Bose-Hubbard Model

One-Sentence Description

The Bose-Hubbard model describes bosons hopping on a lattice while paying an onsite interaction energy for multiple occupancy.

The full lattice treatment is Bose–Hubbard Model, Bose–Hubbard Dimer owns the exact two-mode finite problem, and Bose–Hubbard Chain owns the one-dimensional specialization. This page is the compact model card.

Physical Setup

Choose lattice sites as modes. Bosons can hop between neighboring sites and can pile up on the same site, with onsite repulsion or attraction controlling the cost of multiple occupancy. The repulsive model is a standard minimal description of cold bosonic atoms in optical lattices and of lattice-boson superfluid-Mott competition.

Hilbert Space

The full bosonic Fock space over $L$ lattice modes allows arbitrarily high occupation on each site. For numerical work, one often fixes the total particle number and may also impose a maximum onsite occupation as a controlled truncation.

At fixed total particle number $N$ , a basis is

\lvert n_1,n_2,\ldots,n_L\rangle, \qquad \sum_{j=1}^{L}n_j=N.

Hamiltonian

A common Bose-Hubbard Hamiltonian is

H = -t \sum_{\langle i,j\rangle} \left( b_i^\dagger b_j + b_j^\dagger b_i \right) + \frac{U}{2} \sum_i n_i(n_i-1) - \mu\sum_i n_i.

Here

n_i=b_i^\dagger b_i.

Parameters

Symbol	Meaning
$t$	hopping amplitude
$U$	onsite interaction energy
$\mu$	chemical potential
$L$	number of lattice sites
$N$	total particle number in fixed- $N$ calculations

Solvability

The Bose-Hubbard model is generally not exactly solvable. Important limits are simple: $U=0$ gives free lattice bosons, $t=0$ gives independent sites, and small clusters can be diagonalized. Mean-field theory, strong-coupling expansions, quantum Monte Carlo, tensor networks, and exact diagonalization are standard tools.

Key Observables

Site occupation and number fluctuations.
Condensate fraction or long-range phase coherence.
One-body density matrix.
Compressibility.
Mott gap at integer filling.

What It Teaches

The model teaches competition between delocalization from hopping and localization from onsite repulsion. At integer filling, strong repulsion favors Mott-insulating states; large hopping favors phase coherence and superfluid behavior. Atomic intervals, optical-lattice reduction, and the model-specific mean-field boundary are developed in Bose–Hubbard Model. The Bose–Hubbard Dimer isolates exact number squeezing, pair formation, fragmentation, and Josephson dynamics. Distinct critical behavior at lobe sides and tips is interpreted in Quantum Phase Transitions.

It is a clean example where the ideal Bose gas is insufficient: interactions and lattice discreteness are essential.

Canonical Links

Variants

homogeneous Bose-Hubbard model;
trapped Bose-Hubbard model;
disordered Bose-Hubbard model;
extended Bose-Hubbard model with nearest-neighbor interactions;
multi-component Bose-Hubbard models;
driven or dissipative lattice-boson models.

Common Mistakes

Confusing ideal Bose condensation with Bose-Hubbard superfluidity.
Forgetting the factor $1/2$ in $U n_i(n_i-1)/2$ .
Treating onsite occupation truncation as physical without checking convergence.
Comparing phase diagrams without stating dimension, filling, and lattice geometry.

Quick Check

Why does the onsite interaction use $n_i(n_i-1)$ rather than $n_i^2$ ?

Solution

The interaction counts pairs of bosons on the same site. The number of unordered pairs among $n_i$ bosons is $n_i(n_i-1)/2$ , which vanishes for zero or one boson.

References

M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Physical Review B 40, 546-570, 1989.
I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Reviews of Modern Physics 80, 885-964, 2008.
D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Annals of Physics 315, 52-79, 2005.