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arXiv:2501.14887 · Physical Review A

Strong-coupling random-phase-approximation theory of a Bose gas near the superfluid–Mott-insulator transition: Universal thermodynamics and two-body contact

N. Dupuis, Moksh Bhateja, Adam Rançon

Published Sep 11, 2025• 2 citations
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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 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 quasiparticles 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 $|n\ensuremath{-}{n}_{\mathrm{MI}}|$ of particles (or holes) with respect to the Mott insulator. We define a universal two-body contact ${C}_{\mathrm{univ}}$ that controls the high-momentum tail $\ensuremath{\sim}1/{|\mathbf{k}|}^{4}$ of the singular part ${n}_{\mathbf{k}}^{\mathrm{sing}}$ 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 ${C}_{\mathrm{univ}}^{\mathrm{sd}}$ which controls the high-momentum tail of ${n}_{\mathbf{k}}^{\mathrm{sing}}$ at scales larger than the inverse lattice spacing, and a full contact $C$, which controls the high-momentum tail of the full-momentum distribution ${n}_{\mathbf{k}}$.

DOI
10.1103/zhpw-crqc

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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 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 quasiparticles at the quantum critical point have a quadratic dispersion with an effective mass m∗{m}^{*}m∗ and their mutual interaction is described by an effective sss-wave scattering length a∗{a}^{*}a∗. The singular part of the pressure takes the same form as in a dilute Bose gas, provided we replace the boson mass mmm and the scattering length in vacuum aaa by m∗{m}^{*}m∗ and a∗{a}^{*}a∗, and the density nnn by the excess density ∣n\ensuremath−nMI∣|n\ensuremath{-}{n}_{\mathrm{MI}}|∣n\ensuremath−nMI​∣ of particles (or holes) with respect to the Mott insulator. We define a universal two-body contact Cuniv{C}_{\mathrm{univ}}Cuniv​ that controls the high-momentum tail \ensuremath∼1/∣k∣4\ensuremath{\sim}1/{|\mathbf{k}|}^{4}\ensuremath∼1/∣k∣4 of the singular part nksing{n}_{\mathbf{k}}^{\mathrm{sing}}nksing​ 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 Cunivsd{C}_{\mathrm{univ}}^{\mathrm{sd}}Cunivsd​ which controls the high-momentum tail of nksing{n}_{\mathbf{k}}^{\mathrm{sing}}nksing​ at scales larger than the inverse lattice spacing, and a full contact CCC, which controls the high-momentum tail of the full-momentum distribution nk{n}_{\mathbf{k}}nk​.
Bosonic atoms confined in optical lattices are described by the Bose-Hubbard model and can exist in two different phases, Mott insulator or superfluid, depending on the strength of the system parameters. In the vicinity of the phase boundary, there are degeneracies that occur between every two adjacent Mott lobes. Because of this, nondegenerate perturbation theory fails to give meaningful results for the condensate density: it predicts a phase transition in a point of the phase diagram where no transition occurs. Motivated by this, we develop two different degenerate perturbative methods to solve the degeneracy-related problems. Moreover, we study the one-dimensional repulsively interacting Bose gas under harmonic confinement, with special attention to the asymptotic behavior of the momentum distribution, which is a universal k−4k^{-4}k−4 decay characterized by the Tan's contact. The latter constitutes a direct signature of the short-range correlations in such an interacting system and provides valuable insights about the role of the interparticle interactions. We investigate the system constituted of NNN interacting particles in the strongly interacting limit. In such a regime, the strong interparticle interaction makes the bosons behave similarly to the ideal Fermi gas. Because of the difficulty in analytically solving the system for NNN particles at finite interaction, the Tonks-Girardeau regime provides a favorable scenario to probe the contact. Therefore, we are able to provide an analytical formula for the Tan's contact. Furthermore, we analyze the scaling properties of the Tan's contact in terms of NNN in the high-temperature regime as well as in the strongly interacting regime. Finally, we compare our analytical calculations of the Tan's contact to quantum Monte Carlo simulations and discuss some fundamental differences between the canonical and the grand-canonical ensembles.
We study the quantum phase transitions and identify a tricritical point between a normal Bose superfluid, a superfluid that breaks additional Z(2) Ising symmetry, and a Mott insulator in a recent shaken optical lattice experiment. We show that near the transition between normal and Z(2) symmetry breaking superfluids, bosons can condense into a momentum state with high or even locally maximum kinetic energies due to the interaction effect. We present a general low-energy effective field theory that treats both the superfluid transition and the Ising transition in a uniform framework. Using the perturbative renormalization group method, we find that the critical behavior of the quantum phase transition belongs to a universality class different from that of a dilute Bose gas.
We investigate properties of an ultracold, two-component bosonic gas in a square optical lattice at unit filling. In addition to density-density interactions, the atoms are subject to coherent light-matter interactions that couple different internal states. We examine the influence of this coherent coupling on the system and its quantum phases by using Gutzwiller mean-field theory as well as bosonic dynamical mean-field theory. We find that the interplay of strong interspecies repulsion and coherent coupling affects the Mott insulator to superfluid transition and shifts the tip of the Mott lobe toward higher values of the tunneling amplitude. In the strongly interacting Mott regime, the resulting Bose-Hubbard model can be mapped onto an effective spin Hamiltonian that offers additional insights into the observed phenomena.
We study an extended Bose-Hubbard model with next-nearest-neighbor (NNN) hopping in a shaken optical lattice. We show how mean-field phase diagram evolves with the change of NNN hopping amplitude t2{t}_{2}t2​, which can be easily tuned via shaking amplitude. As t2{t}_{2}t2​ increases, a Z2{Z}_{2}Z2​-symmetry-breaking superfluid (Z2SF)({Z}_{2}\mathrm{SF})(Z2​SF) phase emerges at the bottom of the Mott lobs. The tricritical points between normal superfluid, Z2SF{Z}_{2}\mathrm{SF}Z2​SF, and Mott insulator (MI) phases are identified. We further demonstrate the tricritical point can be tuned to the tip of the Mott lobe, in which case a new critical behavior has been predicted. Within random-phase approximation, excitation spectra in the three phases are obtained, which indicate how the phase transitions occur.
We investigate the thermal physics of a Bose-Hubbard model with Rashba spin-orbit coupling starting from a strong coupling mean-field ground state. The essential role of the spin-orbit coupling $\left(\ensuremath{\gamma}\right)$ is to promote condensation of the bosons at a finite wave vector ${\mathbit{k}}_{0}$. We find that the bosons display either homogeneous or phase-twisted or orbital ordered superfluid phases, depending on $\ensuremath{\gamma}$ and the interspecies interaction strength $\left(\ensuremath{\lambda}\right)$. We show that an increase of $\ensuremath{\gamma}$ leads to suppression of the critical interaction Uc{U}_{c}Uc​ for the superfluid to Mott insulator transition in the ground state and a reduction of the Tc{T}_{c}Tc​ for superfluid to Bose-liquid transition at a fixed interaction strength. We capture the thermal broadening in the momentum distribution function, and the real space profiles of the thermally disordered magnetic textures, including their homogenization for $T\ensuremath{\gtrsim}{T}_{c}$. We provide a Landau theory based description of the ground state phase boundaries and thermal transition scales and discuss experiments which can test our theory.