The Bose–Hubbard model gives a description of the physics of interacting spinless bosons on a lattice.
It is closely related to the Hubbard model that 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 introduced by Gersch and Knollman[1] in 1963 in the context of granular superconductors.
[2][3][4] The Bose–Hubbard model can be used to describe physical systems such as bosonic atoms in an optical lattice,[5] as well as certain magnetic insulators.
If unspecified, typically the phrase 'Bose–Hubbard model' refers to the case where the on-site interaction is repulsive.
symmetry, which means that it is invariant (its physical properties are unchanged) by the transformation
Typically additional terms are included to model interaction between species.
At zero temperature, the Bose–Hubbard model (in the absence of disorder) is in either a Mott insulating state at small
[8] 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.
The superfluid is characterized by long-range phase coherence, a spontaneous breaking of the Hamiltonian's continuous
At non-zero temperature, in certain parameter regimes a regular fluid phase appears that does not break the
[4] The Bose glass is a Griffiths phase, and can be thought of as a Mott insulator containing rare 'puddles' of superfluid.
These superfluid pools are not interconnected, so the system remains insulating, but their presence significantly changes model thermodynamics.
The Bose glass phase is characterized by finite compressibility, the absence of a gap, and by an infinite superfluid susceptibility.
[4] It is insulating despite the absence of a gap, as low tunneling prevents the generation of excitations which, although close in energy, are spatially separated.
The Bose glass has a non-zero Edwards–Anderson order parameter[10][11] and has been suggested (but not proven) to display replica symmetry breaking.
The phase diagram can be determined by calculating the energy of this mean-field Hamiltonian using second-order perturbation theory and finding the condition for which
The local term is diagonal in the Fock basis, giving the zeroth-order energy contribution:
The energy can be expressed as a series expansion in even powers of the order parameter (also known as the Landau formalism):
[4] Ultracold atoms in optical lattices are considered a standard realization of the Bose–Hubbard model.
The ability to tune model parameters using simple experimental techniques and the lack of the lattice dynamics that are present in solid-state electronic systems mean that ultracold atoms offer a clean, controllable realisation of the Bose–Hubbard model.
To see why ultracold atoms offer such a convenient realization of Bose–Hubbard physics, the Bose–Hubbard Hamiltonian can be derived starting from the second quantized Hamiltonian that describes a gas of ultracold atoms in the optical lattice potential.
is a Wannier function for a particle in an optical lattice potential localized around site
[15] The tight-binding approximation significantly simplifies the second quantized Hamiltonian, though it introduces several limitations at the same time: Quantum phase transitions in the Bose–Hubbard model were experimentally observed by Greiner et al.,[9] and density dependent interaction parameters
means that large occupation of a single site is improbable, allowing for truncation of local Hilbert space to states containing at most
The dimension of the full Hilbert space grows exponentially with the number of lattice sites, limiting exact computer simulations of the entire Hilbert space to systems of 15-20 particles in 15-20 lattice sites.
[citation needed] Experimental systems contain several million sites, with average filling above unity.
[citation needed] One-dimensional lattices may be studied using density matrix renormalization group (DMRG) and related techniques such as time-evolving block decimation (TEBD).
[24] Higher dimensions are significantly more difficult due to the rapid growth of entanglement.
Bose–Hubbard-like Hamiltonians may be derived for different physical systems containing ultracold atom gas in the periodic potential.