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Journal of the Physical Society of Japan

Theory of Superfluid Ground State. II

Masakazu Ichiyanagi, Masato Ooya

Published Dec 1, 1976• 1 citation
Publisher

Abstract

A theory is formulated for the ground state of a degenerated Bose system at zero temperature. By making use of the self-consistent field method, we determine the physical excitation of the system. The physical fields are obtained from the particle fields by the unitary transformation. The condensate wavefunction is determined as the expectation value of the slowly-varying field in the superfluid ground state, which is the vacuum state of the rapidly-varying (physical) field operator.

DOI
10.1143/jpsj.41.1870

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A theory is formulated for the ground state of a degenerated Bose system at zero temperature. By making use of the self-consistent field method, we determine the physical excitation of the system. The physical fields are obtained from the particle fields by the unitary transformation. The condensate wavefunction is determined as the expectation value of the slowly-varying field in the superfluid ground state, which is the vacuum state of the rapidly-varying (physical) field operator.
A theory is formulated for the ground state of a degenerate Bose fluid in thermal equilibrium. The fluctuations in the condensate are treated in terms of the fluctuation operators C = a 0 -α, and thir conjugate, α 0 being the annihilation operator for the zero momentum state and the order parameter α the quasiaverage of α 0 . The superfluid ground state is viewed as energy lowering by coherent excitations. The coherent state is defined in terms of the normal modes in the superfluid phase. The superfluid ground state is not the ground state of the type studied by Glassgold and Sauermann.
The theory of the condensate of a weakly interacting Bose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self-consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self-consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a boundary. This condition fully determines the dynamics of the system. These points are illustrated by considering the motion of a foreign ion in a Bose gas, a rotating container of fluid, and the Landau criterion for superfluidity.
The unperturbed Hamiltonian for the Bose-Einstein condensate, which includes not only the first and second powers of the zero mode operators but also the higher ones, is proposed to determine a unique and stationary vacuum at zero temperature. From the standpoint of quantum field theory, it is done in a consistent manner that the canonical commutation relation of the field operator is kept. In this formulation, the condensate phase does not diffuse and is robust against the quantum fluctuation of the zero mode. The standard deviation for the phase operator depends on the condensed atom number with the exponent of $\ensuremath{-}1/3$, which is universal for both homogeneous and inhomogeneous systems.
We conclude our study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. The cornerstone of our fundamental understanding of superfluidity has been the recognition of the central importance of the condensate, a single quantum state which is macroscopically occupied. For the Bose liquid one has a particle condensate, characterized by a non-vanishing expectation value of ψ (r). The existence of the condensate explains all major aspects of superfluid behavior; in particular, we now possess: 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. A precise microscopic theory at T = 0 for a specific model, the dilute Bose gas. Variational calculations of the excitation spectrum of liquid He II (and of the ground state properties as well), 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 qualitative understanding of the role which is played by quantized vortices in both equilibrium and non-equilibrium phenomena.
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.
This chapter considers the factors arising in the zero-temperature frequency representation for the ground state energy. The physical essence of the result, the relation between the symmetry of the non-interacting zero-temperature ground state and the exact ground state. The expectation value of an arbitrary product of operators in the Bose ground state cannot be separated into independent products of contractions and there is no zero-temperature Wick's theorem as in the Fermion case. The point at which our previous derivations for Fermions cannot be generalized to Bosons is the construction of a canonical transformation analogous to such that all the new annihilation operators annihilate the ground state. The Fermion minus signs combined with the factor of N from the color sums do just what is required to generate the correct Bose factors. Zero temperature theory is applied to the dilute Bose gas and to Fermions in one dimension interacting via δ-function forces.