Extended Bose-Hubbard Model

Updated 11 December 2025

The extended Bose-Hubbard model (eBHM) is a quantum lattice model that generalizes the standard Hamiltonian by incorporating nonlocal interactions, density-dependent tunneling, and multibody onsite effects.
It exhibits diverse phases such as superfluid, Mott insulator, density wave, supersolid, and Haldane insulator, each characterized by distinct order parameters and critical phenomena.
Advanced computational methods like Quantum Monte Carlo, DMRG, and mean-field theories, along with experimental platforms like cold atoms and Rydberg-dressed systems, validate its rich phase diagram and tunable interactions.

The extended Bose-Hubbard model (eBHM) generalizes the standard Bose-Hubbard Hamiltonian by including nonlocal interaction terms, typically nearest-neighbor (NN) repulsion, and in some variants density-dependent tunneling, multibody onsite interactions, or explicit pair-hopping. This extension enables a diversity of strongly correlated bosonic lattice phases and underpins much recent progress in synthetic quantum matter, quantum simulation with cold atoms, and studies of quantum phase transitions beyond the Mott insulator–superfluid paradigm.

1. Hamiltonian Structure and Fundamental Terms

The eBHM in its minimal form on a general lattice is

$H = -t\sum_{\langle i, j \rangle}(b_i^\dagger b_j + \mathrm{h.c.}) + \frac{U}{2}\sum_i n_i(n_i-1) + V\sum_{\langle i, j \rangle} n_i n_j - \mu\sum_i n_i$

where:

$b_i^\dagger, b_i$ : bosonic creation and annihilation operators at site $i$ .
$n_i = b_i^\dagger b_i$ : number operator.
$t$ : nearest-neighbor hopping amplitude.
$U$ : on-site interaction energy.
$V$ : nearest-neighbor repulsion (or attraction if negative).
$\mu$ : chemical potential.
Sums $\langle i, j \rangle$ run over nearest-neighbor pairs.

Further generalizations include:

Extended-range density–density interactions ( $V_{ij}$ beyond NN).
Density-dependent (bond–charge) tunneling terms $T$ : $-T\sum_{\langle i,j\rangle}[a_i^\dagger(n_i+n_j)a_j + \text{h.c.}]$ .
Pure three-body interaction terms: $(W/6)\sum_i n_i(n_i-1)(n_i-2)$ .
Pair-hopping amplitudes $t_p$ : $-t_p\sum_{\langle ij \rangle}(a_i^{\dagger 2} a_j^2 + \text{h.c.})$ .
Constraints on maximum onsite occupation (e.g., hard-core or three-body).

These Hamiltonians are relevant for synthetic quantum systems where nonlocal couplings, multi-body effects, or correlated tunneling can be engineered or are intrinsic, such as dipolar gases, Rydberg-dressed atoms, or semiconductor exciton arrays (Kurdestany et al., 2012, Lagoin et al., 2022, Weckesser et al., 30 May 2024).

2. Phase Diagram: Canonical Phases and Competing Orders

The interplay of kinetic energy ( $t$ ), local repulsion ( $U$ ), and NN repulsion ( $V$ ) yields a rich phase structure, displayed in the $(\mu, U/t, V/t)$ plane. The principal phases are:

Phase	Order Parameters	Typical regime
Superfluid (SF)	$\langle b_i\rangle\neq0$ ; off-diagonal long-range order	large $t$ , weak $U,V$
Mott insulator (MI)	Compressible gap $\Delta_c>0$ , uniform density ( $\langle n_i\rangle = n \in \mathbb{Z}$ )	large $U$ , small $t, V < U$
Density wave (DW)	Broken sublattice symmetry, staggered density ( $\langle n_A\rangle \neq \langle n_B\rangle$ )	large $V/U$ , commensurate filling
Supersolid (SS)	Coexisting SF and DW order	intermediate $t$ , large $V/U$
Haldane insulator (HI, 1D)	Nonlocal string order, gapped excitations	intermediate $U, V$ at unit filling
Bose glass (BG), Disordered solid (DS)	Finite compressibility without SF or DW order	introduced by disorder (Lin et al., 2015)

Critical properties, e.g., BKT transitions at phase boundaries (notably in 1D), can be precisely extracted via energy-gap scaling and entanglement entropy analysis (Rossini et al., 2012, Sowiński, 2013). Long-range (checkerboard or stripe) insulators emerge at fractional filling for $V/U\gtrsim 0.02$ –$0.3$ (Lagoin et al., 2022, Kurdestany et al., 2012).

3. Quantum Monte Carlo, DMRG, and Mean-Field Approaches

A suite of computational tools underpins phase diagram and dynamical studies:

Stochastic Series Expansion (SSE) QMC: Accurately treats finite- $T$ and disordered systems, with rare event sampling; efficient for large lattices (Lin et al., 2015, Kawaki et al., 2017).
Density Matrix Renormalization Group (DMRG): Provides zero-temperature and entanglement structure in 1D; resolves MI, HI, DW, and SS phases with high accuracy for large systems (Rossini et al., 2012, Sowiński, 2013).
Self-consistent Mean-Field Theories (MFT): Decouple hopping/density operators; efficiently access large systems and inhomogeneous (trapped) phases (Kurdestany et al., 2012, Kemburi et al., 2011). Capture phase boundaries and shell structures.
Cluster Gutzwiller/Variational Ansatz: Incorporate intersite correlations to improve mean-field results, critical for capturing supersolidity and stripe order (Barbier et al., 2021).

