Bose-Hubbard model

Bose-Hubbard model describes the hopping of bosonic particles in the presence of (repulsive, and usually on-site) interaction. Depending on the ratio between the tunneling term \, t and the interaction term \, U , the macroscopic properties of the system may change from superfluid to Mott insulator.

The Bose-Hubbard model is described by the following Hamiltonian:

H_{BH}=-\frac{t}{2}\sum\limits_{\left\langle i,j \right\rangle }{\left( a_{i}^{\dagger}a_{j}+a_{j}^{\dagger}a_{i} \right)}+\frac{U}{2}\sum\limits_{i}{n_{i}\left( n_{i}-1 \right)},

where \left\langle i,j \right\rangle denotes the sum over nearest-neighbor sites, and n_{i}\equiv a_{i}^{\dagger}a_{i} with bosonic commutation relations, e.g. \left[ a_{i},a_{j}^{\dagger} \right]=\delta _{ij}.

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Physical Realization of the Bose-Hubbard model

The Bose-Hubbard model is nowadays most commonly realized by neutral atomic gases trapped in a optical lattice.

Atom trapped in a Optical lattice

We shall consider a neutral atom interacting with a spatially varying laser field. Let us for the moment assume the frequency of the laser field is comparable but detuned from the transition between an electronic excited state and the ground state. Then, from second-order perturbation theory, the time-average energy shift \, \Delta E is proportional to the field intensity \, I, but inversely proportional to the detuning \Delta \equiv \omega _{laser}-\omega _{eg},

\Delta E\propto \frac{I}{\Delta }.

Therefore, an atom will see an effective potential which is proportional to the intensity \, I and is attractive when \, \Delta <0, repulsive when \, \Delta >0.

A regular "lattice" can be constructed by the interference of pairs of (counter-propagating) polarized light beams. For example, \vec{E}_{1}=\vec{E}_{0}\cos \left( \vec{k}\cdot \vec{r}+\omega t \right) and \vec{E}_{2}=\vec{E}_{0}\cos \left( -\vec{k}\cdot \vec{r}+\omega t \right). The total electric field is simply \vec{E}_{total}=\vec{E}_{1}+\vec{E}_{2}=2\vec{E}_{0}\cos \left( \omega t \right)\cos \left( \vec{k}\cdot \vec{r} \right). We therefore have a spatially varying potential of the form (apart from a constant and consider 1D):

V\left( x \right)=V_{0}\cos \left( 2kx \right),

where V_{0}\propto I/\Delta is a controllable constant. It is straight forward to generalize the analysis into 3D by including two extra pairs of counter propagating beams. Here we note that in real experiments, this potential is superposed by a much slowly varying trapping potential to avoid loss of atoms.

Tight-binding approximation

In principle the width (wavelength of the laser beams) of the wells, and the height (light intensity) of the wells are adjustable. The range of interest for constructing the Bose-Hubbard Hamiltonian is the range where the wavefunction of the system may be described by the localized ground-state wavefunction for each well, and the correction due to the overlap of the wavefunctions nearby. This regime is also known as the tight-binding approximation.

Under this approximation, the hopping (non-interacting) part of the Hamiltonian can be described by

H_{free}=-\frac{t}{2}\sum\limits_{\left\langle i,j \right\rangle }{\left( a_{i}^{\dagger}a_{j}+a_{j}^{\dagger}a_{i} \right)},

where the tunneling term \, t depends on the overlap of the localized wavefunctions \phi _{j}\left( x \right)\equiv \left\langle  x \right|a_{j}^{\dagger}\left| 0 \right\rangle between two wells, and is assumed to be the same throughout the lattice.

Interaction term

For a dilute gas, the effective two-body (s-wave) interaction is the dominate interaction, and is given by

V_{eff}\left( r \right)=\frac{4\pi \hbar ^{2}a_{s}}{m}\delta \left( r \right),

where \, a_{s} is the scattering length. The wavefunction overlap between different wells are exponentially small, compared with the on-site part. We therefore keep only the part where two bosons are in the same well. Provided that the interaction is "weak", compared with the excitation energy of the well (which we shall justify below), the second quantized form of the interaction term is

H_{\operatorname{int}}=\frac{U}{2}\sum\limits_{i}{n_{i}\left( n_{i}-1 \right)}.

Constraint for the scattering length

We shall now consider the condition where higher excitations of the localized states can be neglected. First, we assume each localized well can be approximated by a harmonic potential with energy \hbar \omega _{0}, which depends on the curvature of the bottom part of the well, and the atomic mass. Let us define the length scale \, a_{zp} (called zero-point length) associated with this energy scale by

\hbar \omega _{0}\equiv \frac{\hbar ^{2}}{ma_{zp}^{2}}.

The ground state wavefunction of a harmonic oscillator is given by \psi _{0}\left( x \right)=\left( \frac{1}{\pi } \right)^{1/4}\frac{1}{\sqrt{a_{zp}}}e^{-x^{2}/2a_{zp}^{2}}. Consider two particles in the same state, with \psi \left( x_{1},x_{2} \right)=\psi _{0}\left( x_{1} \right)\psi _{0}\left( x_{2} \right),

U=\frac{4\pi \hbar ^{2}a_{s}}{m}\int{\left| \psi \left( r \right) \right|^{4}dr\approx \frac{\hbar ^{2}a_{s}}{ma_{zp}^{3}}},

apart from some constant. To avoid the excited states, we impose a constraint U\ll \hbar \omega _{0}, which implies that

a_{s}\ll a_{zp}.

This is the result we want in this section.