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Journal of Mathematical Physics

Further Generalized Bose Condensation, Anisotropic ODLRO and Thin 4He Films

Gideon Carmi

Published Jan 1, 1968• 4 citations
Publisher

Abstract

This paper contains a detailed analysis of the Bose condensation of an ideal Bose gas, finite in one dimension and infinite in the other two. Limiting processes, which were treated questionably in previous work, are here re-examined, and some seemingly paradoxical conclusions of the past work are explained. Two new concepts emerge—those of ``further generalized Bose condensation'' (FGBC) and ``anisotropic off-diagonal long-range order'' (AODLRO). The two concepts are shown to be connected to each other, and both are shown to exist at T = 0 for the model discussed.

DOI
10.1063/1.1664472

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•Jan 1, 2004•0 citations
This paper contains a detailed analysis of the Bose condensation of an ideal Bose gas, finite in one dimension and infinite in the other two. Limiting processes, which were treated questionably in previous work, are here re-examined, and some seemingly paradoxical conclusions of the past work are explained. Two new concepts emerge—those of further generalized Bose condensation'' (FGBC) and anisotropic off-diagonal long-range order'' (AODLRO). The two concepts are shown to be connected to each other, and both are shown to exist at T = 0 for the model discussed.
It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off-diagonal long-range order (ODLRO) in the single-particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single-particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box-enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box-enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second-quantizes with respect to Hartree-Fock orbitals rather than free-particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two-particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.
We analyze the concept of generalized Bose–Einstein condensation (g-BEC), known since 1982 for the perfect Bose gas (PBG) in Casimir (or anisotropic) boxes. Our aim is to establish a relation between this phenomenon and two concepts: the concept of long cycles and the off-diagonal-long-range-order (ODLRO), which are usually considered as an adequate way to describe standard BEC on the ground state for cubic boxes. First, we show that these three criteria are equivalent in this latter case. Then, based on a scaling approach, we revise the formulation of these concepts to prove that the classification of g-BEC into three types I, II, III, implies a hierarchy of long cycles (depending on their size scale) as well as a hierarchy of ODLRO which depends on the coherence length of the condensate.
Abstract In this chapter we outline the microscopic theory of an interacting Bose fluid. The aim is to provide a basis for a microscopic understanding of the excitations in superfluid and normal 4He discussed in the following chapters. The microscopic picture can be omitted on a first reading and referred to when desired.We assume that the reader is familiar with the theory of Bose-Einstein condensation in a Bose gas introduced in §7.2 and presented in many texts on statistical mechanics. In §7.2 we also discussed the generalization of Bose condensation to an interacting fluid by Penrose and Onsager (1956).Condensation was identified with macroscopic occupation of one of the eigenstates of the fluid and the appearance of an infinitely long tail in the one-body density matrix, denoted as off-diagonal long range order (ODLRO). For a homogeneous fluid it is natural to take momentum states as the eigenstates and to assume that the macroscopically occupied state is the state with momentum p = 0. In this case the criterion reduces to that for a gas but with a firm foundation.
The Bogoliubov Model of the Weakly Nonideal Bose Gas In this chapter we review the basic theory of Bose-Einstein condensation (BEC) of excitons. To start, this means we must review the basic theory of BEC of any kind of particle, the theory known as the Bogoliubov model, after the foundational contributions made by N. N. Bogoliubov. This model introduces all the strange things associated with BEC: spontaneous symmetry breaking, off-diagonal long-range order (ODLRO), macroscopic occupation of a single quantum state, etc. It is often said that physicists who spend years studying quantum mechanics eventually warp their intuition so much that things like ODLRO seem normal. It is well worth stepping back every now and then to think about just how strange a Bose condensate is. First, consider the idea of spontaneous symmetry breaking. Many systems exist in which the underlying physics, expressed in the Hamiltonian, do not favor one state over another. The a priori probability of occupation of the different states by a particle is equal, i.e., symmetric. Nevertheless, in some cases, thermodynamics requires that a macroscopic number of particles must somehow “choose” one of the states preferentially. If they did not, the system would not be in equilibrium, i.e, could not have a definable temperature. If the particles preferentially choose one state, then the underlying symmetry has been broken. In the context of magnets, the breaking of the symmetry of the magnet (to have all its constituent dipoles lined up one way and not another) is energetically favored, since there are terms in the Hamiltonian that give lower energy for aligned dipoles, e.g., μS 1 · S 2 .
It is shown that the lack of condensation in partially finite geometries is not dependent on the use of periodic boundary conditions or the symmetry-breaking technique for a class of independent-particle Bose-gas models. Condensation in an ideal Bose gas in a finite geometry is studied for one-, two-, and three-dimensional systems. Explicit expressions for the chemical potential in terms of the number of particles, the temperature, and the dimensions of the system are obtained by using a contour-integration technique to express sums as integrals plus corrections. Even though there is no true condensation, the range of the reduced density matrix for an ideal Bose gas in a thick slab (${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}\ifmmode\times\else\texttimes\fi{}{L}_{3}$ with L1{L}_{1}L1​, ${L}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\ensuremath{\infty}>{L}_{3}\ensuremath{\gg}{\ensuremath{\lambda}}_{T}=\mathrm{the}\mathrm{thermal}\mathrm{wavelength}$) is found to go from order $\frac{{\ensuremath{\lambda}}_{T}{T}_{\mathrm{cB}}}{(T\ensuremath{-}{T}_{\mathrm{cB}})}$ to order ${L}_{3}\mathrm{exp}{\frac{1}{2}\sqrt{\ensuremath{\pi}}{\ensuremath{\zeta}}_{\frac{3}{2}}(\frac{{L}_{3}}{{\ensuremath{\lambda}}_{T}})[{(\frac{{T}_{\mathrm{cB}}}{T})}^{\frac{3}{2}}\ensuremath{-}1]}$ as the temperature is lowered through the bulk critical temperature ${T}_{\mathrm{cB}}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\zeta}}_{\ensuremath{\chi}} \mathrm{is}\mathrm{the}\mathrm{Riemann} \ensuremath{\zeta} \mathrm{function}.)$ The range exhibits a spatial asymmetry. The onset of this extremely long-range order is found to be reflected in PVT diagrams and the specific heat in a manner very similar to that in bulk condensation. It is conjectured that in calculations for infinite systems the existence of a long- but finite-range order may be a more relevant criterion for superfluidity and superconductivity than the usual criterion of an infinitely long-range order.
We present a rigorous study of the perfect Bose-gas in the presence of a homogeneous ergodic random potential. It is demonstrated that the Lifshitz tail behaviour of the one-particle spectrum reduces the critical dimensionality of the (generalized) Bose–Einstein Condensation (BEC) to d = 1. To tackle the Off-Diagonal Long-Range Order (ODLRO) we introduce the space average one-body reduced density matrix. For a one-dimensional Poisson-type random potential we prove that randomness enhances the exponential decay of this matrix in domain free of the BEC. To cite this article: O. Lenoble et al., C. R. Physique 5 (2004).