Using renormalization-group arguments we show that the low-temperature thermodynamics of a three-or two-dimensional dilute Bose gas is fully determined by a universal scaling function F d (µ/kBT,g(T )) once the mass m and the s-wave scattering length a d of the bosons are known (d is the space dimension). Here µ and T denote the chemical potential and temperature of the gas, and the temperature-dependent dimensionless interaction constantg(T ) is a function of ma 2 d kBT / 2 . We compute the scaling function F2 using a nonperturbative renormalization-group approach and find that both the µ/kBT andg(T ) dependencies are in very good agreement with recent experimental data obtained for a quasi-two-dimensional Bose gas with or without optical lattice. We also show that the nonperturbative renormalization-group estimate of the Berezinskii-KosterlitzThouless transition temperature compares well with the result obtained from a quantum Monte Carlo simulation of an effective classical field theory.
We present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid-Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the original predictions of Fisher et al. [Phys. Rev. B 40, 546 (1989)] based on simple scaling arguments. At a generic transition point, the critical behavior is mean-field like with logarithmic corrections in two dimensions. In the weakly-correlated superfluid phase (far away from the Mott insulating phase), the renormalization-group flow is controlled by the Bogoliubov fixed point down to a characteristic (Ginzburg) momentum scale kG which is much smaller than the inverse healing length k h . In the vicinity of the multicritical points, when the density is commensurate, we identify a sharp crossover from a weakly-to a strongly-correlated superfluid phase where the condensate density and the superfluid stiffness are strongly suppressed and both kG and k h are of the order of the inverse lattice spacing.
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