The standard perturbation theory for repulsive bosons breaks down at finite temperature in two dimensions because of the vanishing of both the Bose-Einstein condensate (at finite temperature) and the t matrix at long wavelengths and zero particle density. By adopting polar coordinates and using a running coupling constant, we circumvent the above two difficulties. The resulting theory is more straightforward than an earlier one of Popov which depends on an artificial division of momentum space. It can also be applied to three dimensions.
The standard perturbation theory for repulsive bosons breaks down at finite temperature in two dimensions because of the vanishing of both the Bose-Einstein condensate (at finite temperature) and the t matrix at long wavelengths and zero particle density. By adopting polar coordinates and using a running coupling constant, we circumvent the above two difficulties. The resulting theory is more straightforward than an earlier one of Popov which depends on an artificial division of momentum space. It can also be applied to three dimensions.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is Tc\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/na2)], and the condition of validity of the dilute limit is ln ln(1/na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.
The theory of ultracold, dilute Bose gases is the subject of intensive studies, driven by new experimental applications, which also motivate the study of Bose-Einstein condensation (BEC) in low dimensions. From the theoretical point of view there are few, quite special, models in which we are able to prove BEC for interacting bosons. With the aim of studying the condensation problem I considered a simplified model for a three and two dimensional system of bosons interacting with a repulsive short range potential, obtained by introducing an ultraviolet momentum cutoff. I proved that the interacting theory is well defined at all orders in terms of one (in three dimensions) or two (in two dimensions) effective parameters, related to the intensity of the interaction. Moreover the correlations do not exhibit anomalous dimensions, i.e. the model is in the same universality class of the exactly soluble Bogoliubov model. The result is obtained by using Rigorous Renormalization Group techniques and implementing local Ward identities (WI), which reduce the number of independent running couplings, this fact being crucial for the control of the two dimensional theory, where the effective interactions are relevant. Since the ultraviolet momentum cutoff breaks local gauge invariance, the corrections to the formal WI have been considered.
We investigate Bose-Einstein condensation for interacting bosons at zero and nonzero temperature. Functional renormalization provides us with a consistent method to compute the effect of fluctuations beyond the Bogoliubov approximation. For three-dimensional dilute gases, we find an upper bound on the scattering length a which is of the order of the microphysical scale---typically the range of the van der Waals interaction. In contrast to fermions near the unitary bound, no strong interactions occur for bosons with approximately pointlike interactions, thus explaining the high quantitative reliability of perturbation theory for most quantities. For zero temperature we compute the quantum phase diagram for bosonic quasiparticles with a general dispersion relation, corresponding to an inverse microphysical propagator with terms linear and quadratic in the frequency. We compute the temperature dependence of the condensate and particle density n, and find for the critical temperature Tc a deviation from the free theory, $\ensuremath{\Delta}{T}_{c}/{T}_{c}=2.1a{n}^{1/3}$. For the sound velocity at zero temperature we find very good agreement with the Bogoliubov result, such that it may be used to determine the particle density accurately.
Through exact numerical solutions we show Bose-Einstein condensation (BEC) for the one-dimensional (1D) bosons with repulsive short-range interactions at zero temperature by taking a particular large size limit. Following the Penrose-Onsager criterion of BEC, we define condensate fraction by the fraction of the largest eigenvalue of the one-particle reduced density matrix. % We show the finite-size scaling behavior such that condensate fraction is given by a scaling function of one-variable: interaction parameter multiplied by a power of particle number. Condensate fraction is nonzero and constant for any large value of particle number or system size, if the interaction parameter is proportional to the negative power of particle number. %Here the interaction parameter is defined by the coupling constant of the delta-function potentials devided by the density. %With the scaling behavior we derive various themodynamic limits where condensate fraction is constant for any large system size; for instance, it is the case even in the system of a finite particle number.
A Renormalization-Group Study of Interacting Bose–Einstein Condensates: II. Anomalous Dimension η for d≲4 at Finite Temperatures
We study the anomalous dimension η of homogeneous interacting single-component Bose-Einstein condensates at finite temperatures for d≲4 dimensions. This η is defined in terms of the one-particle density matrix ρ(r)≡⟨ψ^†(r1)ψ^(r1+r)⟩ through its asymptotic behavior ρ(r)→N0/V+Cr−d+2−η for r→∞, where N0/V is the condensate density and C is a constant. It is shown that the anomalous dimension is given by η=0.181ε2 to the leading order in ε≡d−4. The change of the prefactor 0.181 from the value 0.02 at the transition point of the O(2) symmetric ϕ4 model is attributed to the emergence of three-point vertices and the anomalous Green's function when N0 acquires a finite value.
We investigate exactly a system of either fermions or bosons interacting in one dimension by a two-body potential V(r)=r2g with periodic boundary conditions. In addition to rederiving known results for correlation functions and thermodynamics in the thermodynamic limit, we present expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant g, as a determinant of order $N\ifmmode\times\else\texttimes\fi{}N$. These concise expressions allow a discussion of the momentum distribution in the thermodynamic limit. In particular, for a case of repulsive bosons, the determinant is evaluated explicitly, exhibiting a weak (logarithmic) singularity at zero momentum, and vanishing outside of a "Fermi" surface.