as it captures phenomena inaccessible to the pure on-site model, such as charge-density wave (CDW) order, phase separation, exotic magnetic phases, multicomponent order parameters, and emergent geometry. The extended Hubbard framework is also fundamental in experimental settings, from quantum dot arrays and molecular crystal modeling to cold-atom optical lattices and solid-state systems.

1. Hamiltonian Structure and Variants

The archetypal extended Hubbard Hamiltonian augments the standard Hubbard model with nearest-neighbor (NN) and, potentially, longer-range and anisotropic terms:

$$H = -\sum_{\langle i,j \rangle,\sigma} t_{ij} (c_{i\sigma}^\dagger c_{j\sigma} + \text{H.c.}) + U \sum_i n_{i\uparrow} n_{i\downarrow} + \sum_{\langle i,j \rangle} V_{ij} n_i n_j + \dots$$

Key terms:

• $t_{ij}$: hopping amplitude, possibly anisotropic or orbital-dependent
• $U$: on-site Coulomb repulsion
• $V_{ij}$: nearest-neighbor (NN) or longer-range density–density interaction
• Additional extensions: next-nearest hopping, exchange, pair hopping, density-assisted hopping, multi-orbital interactions, Hund’s coupling (Georgescu et al., 2015, Tsuchiizu et al., 2011)

Generalizations are common in both the fermionic (electrons, ultracold atoms) and bosonic (Bose–Hubbard) sectors. Extended Hubbard parameters are often determined using ab initio quantum chemistry (Tsuchiizu et al., 2011), mapping to multi-orbital or fragment-localized bases.

2. Correlated Phases and Phase Diagrams

The extended Hubbard model hosts a rich array of phases, determined by the competition between kinetic, on-site, and nonlocal interactions. Determinant Quantum Monte Carlo (DQMC), dynamical mean-field theory (DMFT), variational Monte Carlo (VMC), and analytic strong-coupling approaches constitute the methodological backbone for obtaining these phase diagrams (Sousa-Júnior et al., 2023, Kennedy et al., 2024, Amaricci et al., 2010, Kaneko et al., 2016).

Prototypical phases:

• Mott Insulator (MI): Stabilized by large $U$ at commensurate filling; spin sector may exhibit antiferromagnetism (AFM) (Sousa-Júnior et al., 2023, Kennedy et al., 2024).
• Charge-Density Wave (CDW): Arising for strong repulsive $V$, with alternation in local charge ($n_A \neq n_B$); typically checkerboard on bipartite lattices (Sousa-Júnior et al., 2023, Kennedy et al., 2024).
• Superconductivity (SC): Attractive $U$ supports on-site $s$-wave pairing; extended attraction or repulsion ($V$) modifies the dominant symmetry to $t_{ij}$0- or $t_{ij}$1-wave (and may drive phase transitions between these) (Hutchinson et al., 2019, Sousa-Júnior et al., 2023).
• Supersolid (SS): In extended Bose–Hubbard models, coexistence of superfluid and density-wave order; competition with phase separation (Kimura, 2011).
• Phase Separation (PS): Strong attractive $t_{ij}$2 drives macroscopic segregation into charge-rich and charge-poor domains; related to Wigner–Mott transitions (Amaricci et al., 2010, Loon et al., 2017).
• Exotic Magnetic Order & Frustration: Doped models with significant $t_{ij}$3 produce emergent triangular or kagome lattices, ring-exchange, and frustration (Kaneko et al., 2016).

Tables mapping interaction regimes and their realized phases have been established via unbiased numerics; critical lines (e.g., AFM–CDW, SC–PS) are sharply defined. Example (square lattice at half-filling, $t_{ij}$4), showing some ground-state boundaries (Sousa-Júnior et al., 2023):

$t_{ij}$5	$t_{ij}$6	$t_{ij}$7
1	0.3 ± 0.1	—
2	0.65 ± 0.05	–0.45 ± 0.05
4	1.1 ± 0.1	—

3. Strong Coupling, Renormalization and Emergent Lattices

Strong-coupling regimes ($t_{ij}$8) are accessible to perturbation theory, slave-particle approaches, or VMC. Here, charge order induces effective reduction in the active degrees of freedom: for example, at noninteger filling, CO can transform the original lattice into an emergent frustrated one with antiferromagnetic or ferromagnetic exchange and higher-order interactions (Kaneko et al., 2016). On the honeycomb lattice, for example, CDW leads to an effective triangular lattice of singly occupied “spin” sites with frustrated exchange and possible ring-exchange or chiral terms.

Slave-particle representations handle large-$t_{ij}$9, multi-orbital, and Hund-coupled situations; the generalized formalism unifies slave-rotor, slave-spin, and intermediate models