Two-body contact of a Bose gas near the superfluid--Mott-insulator transition

Abstract

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 $C_{\rm univ}$ can be defined from the singular part $P-P_{\rm MI}$ of the pressure ($P_{\rm MI}$ is the pressure of the Mott insulator). Its expression $C_{\rm univ}=C_{\rm DBG}(|n-n^{\rm MI}|,a^*)$ coincides with that of a dilute Bose gas provided we consider the effective ``scattering length'' $a^*$ of the quasi-particles at the quantum critical point (QCP) rather than the scattering length in vacuum, and the excess density $|n-n^{\rm MI}|$ of particles (or holes) with respect to the Mott insulator. Close to the transition, we find that the singular part $n^{\rm sing}_{\bf k} = n_{\bf k} - n^{\rm MI}_{\bf k}$ of the momentum distribution exhibits a high-momentum tail of the form $Z_{\rm QP} C_{\rm univ}/|{\bf k}|^4$ over a broad region of the Brillouin zone, where $Z_{\rm QP}$ is the quasi-particle 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 $C_{\rm univ}$ can be measured in state-of-the-art experiments on Bose gases in optical lattices and magnetic insulators.