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Progress of Theoretical Physics

A Note on the Pair Theory of a Degenerate System of Bosons

Shuichiro Yamasaki

Published May 1, 1984• 0 citations
Publisher

Abstract

A degenerate system of interacting bosons is analyzed in terms of the quasi-particle operators defined by the Bogoliubov transformation and their canonical products introduced in a previous paper. The analysis leads to a version of the pair theory. It is shown that, contrary to Luban's conclusion, the entropy suffers no discontinuous change when the number of zero-momentum particles decreases to zero and the system becomes non-degenerate.

DOI
10.1143/ptp.71.1083

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A degenerate system of interacting bosons is analyzed in terms of the quasi-particle operators defined by the Bogoliubov transformation and their canonical products introduced in a previous paper. The analysis leads to a version of the pair theory. It is shown that, contrary to Luban's conclusion, the entropy suffers no discontinuous change when the number of zero-momentum particles decreases to zero and the system becomes non-degenerate.
Creation and annihilation operators of quasi-particles are introduced into the description of a finite temperature system of degenerate bosons with the aid of a canonical transformation which generalizes the one due to Bogoliubov. A variational calculation taking into account the interaction between non-zero momentum particles shows that the quasi-particles are phonons in the long wave-length limit. The state of the system, which is approximated by the grand canonical ensemble of the quasi-particles, approaches the Hartree-Fock state when the condensate is depleted completely at the critical temperature.
A finite temperature system of degenerate bosons is studied with the aid of the extended Bogoliubov transformation which has been proposed in a preceding part I. The energies of quasi-particles and the vertex functions describing their interaction are determined in an approximation which includes the lowest order effects of the interaction between non-zero momentum bosons. The quasi-particle energy depends on the distribution of thermal excitations, and is of phonon type in the long wavelength limit. The phonon nature of the spectrum is preserved by the interaction between quasi-particles, because the vertex functions vanish in the long wavelength limit. In the limit as the condensate is depleted completely, the proposed formalism simplifies to the Hartree-Fock theory.
The infrared divergence of the interaction among Bogoliubov's quasi· particles in a degener­ ate boson system can be avoided by introducing improved quasi-particles. A formula is given which relates the single· particle operators ap directly to the improved quasi-particle operators bp ' which have been defined in terms of collective variables in a preceding paper. The operators ap are expressed in terms of bp as a series using N112 as a formal expansion parameter, where N is the number of particles in the system. Terms up to order N312 are given together with a tenn of order N2 which is necessary for the calculation of the ground· state energy to order N2 • The fonnula makes the reason clear why the operators b p enable us to eliminate the ultraviolet divergence haunting collective theories including the quantum hydrodynamics.
The convolution approximations to the three• particle and four-particle structure functions are incorporated in a non-linear canonical transformation applicable to a system of degenerate bosons.The transformation relates the quasi• particle operators bk to the density fluctuations P k and their canonical conjugates 'P k by way of a formal series in powers of N-ll ', where N is the number of particles.Higher order terms in the series require corrections to the convolution approximations.The next order terms are calculated to improve the approximations to the structure functions.The present formalism provides an unambiguous way of evaluating the residual interaction not taken account of in the trial state, the b•vacuum, which admits the convolution approximation or its extension. § 1. Introduction
For strongly pair-correlated systems, a new transformation is derived similar in form to the usual Bogoliubov-Valatin transformation, but with the transformation coefficients replaced by operators. The original fermion operators are now replaced by two sets of independent operators, the pair operators and the quasi-fermions . The transformation conserves the particle number. Among other applications, it is particularly suited for the construction of a microscopic theory of the type of the interacting boson model. The new transformation given here also serves to provide a connection between the usual ph picture of a phonon and the pp picture-of bosons in IBM. However, the boson picture that evolves from the present formulation differs from the ideal boson picture in IBM..