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Physical Review A

Quantum Field Theory for Bosons with Condensed Phase at a Finite Temperature

Robert Mills

Published Jul 1, 1971• 3 citations
Publisher

Abstract

A detailed proof is given for the validity of substituting a $c$ number for the zero-momentum-state operators in the quantum-field-theory approach to the many-boson problem at a finite temperature. We use a modification of the method of Kromminga and Bolsterli, and add slightly to their work for the ground-state case. The case in which the bosons which condense are themselves composite particles, as in superconductivity, is treated in this method by the artifice of coupling to a fictitious elementary boson and then letting the coupling become infinitely weak; the self-consistent formulation of Schrieffer for superconductivity can be obtained in this way.

DOI
10.1103/physreva.4.394

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A detailed proof is given for the validity of substituting a ccc number for the zero-momentum-state operators in the quantum-field-theory approach to the many-boson problem at a finite temperature. We use a modification of the method of Kromminga and Bolsterli, and add slightly to their work for the ground-state case. The case in which the bosons which condense are themselves composite particles, as in superconductivity, is treated in this method by the artifice of coupling to a fictitious elementary boson and then letting the coupling become infinitely weak; the self-consistent formulation of Schrieffer for superconductivity can be obtained in this way.
A self consistent field approach is developed for truncating the infinite set of equations obtained in apply­ ing the equation of motion method to interacting systems. The operator which is assumed to create elementary excita­ tions in the system is expanded in terms of simpler opera­ tors and both the coefficients in the expansion and relations independent of the coefficients are found. The method is applied to a system of Bosons at low temperatures in the approximation that the interaction is both tempera­ ture and density independent. Corrections to the pairing approximation are obtained due both to the inclusion of more terms in the Hamiltonian and the inclusion of collective excitations. It is shown that in the long wavelength limit these corrections can be included by renormalizing both the single particle and pairing energies. The renormalization can be carried out by adding a term to the single particle energy and an extra interaction to the pairing part of the Hamiltonian. This interaction is attractive over part of momentum space and obeys the criterion for a simple Bose condensation. It is shown that no single particle like zero momentum excitation exists. A collective excitation is con­ sidered for the zero momentum excitation and it is found v vi that its energy must be non-zero if it exists at all. The application to superfluid He11 is discussed.
We present a method of treating the assembly of interacting bosons under Bose-Einstein condensation. Without applying the Bogoliubov approximation in which the creation and the annihilation operators of zero-momentum particles are replaced by a c number, we keep the quantum nature of these operators—thus the title, ``Quantum Mechanics.'' The method is a quantum mechanical adaptation of the theory of small oscillation. The oscillation means the fluctuation of the number of condensed particles. The interaction between particles determines the stability of this oscillation. When it is stable and its amplitude is not macroscopic, the Bogoliubov approximation is valid. In this way, our method provides a validity criterion for the Bogoliubov approximation as well as an estimation of the errors thereby committed. We have to note that the excitations associated with the fluctuation of condensed particles can never be obtained within that approximation. Our method is applied to the Huang model, the assembly of bosons interacting through a hard core plus weak attractive potential. Having found that, within the physically accessible range of the particle density, the above-mentioned oscillation is stable, we can conclude that Huang's treatment is well founded. We have discussed the mathematical background of our approximation by invoking the representation theory of canonical variables of an infinitely large system.
We have developed a microscopic theory of superfluidity in a two-dimensional, slightly nonideal Bose gas. The theory is rooted in the fact that a number of bosons are concentrated in the small momentum region at low temperatures when the interaction is weak, in spite of the fact that there is no Bose condensate in the two-dimensional case when T≠0. This fact lets us reduce the description of the system from one employing creation ak+ and annihilation ak operators for bosons to one based on combinations of these operators ρk and φk(ρk = S−1Σqaq+aq+k, φk· = −(i/ns) Σ(kq/k2 + l)aq+aq+k, where n is the boson density, and S is the area of the system). The description in terms of ρk and φk is simple because the hamiltonian is quadratic with respect to ρk and φk. The operators ρk and φk can also be used to express the occupation number Nk = ak+ak and current operators. We find the elementary excitation spectrum (which is identical to the Bogolyubov spectrum) and the momentum distribution of the bosons. We calculate the normal density due to thermal oscillations in the Bose gas and to interactions with impurity centers in the substrate.
We examine in detail the microscopic theories of Beliaev and Hugenholtz and Pines as applied to an interacting system of bosons in its ground state and demonstrate that these theories are strictly applicable to a zero density gas only. After providing a re-normalization of the time dependent perturbation theory of a many fermion system in its ground state, we reformulate the corresponding boson theory without ab initio approximations concerning the k=0 mode.The resulting boson perturbation theory has a diagrammatic structure which is topologically identical with the corresponding fermion theory and provided the implicit assumptions concerning convergence are valid, we conclude that a finite fraction of the particles in an arbitrary interacting Bose gas at T=ok will not condense into the k=0 mode.
We formulate a bosonic dynamical mean-field theory (B-DMFT) which provides a comprehensive, thermodynamically consistent framework for the theoretical investigation of correlated lattice bosons. The B-DMFT is applicable for arbitrary values of the coupling parameters and temperature and becomes exact in the limit of high spatial dimensions ddd or coordination number ZZZ of the lattice. In contrast to its fermionic counterpart, the construction of the B-DMFT requires different scalings of the hopping amplitudes with ZZZ depending on whether the bosons are in their normal state or in the Bose-Einstein condensate (BEC). A detailed discussion of how this conceptual problem can be overcome by performing the scaling in the action rather than in the Hamiltonian itself is presented. The B-DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. It reproduces all previously investigated limits in parameter space, such as the Beliaev-Popov and Hartree-Fock-Bogoliubov approximations, and generalizes the existing mean-field theories of interacting bosons. The self-consistency equations of the B-DMFT are those of a bosonic single-impurity coupled to two reservoirs corresponding to bosons in the condensate and in the normal state, respectively. We employ the B-DMFT to solve a model of itinerant and localized, interacting bosons analytically. The local correlations are found to enhance the condensate density and the BEC transition temperature TBEC{T}_{\text{BEC}}TBEC​. This effect may be used experimentally to increase TBEC{T}_{\text{BEC}}TBEC​ of bosonic atoms in optical lattices.