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Progress of Theoretical Physics

Scaling Laws and Critical Exponents of Condensed Bose Liquid

Shokichi Kanno

Published Feb 1, 1973• 0 citations
Publisher

Abstract

The relations among the critical exponents of condensed Bose liquid are obtained under the assumption of the scaling property for the mean occupation number of particles. Some of these relations are inequalities and may violate the scaling relations of Widom and Kadanoff. This violation is caused by two factors. One is the breakdown of the homogeneity of the one-particle density matrix and the other is nonvanishing condensation density contributing to the total density. The fluctuation of the condensation density is also discussed in connection with the scaling property.

DOI
10.1143/ptp.49.416

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The relations among the critical exponents of condensed Bose liquid are obtained under the assumption of the scaling property for the mean occupation number of particles. Some of these relations are inequalities and may violate the scaling relations of Widom and Kadanoff. This violation is caused by two factors. One is the breakdown of the homogeneity of the one-particle density matrix and the other is nonvanishing condensation density contributing to the total density. The fluctuation of the condensation density is also discussed in connection with the scaling property.
Progress of Theoretical Physics Vol. 48 No. 6 (1972) pp. 2210-2236 Phase Transition in Bose Liquid. I Shokichi Kanno Progress of Theoretical Physics Vol. 49 No. 2 (1973) pp. 401-415 Phase Transition in Bose Liquid. II Shokichi Kanno Progress of Theoretical Physics Vol. 49 No. 2 (1973) pp. 416-423 Scaling Laws and Critical Exponents of Condensed Bose Liquid Shokichi Kanno Progress of Theoretical Physics Vol. 84 No. 3 (1990) pp. 420-424 Condensate Fraction of a Charged Bose Gas at High Density Toshiyuki Nishiyama
In this work, we investigate the critical properties of the Bose-Einstein condensation transition in a model of non-interacting bosons whose one-particle hamiltonian in the tight-binding approximation includes long-range couplings in a linear chain with periodic boundary conditions. The couplings along the chain are described by a kinetic hopping term which is responsible by the quantum mobility of the particles and whose amplitude decays with the distance between the sites as a power-law with a characteristic exponent a. We show that the density of states in the vicinity of the lower band edge of the energy spectrum strongly depends on a. In the re- gime of 1 < α < 2, the density of states vanishes as a power-law, similar to the behavior found for bosons in hypercubic lattices with dimension d = 2/(α -1). We determine the critical properties of the Bose-Einstein condensation transition that takes place in this regime using a hypothesis of a single characteristic length scale and a finite-size scaling analysis. In the regime of 3/2 < α < 2, the critical exponents are quite similar to those that characterize the transition in the corresponding hypercubic lattice. On the other hand, in the regime of 1 < α < 3/2, the exponents obtained from the single length scale hypothesis are quite distinct form those on the hypercubic lattice. This result indicates that the single length scale hypothesis can not correctly capture the critical exponents in the regime where the effective dimensionality of the model is above the upper critical dimension above which the transition has a mean-field critical behavior.
Abstract Some properties of an ideal gas of massive bosons in a mean field potential and, confined between two infinite parallel slabs in a d-dimensional configuration space are investigated systematically. Here, one particle density of states approach is employed to study the critical temperature, shift of density, Casimir effects and critical exponents, starting from the evaluation of the grand canonical free energy in d-dimension. We have found that, the shift of density, Casimir force and the critical temperature depend on the space dimensionality. But the Casimir force decays as an inverse power law of the distance between two slabs in the condensate and, decays exponentially in the non-condensed state situated very close to the point of phase transition. Specifically, this study enabled us to predict the shift of the density of boson in the excited states due to mean field potential, and also the dimensional dependence of the critical exponent. The form of the critical exponent is found to be 1 d − 2 for the imperfect Bose gas. This leads to a value of critical exponent 1 for d = 3. Most interestingly, in the limit when the mean field potential goes to zero the resulting expressions for the above properties coincide with those of the ideal Bose gas.
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 show that the condensate occupation of a superfluid Bose liquid quite generally exhibits anomalously large fluctuations at finite temperatures $T\ensuremath{\ne}0.$ In three dimensions, the variance $〈\ensuremath{\delta}{N}_{0}^{2}〉$ of the number N0{N}_{0}N0​ of particles in the condensate scales nonlinearly with volume V like T2V4/3{T}^{2}{V}^{4/3}T2V4/3 at low T, generalizing the result obtained by Giorgini, Pitaevskii, and Stringari for a weakly interacting Bose gas. In two dimensions there is only a quasicondensate whose fluctuations are of the same order as the mean value.
We derive via diagrammatic perturbation theory the scaling behavior of the condensate and superfluid mass density of a dilute Bose gas just below the condensation temperature, T(c). Sufficiently below T(c) particle excitations are described by mean field (Bogoliubov). Near T(c), however, mean field fails, and the system undergoes a second order phase transition, rather than first order as predicted by Bogoliubov theory. Both condensation and superfluidity occur at the same T(c), and have similar scaling functions below T(c), but different finite size scaling at T(c) to leading order in the system size. A self-consistent two-loop calculation yields the condensate fraction critical exponent, 2beta approximately 0.66.