|
The Bose–Hubbard model gives a 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 model was first introduced by Gersch H., Knollman G in 1963; the name Bose refers to the fact that the particles in the system are bosonic. The Bose–Hubbard model can be used to model systems such as bosonic atoms on an optical lattice. 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 Hamiltonian == The physics of this model is given by the Bose–Hubbard Hamiltonian: . Here i is summed over all lattice sites, and denotes summation over all neighboring sites i and j, while and gives the number of particles on site i. Parameter is the hopping amplitude, describing mobility of bosons in the lattice. Parameter describes the on-site interaction, repulsive, if , and attractive for . is the chemical potential. The dimension of the Hilbert space of the Bose–Hubbard model grows exponentially with respect to the number of particles ''N'' and lattice sites L. It is given by: . The model can also be considered with fermionic atoms, then it is called Fermi–Hubbard Model. In such a case: The different results stem from different statistics of fermions and bosons. Analogous hamiltonian may be formulated to describe mixtures of different atom species (Bose–Fermi mixtures being a prominent - even with differing statistics example due to large scientific interest). Then the Hilbert space is simply the tensor product of Hilbert spaces of the relevant species. Typically additional terms need to be included to model interaction between species. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bose–Hubbard model」の詳細全文を読む スポンサード リンク
|