We present a strong-coupling expansion of the Bose-Hubbard model based on a mean-field treatment of the hopping term, while onsite fluctuations are taken into account exactly. This random phase approximation describes the universal features of the generic Mott-insulator–superfluid transition (induced by a density change) and the superfluid state near the phase transition. The critical quasiparticles at the quantum critical point have a quadratic dispersion with an effective mass $m^{*}$ and their mutual interaction is described by an effective $s$-wave scattering length $a^{*}$. The singular part of the pressure takes the same form as in a dilute Bose gas, provided we replace the boson mass $m$ and the scattering length in vacuum $a$ by $m^{*}$ and $a^{*}$, and the density $n$ by the excess density $|n-n_{\mathrm{MI}}|$ of particles (or holes) with respect to the Mott insulator. We define a universal two-body contact $C_{\mathrm{univ}}$ that controls the high-momentum tail $\sim 1/{\left|\mathbf{k}\right|}^4$ of the singular part $n_{\mathbf{k}}^{\mathrm{sing}}$ of the momentum distribution. We also apply the strong-coupling RPA to a lattice model of hard-core bosons and find that the high-momentum distribution is controlled by a universal contact, in complete agreement with the Bose-Hubbard model. Finally, we discuss a continuum model of bosons in an optical lattice and define two additional two-body contacts: a short-distance universal contact $C_{\mathrm{univ}}^{\mathrm{sd}}$ which controls the high-momentum tail of $n_{\mathbf{k}}^{\mathrm{sing}}$ at scales larger than the inverse lattice spacing, and a full contact $C$, which controls the high-momentum tail of the full-momentum distribution $n_{\mathbf{k}}$.

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• Bose gases
• Cold atoms & matter waves
• Mott-superfluid transition