Extended Bose-Hubbard Model Overview

• The extended Bose-Hubbard model is a framework that expands the standard Bose-Hubbard model by including nonlocal interaction and additional hopping terms.
• It explains complex quantum phases such as Mott insulators, supersolids, and topologically nontrivial states through the interplay of kinetic energy and both local and nonlocal interactions.
• It guides experimental realizations in ultracold atomic gases, dipolar systems, and engineered solid-state arrays, providing a versatile platform for quantum simulations.

The extended Bose-Hubbard model (eBHM) generalizes the canonical Bose-Hubbard model by incorporating interaction terms and hopping processes beyond the standard on-site repulsion and nearest-neighbor tunneling. This extension enables the exploration of a broad spectrum of strongly correlated quantum phases, including density waves, supersolids, topologically nontrivial states, and nonergodic phases driven by competition between kinetic, local, and nonlocal interaction terms. The eBHM serves as an essential theoretical framework for describing ultracold atomic gases in optical lattices with strong dipolar interactions, polar molecules, Rydberg-dressed gases, solid-state arrays, and other platforms where off-site couplings are non-negligible.

1. Fundamental Structure and Terms of the Extended Bose-Hubbard Model

The generic eBHM Hamiltonian on a lattice of $L$ sites is given by

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

Here,

• $t$ is the single-particle nearest-neighbor (NN) hopping amplitude,
• $b_i^\dagger$ ($b_i$) creates (annihilates) a boson at site $i$,
• $n_i = b_i^\dagger b_i$ is the number operator,
• $U$ is the on-site interaction strength,
• $V$ is the NN interaction (off-site, e.g. dipolar or long-range),
• $\mu$ is the chemical potential.

Ellipses denote further terms that may arise in specific contexts:

• Next-nearest-neighbor (NNN) hopping: $$-J_2 \sum_{\langle\langle i, j \rangle\rangle} (b_i^\dagger b_j + \mathrm{h.c.})$$
• Density-dependent tunneling: $$-T \sum_{\langle i, j \rangle} [ b_i^\dagger (n_i + n_j) b_j + \mathrm{h.c.}]$$
• Multi-body (e.g. three-body) on-site interaction: $+\frac{W}{6} \sum_i n_i(n_i-1)(n_i-2)$
• Higher-order interaction and hopping terms, especially in long-range interacting or Rydberg-dressed systems.

These extensions are crucial for capturing interactions in ultracold systems with large dipole moments, strong polarizability, or synthetic gauge fields.

2. Quantum Phases and Phase Diagrams

The competition between kinetic and various interaction terms in the eBHM leads to a rich phase diagram, surpassing that of the standard Bose-Hubbard model.

• Mott insulator (MI): Uniform integer occupation; occurs at strong $U$ and commensurate filling.
• Charge-density-wave (CDW) or Density-wave (DW) insulator: Alternating higher and lower occupation across sublattices, stabilized by $V$ (or further range interactions).
• Superfluid (SF): Delocalized phase with long-range phase coherence; dominates at large $t$.
• Supersolid (SS): Simultaneous density modulation and nonzero condensate order ($\psi_A \neq \psi_B$, $\rho_A \neq \rho_B$); arises for appropriate $U$, $V$, and $t$ (Iskin, 2011, Kurdestany et al., 2012).
• Pair superfluid/supersolid: Coherent condensation of boson pairs, enabled by explicit pair hopping (Wang et al., 2013).
• Haldane-insulator (HI): In 1D with large $V$ and $U$, protected by a nonlocal string order; haLLMark of topological order (Fraxanet et al., 2021).
• Topologically protected edge states and critical points: In specific dimerized or strongly correlated 1D regimes, bulk-edge correspondence yields localized edge states at topological transitions (Fraxanet et al., 2021, Zhang et al., 17 Oct 2024).
• Insulating checkerboard (CB) order at half filling: Stabilized in 2D by strong off-site (dipolar) repulsion, observed as an incompressible lobe in the filling vs. chemical potential plane (Lagoin et al., 2022).

The phase boundaries are modified by the inclusion of terms such as NNN hopping (e.g., shaken lattices (Miao, 2015)), density-dependent tunneling (Maik et al., 2013), and three-body interactions (Sowiński, 2013). For example, the inclusion of strong NN repulsion $V$ can dramatically enlarge the SS region, especially when $2dV > U$ on a $d$-dimensional hypercubic lattice (Iskin, 2011).

3. Spectral Statistics, Integrability, and Quantum Chaos

The eBHM provides a platform for studying quantum chaos and ergodicity breaking in many-body systems (Kollath et al., 2010).

