Bose–Hubbard Model

The Bose–Hubbard model describes bosons hopping on a lattice while paying an energy for sharing a site. Unlike spin-$1/2$ fermions in a single orbital, each site can hold any nonnegative integer number of bosons. Its atomic limit explains how integer-density plateaus and particle-addition gaps arise before one studies the collective superfluid–Mott transition.

Bosons on a lattice

Take a uniform hypercubic lattice in $d_s$ spatial dimensions, with nearest-neighbor hopping and no disorder or trap. The grand-canonical Hamiltonian is

$$\begin{aligned} H={}&-J\sum_{\langle ij\rangle} (b_i^\dagger b_j+\mathrm{h.c.})\\ &+\frac U2\sum_i n_i(n_i-1)-\mu\sum_i n_i, \qquad U>0,\quad J\ge0. \end{aligned}$$

Here $[b_i,b_j^\dagger]=\delta_{ij}$, $n_i=b_i^\dagger b_i$, and each unoriented bond appears once. The spatial dimension, lattice spacing, boundary conditions, density or chemical potential, and temperature are part of the physical specification. Global phase rotations conserve total boson number.

An optical-lattice realization reduces to this single-band Hamiltonian only when excited bands and longer-range interaction and hopping terms are adequately suppressed Bloch, Dalibard, and Zwerger 2007, §IV.1.

Exact atomic occupations and gaps

At $J=0$, sites decouple. A site with occupation $n$ has energy

$$E_n=\frac U2 n(n-1)-\mu n.$$

For an integer $n_0\ge1$, comparing $E_{n_0}$ with both neighboring occupations gives

$$U(n_0-1)<\mu<Un_0.$$

Inside this interval, the unique on-site ground state has $n_0$ bosons. The costs to add or remove one particle are

$$\Delta_p=Un_0-\mu, \qquad \Delta_h=\mu-U(n_0-1).$$

Both are positive. Thus the zero-temperature density stays constant as $\mu$ varies within the interval, and the compressibility $\partial n_0/\partial\mu$ vanishes. At its midpoint, both gaps equal $U/2$, while a neutral particle–hole pair costs $U$.

The vacuum is a separate case: $n_0=0$ minimizes the energy for $\mu<0$, and no hole excitation exists. At an interval boundary, adjacent occupations are degenerate, so nondegenerate perturbation theory about a unique atomic product state fails.

Hopping and collective behavior

A small hopping amplitude lets an added particle or hole propagate. Its matrix element is enhanced by bosonic occupation factors: $J(n_0+1)$ for the particle and $Jn_0$ for the hole. Consequently, gaps decrease as their defect bands widen. The finite-$J$ Mott ground state already contains virtual number fluctuations; it is not exactly the atomic product state.

At nonzero density, the competition with phase coherence leads to Mott regions at integer filling and superfluid behavior outside them in the appropriate ground-state phase diagram Bloch, Dalibard, and Zwerger 2007, §IV.2. The empty vacuum is a separate region. A first-order defect estimate does not accurately locate every critical boundary.

Density, compressibility, particle-addition gaps, and response to a boundary-phase twist probe distinct aspects of the phases. Finite size, finite temperature, and low spatial dimension affect how order and stiffness are identified. A trap can create regions at different local densities, while disorder changes the uniform-model problem.

Develop the calculation

The Bose–Hubbard Model and Controlled Limits develops the defect calculation. Superfluid–Mott Criticality distinguishes the lobe tip from a generic density-changing boundary. Compare the continuum Weakly Interacting Bose Gas and the spinful Fermi–Hubbard Model.

References

• Bloch, Immanuel, Jean Dalibard, and Wilhelm Zwerger. Many-Body Physics with Ultracold Gases. Preprint (2007), arXiv:0704.3011v2. Open text.