Bose-Hubbard Model
Introduction
The Bose-Hubbard model is a cornerstone of theoretical physics, particularly in the study of quantum many-body systems and ultracold atomic gases. It describes the behavior of interacting bosons on a lattice, capturing the competition between kinetic energy (boson hopping) and potential energy (on-site interactions). This model is widely used to understand phenomena such as quantum phase transitions, superfluidity, and Mott insulation.
The Bose-Hubbard model is defined by the following Hamiltonian:
$$ H = -t \sum_{\langle i,j \rangle} \left( b_i^\dagger b_j + \text{h.c.} \right) + \frac{U}{2} \sum_i n_i (n_i - 1) - \mu \sum_i n_i, $$
where:
- $t$ is the hopping amplitude between nearest-neighbor sites $\langle i,j \rangle$,
- $U$ is the on-site interaction strength, representing the energy cost of having multiple bosons on the same site,
- $\mu$ is the chemical potential, controlling the total number of bosons in the system,
- $b_i^\dagger$ and $b_i$ are the bosonic creation and annihilation operators at site $i$,
- $n_i = b_i^\dagger b_i$ is the number operator, representing the boson occupation at site $i$.
The first term in the Hamiltonian describes the kinetic energy of bosons hopping between lattice sites, favoring delocalization and the formation of a superfluid phase. The second term represents the on-site interaction energy, which penalizes multiple bosons occupying the same site and favors localization. The third term, involving the chemical potential $\mu$, controls the overall particle density in the system.
Phenomena
The Bose-Hubbard model exhibits a rich phase diagram, with two primary phases:
- Superfluid phase: At small $U/t$, bosons delocalize across the lattice, forming a coherent superfluid state with long-range phase coherence.
- Mott insulating phase: At large $U/t$, bosons localize at individual lattice sites due to strong repulsive interactions, leading to a gapped insulating state with integer boson occupancy per site.
The transition between these phases is a paradigmatic example of a quantum phase transition, driven by quantum fluctuations rather than thermal effects. This transition has been experimentally observed in ultracold atomic gases trapped in optical lattices, making the Bose-Hubbard model a key tool for understanding and simulating quantum many-body phenomena.
The Bose-Hubbard model is also closely related to other models in condensed matter physics, such as the Josephson junction array and the XY model, and serves as a foundation for studying more complex systems, including disordered and long-range interacting bosonic systems.