The two-body contact is a fundamental quantity of a dilute Bose gas that relates the thermodynamics to the short-distance two-body correlations. For a Bose gas in an optical lattice, near the superfluid–Mott-insulator transition, we show that a “universal” contact $C_{\mathrm{univ}}$ can be defined from the singular part $P-P_{\mathrm{MI}}$ of the pressure ($P_{\mathrm{MI}}$ is the pressure of the Mott insulator). Its expression $C_{\mathrm{univ}}=C_{\mathrm{DBG}}\left(\right|n-n_{\mathrm{MI}}|,a^{*})$ coincides with that of a dilute Bose gas provided we consider the effective “scattering length” $a^{*}$ of the quasiparticles at the quantum critical point (QCP) rather than the scattering length in vacuum, and the excess density $|n-n_{\mathrm{MI}}|$ of particles (or holes) with respect to the Mott insulator. Close to the transition, we find that the singular part $n_{\mathbf{k}}^{\mathrm{sing}}=n_{\mathbf{k}}-n_{\mathbf{k}}^{\mathrm{MI}}$ of the momentum distribution exhibits a high-momentum tail of the form $Z_{\mathrm{QP}}{C}_{\mathrm{univ}}/{\left|\mathbf{k}\right|}^4$ over a broad region of the Brillouin zone, where $Z_{\mathrm{QP}}$ is the quasiparticle weight of the elementary excitations at the QCP. Our results demonstrate that the notion of contact extends to strongly correlated lattice bosons, and we argue that the contact $C_{\mathrm{univ}}$ can be measured in state-of-the-art experiments on Bose gases in optical lattices and magnetic insulators.

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