Bose–Hubbard model

The Bose–Hubbard model gives an approximate description of the physics of interacting bosons on a lattice. It is closely related to the Hubbard model which originated in solid-state physics as an approximate description of superconducting systems and the motion of electrons between the atoms of a crystalline solid. The name Bose refers to the fact that the particles in the system are bosonic; the model was first introduced by H. A. Gersch and G. C. Knollman* in 1963, The Bose–Hubbard model can be used to study systems such as bosonic atoms on an optical lattice. In contrast, the Hubbard model applies to fermionic particles such as electrons, rather than bosons. Furthermore, it can also be generalized and applied to Bose–Fermi mixtures, in which case the corresponding Hamiltonian is called the Bose–Fermi–Hubbard Hamiltonian.

The physics of this model is given by the Bose–Hubbard Hamiltonian:

$H=-t\sum _{\left\langle i,j\right\rangle }\left(b_{i}^{\dagger }b_{j}+b_{j}^{\dagger }b_{i}\right)+{\frac {U}{2}}\sum _{i}{\hat {n}}_{i}\left({\hat {n}}_{i}-1\right)-\mu \sum _{i}{\hat {n}}_{i}$.

Here i is summed over all lattice sites, and $\left\langle i,j\right\rangle$ denotes summation summed over all neighboring sites i and j.. $b_{i}^{\dagger }$ and $b_{i}^{}$ are bosonic creation and annihilation operators. ${\hat {n}}_{i}=b_{i}^{\dagger }b_{i}$ gives the number of particles on site i. Parameter t is the hopping matrix element, signifying mobility of bosons in the lattice. Parameter U describes the on-site interaction, if U>0 it describes repulsive interaction, if U<0 then the interaction is attractive. $\mu$ is the chemical potential.

The dimension of the Hilbert space of the Bose–Hubbard model grows exponentially with respect to the number of atoms N and lattice sites L. It is given by: $D_{b}={\frac {(N_{b}+L-1)!}{N_{b}!(L-1)!}}$ while that of Fermi–Hubbard Model is given by: $D_{f}={\frac {L!}{N_{f}!(L-N_{f})!}},$ in which the Pauli exclusion principle. The different results stem from different statistics of fermions and bosons. For the Bose–Fermi mixtures, the corresponding Hilbert space of the Bose–Fermi–Hubbard model is simply the tensor product of Hilbert spaces of the bosonic model and the fermionic model.

In the calculation of low energy states the term proportional to $n^{2}U$ means that large occupation of a single site is improbable, allowing for truncation of local Hilbert space to states containing at most $d<\infty$ particles. Then the local Hilbert space dimension is $d+1.$ The dimension of full Hilbert space grows exponentially with number of sites in the lattice, therefore computer simulations simulations are limited to study of systems of 15-20 particles in 15-20 lattice sites. Experimental systems contain several millions lattice sites, with average filling above unity.

At zero temperature, the Bose–Hubbard model (in the absence of disorder) is in either a Mott insulating (MI) state at small $t/U$, or in a superfluid (SF) state at large $t/U$, or in a supersolid (SS) phase where both solid and superfluid (diagonal and off-diagonal) orders coexist. The Mott insulating phases are characterized by integer boson densities, by the existence of an energy gap for particle-hole excitations, and by zero compressibility. In the presence of disorder, a third, ‘‘Bose glass’’ phase exists. The Bose glass phase is characterized by a finite compressibility, the absence of a gap, and by an infinite superfluid susceptibility.. It is insulating despite the presence of a gap, as low tunneling prevents the generation of excitations which, although close in energy are spatially separated.

Quantum phase transitions in the Bose–Hubbard model were experimentally observed by Greiner et al. in Germany.

The Bose–Hubbard model is also of interest to those working in the field of quantum computation and quantum information. Entanglement of ultra-cold atoms can be studied using this model.

For numerical simulation of this model, an algorithm of exact diagonalization is presented in this paper.

See also

• Jaynes-Cummings-Hubbard model