Strong-coupling RPA theory of a Bose gas near the superfluid--Mott-insulator transition: universal thermodynamics and two-body contact

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 (RPA) 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 quasi-particles at the quantum critical point have a quadratic dispersion with an effective mass $m^{\ast }$ and their mutual interaction is described by an effective $s$-wave scattering length $a^{\ast }$. 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 $m^{\ast }$ and $a^{\ast }$, and the density $n$ by the excess density $\mathrm{\mid }n-n_{\mathrm{M}\mathrm{I}}\mathrm{\mid }$ of particles (or holes) with respect to the Mott insulator. We define a universal'' two-body contact $C_{\rm univ}$ that controls the high-momentum tail $\sim 1/|{\bf k}|^4$ of the singular part $n^{\rm sing}_{\bf k}$ 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 $C_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}^{\mathrm{s}\mathrm{d}}$ which controls the high-momentum tail of $n_{\mathbf{k}}^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ at scales larger than the inverse lattice spacing, and a ``full’’ contact $C$ which controls the high-momentum tail of the full momentum distribution $n_{\mathbf{k}}$.