The Bose–Hubbard Model and Controlled Limits

The Bose–Hubbard model has controlled superfluid and atomic Mott limits, but a finite interaction ratio alone does not identify either phase. At integer filling the atomic limit has particle and hole gaps; hopping closes them, and the thermodynamic combination of compressibility, stiffness, and gap distinguishes the phases before any universality claim is made.

Required background. Use phase and stiffness and the general notions of critical surfaces and universality. Helpful background. Quantum-simulation validity and lattice error budgets govern platform and numerical certification.

Bose–Hubbard Hamiltonian and exact limits

On a regular lattice with coordination $z$,

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

At $t=0$, the on-site energy is $E_n=Un(n-1)/2-\mu n$. Occupation $n_0$ minimizes it when

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

The particle and hole costs are

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

Both are positive inside an atomic Mott interval, giving zero compressibility away from its boundaries. To first order in hopping, a single particle defect moves with amplitude $t(n_0+1)$ and a hole with amplitude $tn_0$, so their band minima shift by $-zt(n_0+1)$ and $-ztn_0$, respectively. The strong-coupling estimate fails when defects proliferate, but it correctly anchors the lobe shape.

For $t/U\gg1$ at noninteger density, phase coherence and a gapless mode are expected in dimensions that support long-range or algebraic order. A mean-field decoupling $b_i^\dagger b_j\to\psi^*b_j+b_i^\dagger\psi-|\psi|^2$ gives a qualitative boundary, not a dimension-independent control theorem. The model and phase structure were established systematically by Fisher et al. 1989, pp. 546–570; the superfluid-to-Mott crossover was realized in a three-dimensional optical lattice by Greiner et al. 2002, pp. 39–44.

What establishes a Mott regime

A thermodynamic Mott claim requires integer filling, a charge gap, and vanishing compressibility $\kappa=\partial n/\partial\mu$ and stiffness. Finite systems have avoided crossings and discrete addition energies, so each quantity needs size scaling. In a trap, shells at different local chemical potential coexist. Disorder can insert a compressible Bose-glass regime. Higher bands and density-assisted hopping test whether the one-band Hamiltonian remains adequate.

Critical scaling at the lobe tip and edge is not the same; the next page derives the two continuum actions.

Exercises

Find the atomic Mott interval and the minimum particle–hole excitation energy at its midpoint.

Solution

The interval is $U(n_0-1)<\mu<Un_0$. At its midpoint $\mu=U(n_0-1/2)$, both $\Delta_p$ and $\Delta_h$ equal $U/2$; a neutral particle–hole pair costs $U$ in the atomic limit.

References

• Michael P. A. Fisher, Peter B. Weichman, Geoffrey Grinstein, and Daniel S. Fisher, “Boson Localization and the Superfluid–Insulator Transition,” Physical Review B 40 (1989) 546–570, doi:10.1103/PhysRevB.40.546.
• Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hänsch, and Immanuel Bloch, “Quantum Phase Transition from a Superfluid to a Mott Insulator in a Gas of Ultracold Atoms,” Nature 415 (2002) 39–44, doi:10.1038/415039a.