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 atoms in the systems are bosonic. 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. Further more, it can also be applied to Bose-Fermi mixtures, in which case the corresponding Hamilitonian is called Bose-Fermi-Hubbard Hamiltonian.

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

$H\; =\; -t\; sum\_\{\; leftlangle\; i,\; j\; ight\; angle\; \}\; left(\; b^\{dagger\}\_i\; b\_j\; +\; b^\{dagger\}\_j\; b\_i\; ight)\; +\; frac\{U\}\{2\}\; sum\_\{i\}\; hat\{n\}\_i\; left(\; hat\{n\}\_i\; -\; 1\; ight)\; -\; mu\; sum\_i\; hat\{n\}\_i$.

Here $i^\{\}\_\{\}$ is summed over all lattice sites, and $leftlangle\; i,\; j\; ight\; angle$ is summed over all neighboring sites. $b^\{dagger\}\_i$ and $b^\{\}\_i$ are bosonic creation and annihilation operators. $hat\{n\}\_i\; =\; b^\{dagger\}\_i\; b\_i$ gives the number of particles on site $i^\{\}\_\{\}$. $t^\{\}\_\{\}$ is the hopping matrix element, $U^\{\}\_\{\}\; >\; 0$ is the on site repulsion, and $mu^\{\}\_\{\}$ 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 Pauli exclusion principle is already taken into account.

For the Bose-Fermi Mixtures, the corresponding Hilbert space of the Bose-Fermi Hubbard Model is simply the product of Hilbert Spaces of bosonic model and the fermionic model. In 3-dimension the Hilbert space will grow far quicker. Hence, it is a computationally demanding task to model or simulate such systems and thus work in this area is apparently restricted to systems no bigger than 20 atoms and 20 lattice sites.

At zero temperature the Bose Hubbard model (in the absence of disorder) is either in a Mott insulating (MI) state at small $t$ or in a superfluid (SF) state at large $t$. 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. This phase is insulating because of the localization effects of the randomness. The Bose glass phase is characterized by a finite compressibility, the absence of a gap, and by an infinite superfluid susceptibility. Quantum phase transitions in the Bose-Hubbard Model were experimentally observed by Greiner et al [Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W Hänsch, and Immanuel Bloch, "Quantum phase transition from a superfluid to a Mott-insulator in a gas of ultracold atoms", "Nature" 415, 39 (2002).] in Germany. The Bose-Hubbard Model is also of interest to those working in the field of quantum computation and quantum information. Entanglement with ultra-cold atoms can be studied using this model. [O Romero-Isart, K Eckert, C Rodó, and A Sanpera, "Transport and entanglement generation in the Bose–Hubbard model", "J. Phys. A: Math. Theor." 40, 8019 (2007).]

References

* M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, "Phys. Rev. B" 40, 546 (1989).