The correspondence between fermi-sea/bose-condensate displacements and the number-conserving product of two fermi/bose fields is generalised to finite temperatures. It is shown that the straightforward generalisation that involves making the sea-bosons participate in the thermodynamic averaging does not work out. A remedy is found which involves treating the sea-displacement bosons at zero temperature while all finite temperature effects have to be lumped into the coefficients. We also show that all the finite temperature dynamical four-point and six-point functions come out correctly as do some of the commutation rules involving the fermion/boson product that go through unscathed. It is also shown that this unusual prescription of leaving out the sea-bosons from the thermodynamic averaging does in fact reproduce the correct RPA dielectric function at finite temperature. This article is not however, critic-proof.
The theoretical description of trapped weakly-interacting Bose-Einstein condensates is characterized by a large number of seemingly very different approaches which have been developed over the course of time by researchers with very distinct backgrounds. Newcomers to this field, experimentalists and young researchers all face a considerable challenge in navigating through the ‘maze’ of abundant theoretical models, and simple correspondences between existing approaches are not always very transparent. This Tutorial provides a generic introduction to such theories, in an attempt to single out common features and deficiencies of certain ‘classes of approaches ’ identified by their physical content, rather than their particular mathematical implementation. This Tutorial is structured in a manner accessible to a non-specialist with a good working knowledge of quantum mechanics. Although some familiarity with concepts of quantum field theory would be an advantage, key notions such as the occupation number representation of second quantization are nonetheless briefly reviewed. Following a general introduction, the complexity of models is gradually built up, starting from the basic zero-temperature formalism of the Gross-Pitaevskii equation. This structure enables readers to probe different levels of theoretical developments (mean-field, number-conserving and stochastic) according to their particular needs. In addition to its ‘training element’, we hope that this Tutorial will prove useful to active researchers in this field, both in terms of the correspondences made between different theoretical models, and as a source of reference for existing and developing finite-temperature theoretical models.
An attempt is made to generalise the ideas introduced by Haldane and others regarding Bosonizing the Fermi surface. The present attempt involves introduction of Bose fields that correspond to displacements of the Fermi sea rather than just the Fermi surface. This enables the study of short wavelength fluctuations of the Fermi surface and hence the dispersion of single particle excitations with high energy. The number conserving product of two Fermi fields is represented as a simple combination of these Bose fields. It is shown that most(!) commutation rules involving these number conserving products are reproduced exactly, as are the dynamical correlation functions of the free theory. Also the work of Sharp, Menikoff and Goldin has shown that the field operator may be viewed as a unitary representation of the current algebra. An explicit realisation of this unitary representation is given in terms of canonical conjugate of the density operator.
Based on a class of exactly solvable models of interacting Bose and Fermi liquids, we compute the single-particle propagators of these systems exactly for all wavelengths and energies and in any number of spatial dimensions. The field operators are expressed in terms of Bose fields that correspond to displacements of the condensate in the Bose case and displacements of the Fermi sea in the Fermi case. Unlike some of the previous attempts, the present attempt reduces the answer for the spectral function in any dimension in both Fermi and Bose systems to quadratures. It is shown that when only the lowest-order sea-displacement terms are included, is the random-phase approximation in its many guises recovered in the Fermi case, and Bogoliubov's theory in the Bose case. The momentum distribution is evaluated using two different approaches, exact diagonalization and the equation of motion approach; the novelty being, of course, the exact computation of single-particle properties including short-wavelength behavior.
We consider a dilute homogeneous mixture of bosons and spin-polarized fermions at zero temperature. We first construct the formal scheme for carrying out systematic perturbation theory in terms of single particle Green's functions. We especially focus on the description of the boson-fermion interaction. To do so we need to introduce the renormalized boson-fermion
T matrix, which we determine to second order in the boson-fermion
s-wave scattering length. We also discuss how to incorporate the usual boson-boson
T matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to
aBFkF. The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of
6Li and
7Li and of
3He and
4He. Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length
aBF.