BoseHubbardmalli

BoseHubbardmalli is a term used in some Finnish-language scientific literature to refer to the Bose-Hubbard model, a canonical lattice model for interacting bosons in condensed-matter and quantum simulation research. The model captures the competition between kinetic energy from hopping between neighboring lattice sites and on-site repulsion between multiple bosons occupying the same site.

The Hamiltonian is typically written as H = -t ∑⟨i j⟩ (b_i† b_j + b_j† b_i) + (U/2) ∑_i n_i(n_i - 1) - μ ∑_i n_i, where t is the hopping amplitude, U is the on-site interaction strength, μ is the chemical potential, b_i† and b_i are bosonic creation and annihilation operators on site i, and n_i = b_i† b_i is the number operator. The model is defined on a lattice of arbitrary dimension and geometry, with the sum over ⟨i j⟩ indicating nearest-neighbor pairs.

At zero temperature, the Bose-Hubbardmalli exhibits a quantum phase transition between a superfluid phase at large t/U and a Mott insulator phase at commensurate (integer) fillings for large U/t. The superfluid phase features long-range phase coherence and delocalized particles, while the Mott insulator is gapped with fixed integer occupation per site and suppressed number fluctuations.

Experimental realizations are achieved with ultracold atoms in optical lattices, where lattice depth tunes t and interatomic interactions U can be adjusted via Feshbach resonances. The Mott-insulating regime is detected through suppressed number fluctuations and the disappearance of sharp interference peaks in time-of-flight measurements.

Extensions of the model include the extended Bose-Hubbard model with nearest-neighbor interactions V, disorder-induced Bose glass phases, finite-temperature effects, and multi-band or dipolar generalizations. The BoseHubbardmalli is studied with numerical methods such as quantum Monte Carlo, density-matrix renormalization group in one dimension, exact diagonalization, and various mean-field approaches.