• Level spacing statistics: For intermediate $U\sim t$, the spectrum shows level spacing statistics close to the Gaussian Orthogonal Ensemble (GOE), indicating quantum chaotic behavior. In the integrable limits ($U=0$, $t=0$), the statistics become Poissonian.
• Ratio of consecutive level spacings: The statistic $$r_n = \min(\delta_n, \delta_{n-1})/\max(\delta_n, \delta_{n-1})$$ provides a robust discriminator: $\langle r \rangle_{\text{GOE}} \simeq 0.53$, $\langle r \rangle_{P} \simeq 0.386$.
• Perturbations: NNN hopping and nonlocal interactions drive the system away from integrability, with finite-size scaling indicating that even small perturbations can render the spectrum globally chaotic in the thermodynamic limit for generic parameter values.

A cutoff on maximum on-site occupancy (as used in numerics) can shift the spectral statistics towards GOE, even in instances that would otherwise display Poissonian statistics.

4. Effects of Disorder and External Fields

The stability of the eBHM's exotic phases under disorder and artificial gauge fields has been systematically explored.

• Disorder: Weak bounded disorder preserves supersolid order on a simple cubic lattice (Kemburi et al., 2011, Lin et al., 2015). As disorder increases, a percolation transition can cause solid regions to develop supersolidity, as locally supersolid regions connect and establish global phase coherence. There exists a narrow parameter regime where disorder enhances the critical temperature for supersolidity via percolation-enhanced superfluidity.
• Artificial Gauge Fields: Inclusion of a magnetic flux (e.g., Peierls substitution in hopping) reduces the effective kinetic energy, enlarging insulating MI and CDW/AMI lobes in phase diagrams (Iskin, 2011). Analytical mean-field boundaries for superfluid-insulator transitions include explicit dependence on the minimal single-particle Hofstadter spectrum energy.

5. Experimental Realizations and Extensions

The eBHM formalism has been experimentally realized and proposed across multiple platforms:

• Dipolar gases in optical lattices: With erbium atoms (Baier et al., 2015), KRb molecules, or polar molecules, both on-site $U$ and NN $V$ can be tuned. Control of dipole orientation and trapping geometry enables tuning of anisotropic interactions and density-assisted tunneling.
• Artificial gauge fields and shaken lattices: Synthetic fields and Floquet engineering allow for the implementation of NNN hopping and symmetry-breaking superfluid phases (e.g., $Z_2$ superfluid, (Miao, 2015)).
• Semiconductor arrays (dipolar excitons): Arrays of gate-defined quantum wells enable the realization of eBHM in solid-state systems (Lagoin et al., 2022), with checkerboard insulators observed at half filling.
• Rydberg-dressed atoms: Molecular dressing introduces tunable, range-selective interactions, enabling cluster Gutzwiller realizations of eBHM with supersolid order (Barbier et al., 2021).

Further, extended models can serve as effective spin-$1$ quantum magnets with three-body constraints or map to $S=1$ pseudospin models relevant for strongly correlated electrons in cuprates (Panov et al., 2019).

6. Topological Phenomena and Quench Dynamics

The inclusion of interaction-induced topology in the eBHM is a frontier of current research (Fraxanet et al., 2021, Zhang et al., 17 Oct 2024):

• Topological quantum critical points (TQCP): At transitions between Haldane insulator and charge-density-wave phases (or their dimerized analogues), localized edge states survive at criticality due to the protection afforded by a finite charge gap, even when the bulk gap closes (Fraxanet et al., 2021).
• String order: Exotic phases exhibit long-range string order correlators distinguishing topologically nontrivial from trivial insulators.
• Quantum slinky dynamics: For resonant $U=V$ and strong interactions, the restricted dynamics of boson clusters is mapped onto generalized $n$-unit-cell Su-Schrieffer-Heeger (SSH) models, with nontrivial Zak phase and emergent topological edge states manifest as $n$-boson bound states at chain boundaries. These can be dynamically detected via edge oscillation observables in quench experiments (Zhang et al., 17 Oct 2024).
• Super-Tonks-Girardeau quench: Quenching from strong repulsive to strong attractive interactions in the presence of nonlocal (NN) attraction can induce evaporation dynamics of self-bound droplets, providing a probe for the underlying phase diagram (Marciniak et al., 2023).

7. Influence of Beyond-Standard Terms and Future Directions

The eBHM admits a diversity of modifications not present in the canonical model, including density-dependent tunneling, multi-body and correlated hopping, and explicit pair or spin-exchange channels (Wang et al., 2013, Maik et al., 2013).

Theoretical advances continue to clarify:

• The role of these terms in stabilizing new quantum phases (e.g., pair-supersolids, incommensurate supersolids, disorder-induced states).
• The correspondence between ground-state order, excitations (e.g., roton minima, quadratic vs. linear gapless modes), and experimental signatures (modulation spectroscopy, edge microscopy).
• The mapping to effective spin models and the emergence of topological order, as well as the dynamical response to external perturbations or quenches.

The extended Bose-Hubbard model thus remains a cornerstone for the paper and engineering of strongly correlated, topological, and nonequilibrium quantum matter in lattice systems with tunable interactions and dimensionality.