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arXiv:cond-mat/0606363 · Laser Physics

Nonequivalent operator representations for Bose-condensed systems

Vyacheslav I. Yukalov

Published Mar 1, 2006• 41 citations
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Abstract

The necessity of accurately taking into account the existence of nonequivalent operator representations associated with canonical transformations is discussed. It is demonstrated that Bose systems in the presence of the Bose-Einstein condensate and without it correspond to different Fock spaces, orthogonal to each other. A composite representation for the field operators is constructed allowing for a self-consistent description of Bose-condensed systems. Equations of motion are derived from the given Hamiltonian, which guarantees the validity of conservation laws and thermodynamic self-consistency. At the same time, the particle spectrum obtained either from diagonalizing this Hamiltonian or from linearizing the field-operator equations of motion has no gap. The condition of the condensate existence assures the absence of the gap in the spectrum, irrespectively to the approximation involved. The suggested self-consistent theory is both conserving and gapless.

DOI
10.1134/s1054660x06030145

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The necessity of accurately taking into account the existence of nonequivalent operator representations associated with canonical transformations is discussed. It is demonstrated that Bose systems in the presence of the Bose-Einstein condensate and without it correspond to different Fock spaces, orthogonal to each other. A composite representation for the field operators is constructed allowing for a self-consistent description of Bose-condensed systems. Equations of motion are derived from the given Hamiltonian, which guarantees the validity of conservation laws and thermodynamic self-consistency. At the same time, the particle spectrum obtained either from diagonalizing this Hamiltonian or from linearizing the field-operator equations of motion has no gap. The condition of the condensate existence assures the absence of the gap in the spectrum, irrespectively to the approximation involved. The suggested self-consistent theory is both conserving and gapless.
Bose atoms in optical lattices are considered at low temperatures and weak interactions, when Bose-Einstein condensate is formed.A self-consistent approach, based on the use of a representative statistical ensemble, is employed, ensuring a gapless spectrum of collective excitations and the validity of conservation laws.In order to show that the approach is applicable to both weak and tight binding, the problem is treated in the Bloch as well as in the Wannier representations.Both these ways result in similar expressions that are compared for the self-consistent Hartree-Fock-Bogolubov approximation.A convenient general formula for the superfluid fraction of atoms in an optical lattice is derived.
The existence of a Bose-Einstein condensation in an interacting many-boson system at T = 0°K is proved under certain conditions on the particle density and the interparticle potential. Starting with the tentative assumption that the condensation exists, we study the fluctuation in the occupation number of the condensate with due regard to its interactions (1) with particles outside the condensate as well as (2) with the fluctuation itself. If the condensate fluctuation has a normalizable ground state, then the assumed existence of the condensation is tenable. For the case of the pair-Hamiltonian model satisfying the conditions for condensation, the interactions of the second category are of no importance. In the limit of infinite volume, this Hamiltonian can be diagonalized in an irreducible representation of a Bose-field operatorφ(x), whereφ(x) has nonvanishing ground state expectation value, in accordance with the usual c-number replacement of creation and destruction operators for the condensate particles. The full Hamiltonian for a system of pairwise interacting bosons is studied only in a low-density limit. Bose-Einstein condensation exists when the over-all space integral of the interparticle potential is positive. In this case the interactions of the second category play an important role in ensuring a normalizable ground state for the condensate fluctuation. There is an indication that in the limit of infinite volume the Hamiltonian cannot be diagonalized in any irreducible representation of the field operatorφ. Yet the c-number replacement of the condensate operators is legitimate as far as states of particles outside the condensate are concerned. Some speculations are made as to what may happen for systems of moderate density.
Abstract The Hohenberg–Martin dilemma of conserving versus gapless theories for systems with Bose–Einstein condensate is considered. This dilemma states that, generally, a theory characterizing a system with broken global gauge symmetry, which is necessary for Bose–Einstein condensation, is either conserving, but has a gap in its spectrum, or is gapless, but does not obey conservation laws. In other words, such a system either displays a gapless spectrum, which is necessary for condensate existence, but is not conserving, which implies that it corresponds to an unstable system, or it respects conservation laws, describing a stable system, but the spectrum acquires a gap, which means that the condensate cannot appear. An approach is described, resolving this dilemma, and it is shown to give good quantitative agreement with experimental data. Calculations are accomplished in the Hartree–Fock–Bogolubov approximation.
We present a self-consistent ttt-matrix theory for Bose-Einstein-condensed systems within the Hartree-Fock-Bogoliubov (HFB) approximation. Using the Lippmann-Schwinger equation for a ttt matrix describing the collision between two particles via an interparticle potential, we derive a set of equations for the normal and anomalous self-energies in the HFB approximation expressed in terms of the ttt matrix. These equations are solved for a hard-sphere potential. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low densities and to the Brueckner-Sawada theory at high densities. The spectrum is gapless and linear in small momentum, but does not have any roton minimum in the large-momentum region even for high densities such as those of 4{}^{4}4He.
Here, it was shown that the chain of equations, which is similar to the Bogoliubov chain in the theory of classical gases, may be obtained for the quasi-averages of the products of field operators in the system of Bose particles. The closed system of dynamic equations, which accounts for both the one-particle condensate and pair correlations, was obtained at zero temperature for the case where it is sufficient to confine ourselves to taking into account the quasi-averages of one field operator and the products of two field operators. The spectrum of excitations in a spatially homogeneous state was studied, and it was shown that this spectrum has two branches: the sound branch and the branch with an energy gap at zero momentum. The possibility of the existence of quasiparticle excitations with an energy gap in superfluid helium was discussed in connection with experiments on the absorption of microwave radiation.