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

TL;DR

The paper develops a strong-coupling RPA framework for the Bose-Hubbard model to capture universal features of the Mott insulator–superfluid transition driven by density. It identifies emergent quasiparticles with an effective mass $m^*$ and an effective scattering length $a^*$ that govern the singular part of the pressure and the high-momentum tail of the momentum distribution via a universal two-body contact $C_{ m univ}$. The approach reproduces the Bogoliubov-like thermodynamics in the superfluid near the transition and connects to a universal scaling framework with $z=2$ and $ u=1/2$ characteristic of the dilute Bose gas universality class. Extensions to hard-core bosons and to bosons in an optical lattice yield consistent definitions of short-distance and full contacts, highlighting both universal behavior and lattice-specific corrections. Overall, the work provides a tractable route to quantify universal thermodynamics and high-momentum signatures in strongly correlated Bose gases near the MI-SF transition, with potential experimental observables in optical lattices and related systems.

Abstract

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 $m^*$ 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 $|n-n_{\rm MI}|$ 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 $C_{\rm univ}^{\rm sd}$ which controls the high-momentum tail of $n^{\rm sing}_{\bf k}$ at scales larger than the inverse lattice spacing, and a ``full'' contact $C$ which controls the high-momentum tail of the full momentum distribution $n_{\bf k}$.

Figures (6)

• Figure 1: Phase diagram of the three-dimensional Bose-Hubbard model obtained from the criterion $G_{\rm loc}^{-1}(i\omega_n=0)+D=0$ ($D=-t_{{\bf k}=0}=6t$). Each Mott lobe is labeled by the integer $n_{\rm MI}$ giving the mean number of bosons per site. The trivial Mott insulator $n_{\rm MI}=0$ corresponds to the vacuum.
• Figure 2: Excitation energies in the Mott insulator ($\mu=0.9\,\mu_+$, left) and the superfluid state ($\mu=1.1\,\mu_+$, right) 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 (\ref{['cdef']}) in the superfluid state). $\Gamma=(0,0,0)$, R$=(\pi,\pi,\pi)$ and X$=(\pi,0,0)$.
• Figure 3: Effective mass $m^*_\alpha/m_{\rm lat}=1/Z^\alpha_{\rm QP}$ (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 $a_{\rm lat}$ of the bosons moving on the lattice [Eq. (\ref{['alat']})].
• Figure 4: Momentum distribution $n_{\bf k}=-{\cal S}(-{\cal E}_{\bf k}^+)-{\cal S}(-{\cal E}_{\bf k}^-)$, along the Brillouin zone diagonal ${\bf k}=(k,k,k)$, for the particle-doped Mott insulator $n_{\rm MI}=1$: $\mu=1.000005\mu_+$ ($n=1.0003$) and $D=D_c/2$. The contribution $n_0{\cal V} \delta_{{\bf k},0}$ of the condensate is not taken into account. The gapless band $-{\cal E}_{\bf k}^-$ gives a significant contribution only near $k=0$ (see left inset). For larger values of $k$, the momentum distribution is essentially due to the gapped band $-{\cal E}_{\bf k}^+$ whose contribution is very close to $n_{\bf k}^{\rm MI}=-{\cal S}_{\rm MI}(E_{\bf k}^-)$ (see right inset).
• Figure 5: 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 $D=D_c/2$. Top: $\mu=1.000005\mu_+$ ($n=1.0003$); Bottom: $\mu=0.99944\mu_-$ ($n=0.99972$). The figure also shows $Z^\alpha_{\rm QP} n^{\rm Bog}_{\alpha,{\bf k}}$ [Eq. (\ref{['nkbog']})] and $Z_{\rm QP}^+ C_{\rm univ}/{\cal V}|{\bf k}|^4$ where $C_{\rm univ}={\cal V}[m^*_+(\mu-\mu_+)]^2$ is the contact. The dash-dotted (blue) line corresponds to the contribution of the gapless band with negative energy $-{\cal E}^-_{\bf k}$ (top) or positive energy ${\cal E}^-_{\bf k}$ (bottom); see Eqs. (\ref{['nksingpart']}) and (\ref{['nksinghole']}). The vertical dotted line shows the momentum scale $k_*/\sqrt{3}$ where $k_*=2(m^*_\alpha|\mu-\mu_\alpha|)^{1/2}$.