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.

Figures (4)

• Figure 1: Phase diagram of the three-dimensional Bose-Hubbard model obtained from the criterion $G_{\rm loc}^{-1}(i\omega_n=0)+D=0$ ($D=6t$). Each Mott lobe is labeled by the integer $n_{\rm MI}$ giving the mean number of bosons per site.
• Figure 2: Excitation energies in the Mott insulator ($\mu=0.9\,\mu_+$, left) and in the superfluid state ($\mu=1.1\,\mu_+$, right), along the Brillouin zone diagonal ${\bf k}=(k,k,k)$, for $n_{\rm MI}=1$. The dashed lines show the approximate low-energy forms, valid near ${\bf k}=0$, $E_{\bf k}^+= {\bf k}^2/2m^*_+ + \mu_+ - \mu$ and ${\cal E}_{\bf k}^-=c|{\bf k}|$ (with $c$ the sound velocity in the superfluid state).
• Figure 3: Effective mass $m^*_\alpha$ (left) and effective scattering length $a^*_\alpha$ (right) vs $D/D_c$ at the quantum critical point between the Mott insulator $n_{\rm MI}=1$ and the superfluid state, obtained from strong-coupling RPA, nonperturbative functional renormalization group (FRG) Rancon12d and quantum Monte Carlo simulations (QMC) Capogrosso07. The green dotted line in the right panel shows the (vacuum) scattering length $a_{\rm lat}\simeq 1/[8\pi(t/U+0.1264)]$ of the bosons moving on the lattice Rancon11b. The strong-coupling RPA is reliable for the effective mass but less so for the effective scattering length, even if the general trend is correct.
• Figure 4: Singular part $n^{\rm sing}_{\bf k}=n_{\bf k}-n_{\bf k}^{\rm MI}$ of the momentum distribution, along the Brillouin zone diagonal ${\bf k}=(k,k,k)$, for $D=D_c/2$ and $\mu=1.000005\mu_+$ ($n=1.0003$), compared to $-{\cal S}(-{\cal E}_{\bf k}^-)$ (spectral weight of the negative gapless energy branch), $Z^+_{\rm QP}n_{\bf k}^{\rm Bog}$ [Eq. (\ref{['nkbog']})] and $Z_{\rm QP}^+ C_{\rm univ}/{\cal V}|{\bf k}|^4$ where $C_{\rm univ}$ is the contact defined from the pressure [Eq. (\ref{['Cmu']})]. The vertical dotted line shows the momentum scale $k_*/\sqrt{3}$ where $k_*=2(m^*_\alpha|\mu-\mu_\alpha|)^{1/2}$.