The renormalization-group approach is extended to a Bose system within the framework of perturbation theory. Recursion relations are obtained for the parameters of the Hamiltonian and the correlation length index $\ensuremath{\nu}$ is evaluated. The values of $\ensuremath{\nu}$ and of the critical exponent $\ensuremath{\eta}$ (evaluated in an earlier paper) agree with those derived for a classical system characterized by a two-component (real) order parameter. The agreement suggests that the universality hypothesis holds irrespective of the quantum or classical nature of a phase transition.
Mapping of a Bose system of interacting particles onto an effective Bose system having an upper cutoff momentum is derived by integrating out momenta between infinity and p0 where p0 is small compared to the inverse de Broglie wavelength λ-1 of the particles. The mapping, within the ∊-expansion scheme, is a general result independent of any argument specific to the critical region. The cross-over behaviour of pressure and density is worked out in the normal phase as well as in the condensed phase. The results are manifestly independent of the cutoff p0 as expected on physical grounds.
We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature
T and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-
1/2 quantum
XY model with coupling
J in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.
The crossover from ideal-Bose-gas behavior to nonclassical, XY-like criticality in a dilute, interacting Bose fluid is considered in detail with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}$(T), below the transition. A general discussion of crossover scaling in constrained systems shows that ${\ensuremath{\rho}}_{s}$ and other temperature-dependent thermodynamic functions at controlled overall density can be expressed in terms of universal scaling functions. The finite-temperature Green's function formalism is used to calculate the associated linear scaling fields and the crossover exponent exactly for all dimensionalities d>2, scaling predictions, including logarithmic singularities just at d=3, being checked via second-order computations. A mapping of the Bose problem onto the classical
s4 spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio
m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio
a∗/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.
The critical behavior in the condensed phase of a Bose system is studied by the renormalization-group method using Bogolubov's well-known prescription for eliminating the condensate operators. Scaling relations and an equation of state in the Griffiths form are derived with explicit expressions for the critical exponents and the scaling function up to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$. The results strengthen the conclusion reached in an earlier paper, namely, that critical behavior in a given symmetry class is independent of the classical or quantum nature of a phase transition.
We investigate the single-particle spectral density of interacting bosons within the nonperturbative functional renormalization group technique. The flow equations for a Bose gas are derived in a scheme which treats the two-particle density-density correlations exactly but neglects irreducible correlations among three and more particles. These flow equations are solved within a truncation which allows to extract the complete frequency and momentum structure of the normal and anomalous self-energies. Both the asymptotic small momentum regime, where the perturbation regime fails, as well as the perturbative regime at larger momenta are well described within a single unified approach. The self-energies do not exhibit any infrared divergences, satisfy the U(1) symmetry constraints, and are in accordance with the Nepomnyashchy relation, which states that the anomalous self-energy vanishes at zero momentum and zero frequency. From the self-energies we extract the single-particle spectral density of the two-dimensional Bose gas. The dispersion is found to be of the Bogoliubov form and shows the crossover from linear Goldstone modes to the quadratic behavior of quasifree bosons. The damping of the quasiparticles is found to be in accordance with the standard Beliaev damping. We furthermore recover the exact asymptotic limit of the propagators derived by Gavoret and Nozi\`eres and discuss the nature of the nonanalyticities of the self-energies in the very small momentum regime.