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Physical Review A

Systematic Approach to the Bose Liquid

Moshe Schwartz

Published Jul 1, 1970• 8 citations
Publisher

Abstract

An approximation scheme for a Bose liquid is presented, based on small fluctuations in density and currents. The method of obtaining the elementary excitations and their interactions in any order of the approximation is outlined. Also it is shown that the first-order calculations are in agreement with the calculations of Feynman, Bogoliubov, and Pitayevski. Second-order calculations agree with the improved results of Feynman and Cohen.

DOI
10.1103/physreva.2.230

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AIP conference proceedings•Jan 1, 2006•0 citations
An approximation scheme for a Bose liquid is presented, based on small fluctuations in density and currents. The method of obtaining the elementary excitations and their interactions in any order of the approximation is outlined. Also it is shown that the first-order calculations are in agreement with the calculations of Feynman, Bogoliubov, and Pitayevski. Second-order calculations agree with the improved results of Feynman and Cohen.
An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.
We present an extension of the well-known Bogoliubov theory to treat low-dimensional degenerate Bose gases in the limit of weak interactions and low density fluctuations. We use a density-phase representation and show that a precise definition of the phase operator requires a space discretization in cells of size l. We perform a systematic expansion of the Hamiltonian in terms of two small parameters, the relative density fluctuations inside a cell and the phase change over a cell. The resulting macroscopic observables can be computed in one, two, and three dimensions with no ultraviolet or infrared divergence. Furthermore, this approach exactly matches Bogoliubov's approach when there is a true condensate. We give the resulting expressions for the equation of state of the gas, the ground state energy, and the first order and second order correlation functions of the field. Explicit calculations are done for homogeneous systems.
The quantum field Hamiltonian expressed in terms of density and current density variables has been employed together with the equal-time commutation relations among these variables to find the ground state energy and the density fluctuation excitation spectrum of a system of interacting bosons at T = 0°K. The approximation involved consists in assuming that the density fluctuation in space is small compared with the average density. The results easily obtained in the lowest-order approximation agree with those of Bogoliubov. However, in our treatment no condensation of particles in zero-momentum state is assumed or apparent. A connection between the present treatment and the quantum hydrodynamic approach to the irrotational flow of a Bose liquid has been made.
The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in rs34{{r}_{s}}^{\frac{3}{4}}rs​43​. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.
This chapter concludes the study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. A clear understanding of the static and the long wavelength dynamic properties of an equilibrium Bose liquid, including a microscopic basis for the two-fluid model. Variational calculations of the excitation spectrum of liquid He II, which are in good quantitative agreement with experiment. A satisfactory phenomenological treatment of the interactions between elementary excitations which give rise to various irreversible phenomena. A microscopic theory of the ground state and elementary excitation spectrum of liquid He II; in fact, even the theory of the simple dilute Bose gas has not yet been extended to treat the system at finite temperatures, and under non-equilibrium situations. A clear understanding of a number of "vortex" properties, including the critical current for superfluid flow in thin capillaries.
Until now, most of what has been said about excitations in quantum liquids has concerned the dynamic structure function, which is observable by means of neutron scattering. The dynamic structure function can be calculated using standard linear response theory. However, at this level one needs only transition densities 〈0|ρ̂|n〉 or transition currents 〈0|ĵ|n〉, which are oscillatory in time and hence do not describe mass transport. In this work we go a step further and study transport currents in excited states, 〈n|ĵ|n〉, which requires the calculation of 〈n|ĵ|n〉 to second order. For that purpose, we take a well‐tested microscopic theory of inhomogeneous quantum liquids and extend it to find the currents formed when helium atoms scatter off a helium slab or when excitations evaporate atoms (a setup experimented by A. F. G. Wyatt’s group in Exeter). Current conservation was already a major theoretical problem encountered by R. Feynman and led him to introduce backflow corrections. We show that perfect current conservation is expected only for exact solutions of the time‐dependent many‐body Schrödinger equation. This is the first extensive theoretical study of transport phenomena in a quantum liquid based on an accurate microscopic theory.