The well{known results concerning a dilute Bose gas with the short{range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method as basic to much of the present understanding of this subject and nonrelevant behaviour of the pair distribution function at small boson separations.The aim of our paper is to propose a new way of treating the dilute Bose gas with an arbitrary strong interaction.Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials like the Lennard{Jones one.All the consideration concerns the zero temperature.
The well{known results concerning a dilute Bose gas with the short{range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method as basic to much of the present understanding of this subject and nonrelevant behaviour of the pair distribution function at small boson separations.The aim of our paper is to propose a new way of treating the dilute Bose gas with an arbitrary strong interaction.Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials like the Lennard{Jones one.All the consideration concerns the zero temperature.
The well-known results concerning a dilute Bose gas with the short-range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method being basic to much of the present understanding of this subject. The aim of our paper is to propose another way of treating the dilute Bose gas with an arbitrary strong interaction. Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials of the Lennard-Jones type. The derived expansion of the condensate depletion in powers of the boson density n=N/V reproduces the familiar result, while the expansion for the mean energy per particle is of the form epsilon=2 pi planck(2)an/m[1+128/(15 square root of [pi]) square root of [na(3)](1-5b/8a)+...], where a is the scattering length and b> or =0 stands for one more characteristic length depending on the shape of the interaction potential (in particular, for the hard spheres a=b). All the consideration concerns the zero temperature.
The thermodynamics of a homogeneous dilute Bose gas with an arbitrarily strong repulsion between particles is investigated on the basis of the exact relation connecting the pair correlation function with the in-medium pair wave functions and occupation numbers. It is shown that the effective-interaction scheme, which is reduced to the Bogoliubov model with the effective pairwise potential, is not acceptable for investigating the short-range particle correlations in a dilute strongly interacting Bose gas. In contrast to this scheme, our model is thermodynamically consistent and free of the ultraviolet divergences due to accurate treatment of the short-range boson correlations. An equation for the in-medium scattering amplitude is derived that makes it possible to find the in-medium renormalization for the pair wave functions at short boson separations. Low-density expansions for the main thermodynamic quantities are reinvestigated on the basis of this equation. In addition, the expansions are found for the interaction and kinetic energies per particle. It is demonstrated that for a many-boson system of hard spheres the interaction energy is equal to zero for any boson density. The exact relationship between the chemical potential and in-medium pair wave functions is also established.
The low-density expansions for the energy, chemical potential, and condensate depletion of the homogeneous dilute dipolar Bose gas are obtained by regularizing the dipole-dipole interaction at long distances. It is shown that the leading term, proportional to the density, allows a simple physical interpretation and consistently describes the thermodynamic stability of the system. The long-range asymptotics are obtained analytically for the normal and anomalous one-particle correlation functions and the pair distribution function. We discuss the properties of the two-body scattering with zero relative momentum for the dipole-dipole interaction; in particular, we derive the asymptotics of the wave function and a correction to the scattering length for small values of the dipolar range. We show how the density expansions can be derived within the Bogoliubov model of weakly interacting particles without any divergence from the assumption of universality of the expansions at low densities.
We study the properties of quantum gases which are dilute in terms of the interactions but degenerate; this may be the case if the thermal wavelength of the particles is much larger than the interatomic potential range. We assume that the potential contains a short range repulsive part, which creates strong correlations between particles that cannot be treated by ordinary perturbation theory, even if the region of configuration space where they are dominant is relatively small. Our method combines two basic ingredients: Ursell operators, which generalize the Ursell functions that are usually used to generate quantum cluster expansions, but are not in themselves symmetrized (the operators can be related to an auxiliary system of distinguishable particles); in a distinct step, an exact treatment of particle indistinguishability is given through the inclusion of exchange cycles of arbitrary length. For non-condensed boson systems, as well as for fermion systems, the method provides a compact expression of the partition function that generalizes the well-known Beth–Uhlenbeck formula to arbitrary degeneracy. For the study of the Bose–Einstein condensation, our treatment retains more information on the short range correlations than, for instance, the Hartree–Fock method. It also introduces a natural and explicit distinction between particles and quasiparticles; the latter may condense into a single one-particle state, which obeys a non-linear equation somewhat similar to the Gross–Pitaevskii equation.
A new approach developed in Ref.1 for investigating the spatial boson correlations at low temperatures is considered in the particular case of the Coulomb interaction potential. This approach is based on integro–differential equations for the radial distribution function and can be used beyond the weak coupling regime. PACS: 05.30.Jp, 67.90.+z Description of the charged Bose liquid, or, in other words, of the strongly correlated many–boson system with the Coulomb interparticle potential, is a long–standing problem of the condensed matter physics. Recently, interest in this problem has been renewed in the connection with the noteworthy results of applying the Bose and Bose–Fermi liquid models [2] to the explanation of various phenomena inherent in the superconductors with high critical temperatures. These models are based on the Bogoliubov approach [3] implying the weak coupling regime. For the system of charged bosons the weak coupling approximation can be used at small values of the Brueckner parameter rS ∼ 0.1. However, questions requiring detailed investigations often concern the materials with relatively low densities of charge carriers which correspond to rS−values even larger than those of the electron gas at 1 the metallic densities. For example, there are the results [4] testifying to the possibility of forming the so–called large bipolaron (e.g. the two–electron cluster with a large radius of electron–electron separation, see papers in Ref. [5]) inside the polaron environment at the electron densities ∼ 10 19 cm −3 close to the carrier densities of the heavily doped degenerate polar semiconductors. To investigate the possible phase transition of the charge carrier system from the polaron state to the bipolaron phase at these densities, one should operate with the charged Bose liquid of large bipolarons for rS ≫ 1. Indeed, the ”large–bipolaron“ Bohr radius can be expressed [4] as abip =
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.