Order parameters such as superfluid stiffness (via winding numbers), static structure factors, nonlocal string correlators, and compressibility are standard diagnostic tools (Kemburi et al., 2011, Rossini et al., 2012, Kawaki et al., 2017, Lin et al., 2015).

4. Disorder Effects and Percolation-Enhanced Supersolidity

Disorder, modeled as random site chemical potentials $\varepsilon_i$ , introduces glassy and compressible phases. Key findings:

Moderate disorder ( $\Delta/U\lesssim0.1$ ) does not destroy the cubic-lattice supersolid or solid states (Kemburi et al., 2011, Lin et al., 2015).
Increasing disorder transforms the solid into a supersolid via a percolation mechanism: locally shifted regions become superfluid-active, and when these percolate ( $p>p_c$ ), a macroscopic supersolid emerges. Percolation threshold estimates ( $p_c\approx0.31$ for SC lattice) yield disorder-induced supersolid boundary at $\Delta_c\sim 0.13 U$ (MFT), or $\Delta/U\simeq 0.4$ -0.6 (QMC) (Kemburi et al., 2011).
Disorder can enhance the critical temperature of the supersolid in a narrow parameter window, but generally suppresses it at large disorder strength (Lin et al., 2015).

The coexistence or tension of glassy (compressible, non-superfluid) and superfluid/solid order reflects strong competition between disorder and interaction-induced phase locking.

5. Novel Phases via Extended Interactions and Constraints

The eBHM supports phases absent in the standard model due to:

Pair Superfluid and Pair Supersolid: Realized with explicit pair-hopping or strong three-body constraints; on triangular lattices or in models with limited occupation ( $n_i\leq2$ ), the phase diagram is further enriched. For example, pure three-body onsite interactions ( $W>0$ ) yield Mott lobes only for even/odd fillings ( $\rho_0\geq2$ ), with lobe widths increasing with $\rho_0$ (Sowiński, 2013, Panov et al., 2019).
Haldane Insulator: Robust in 1D for intermediate $U,V$ , characterized by string order and separated from MI/DW by closing of charge or neutral gaps (Rossini et al., 2012, Kawaki et al., 2017).
Supersolid Phases in Higher Dimensions: Emergent for strong off-site repulsions or Rydberg-dressed interactions, observed in cold atom and dipolar systems (Barbier et al., 2021, Weckesser et al., 30 May 2024).

In models with density-dependent tunneling, the effective kinetic bandwidth is renormalized, dramatically moving phase boundaries and suppressing/reinforcing charge order and phase separation (Maik et al., 2013).

6. Experimental Realizations and Applications

The eBHM framework underpins multiple experimental platforms:

Cold Atom Lattices: Magnetic atoms (Er, Dy), polar molecules, and Rydberg-dressed bosons allow tuning of on-site $U$ , off-site $V$ , and density-dependent hopping. Direct measurement of nonlocal interactions, anisotropy effects, and SF-MI transition shifts have been reported in erbium atom lattices (Baier et al., 2015).
Rydberg Dressing: Enables tunable-range soft-core repulsions and kinetic constraints, allowing observation of repulsively bound pairs, hard-rod dynamics and ordering phenomena in 1D (Weckesser et al., 30 May 2024, Barbier et al., 2021).
Dipolar Exciton Arrays: Weakly dissipative platforms (hundred-site 2D lattices) with strong NN repulsion exhibit checkerboard ordering and incompressibility at half filling (Lagoin et al., 2022).
Disorder and Quantum Simulation: Controlled disorder enables paper of glass phases, percolation phenomena, and disorder-stabilized supersolidity.

Key experimental implications include observation of shell structures, in-situ imaging of density waves, momentum peaks (Bragg, TOF), and spectroscopic signatures (collapse–revival, modulation). Realistic control over parameter regimes allows direct testing of numerically obtained phase diagrams (Kurdestany et al., 2012, Lin et al., 2015).

7. Extensions: Topological Order, Quenches, and Higher Dimensions

Advanced studies employ extensions of the eBHM to explore topological phases and nonequilibrium phenomena:

Symmetry-Protected Topological Phases: Dimerized chains with constraints exhibit Haldane, generalized valence bond, and SPT phases signaled by Berry phase, entanglement spectrum, and topological pumps with quantized many-body Chern numbers. Bulk–edge correspondence is manifest in center-of-mass dynamics during adiabatic cycles (Kuno et al., 2021).
Fractional Quantum Hall Analogy: Strong lattice tilts in 1D eBHM lead to emergent dipole-conserving Hamiltonians, supporting bosonic Laughlin-like ground states, entanglement signatures, and robust bulk gaps (Sable et al., 31 Oct 2025).
Quench Dynamics: Sudden interaction sign flips (super-Tonks-Girardeau quenches) in eBHM reveal broadening/evaporation of self-bound droplets and serve as dynamical probes of phase diagram boundaries (Marciniak et al., 2023).

These directions illustrate the continued centrality of the eBHM as a testbed for strongly correlated, topological, and disordered quantum systems.

References: