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Physical Review Letters

Impossibility of Bose Condensation or Superconductivity in Partially Finite Geometries

David A. Krueger

Published Sep 4, 1967• 24 citations
Publisher

Abstract

The vanishing of the quasiaverages usually associated with superfluidity and superconductivity is shown for arbitrarily interacting Bose and Fermi systems which are confined to a geometry with one or more dimensions finite while one or more dimensions extend to infinity. However, it is suggested that these partially finite geometries are anomalous and are not good approximations to thin film and pore geometries found in the laboratory. The conditions on the box size which are necessary and sufficient for condensation to occur in an ideal Bose gas are also given.

DOI
10.1103/physrevlett.19.563

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The vanishing of the quasiaverages usually associated with superfluidity and superconductivity is shown for arbitrarily interacting Bose and Fermi systems which are confined to a geometry with one or more dimensions finite while one or more dimensions extend to infinity. However, it is suggested that these partially finite geometries are anomalous and are not good approximations to thin film and pore geometries found in the laboratory. The conditions on the box size which are necessary and sufficient for condensation to occur in an ideal Bose gas are also given.
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.
It is suggested, on the basis of soluble models, that generalized Bose condensation results from a broken symmetry associated with a nonvanishing pairing amplitude with very large low-momentum components, leading to a nonzero ``pairing density'' pc as well as a nonzero generalized condensate density ρc. For systems of interacting bosons it is proved that (1) nonzero pc implies nonzero ρc, and (2) pc = 0 in one or two dimensions and, more generally, for geometries finite in one dimension and infinite in the other two (films) or finite in two dimensions and infinite in one (pores). It is pointed out that this does not exclude superfluidity in such geometries, but does show the need of a new mechanism to explain it.
Fluctuations of the number of condensed atoms in a finite-size, weakly interacting Bose gas confined in a box potential are investigated for temperatures up to the critical region. The canonical partition functions are evaluated using a recursive scheme for smaller systems, and a saddle-point approximation for larger samples that allows to treat realistic size systems containing up to N ∼10 5 particles. We point out the importance of particle-number constraint and interactions between out of condensate atoms for the statistics near the critical region. For sufficiently large systems, the crossover from the anomalous to normal scaling of the fluctuations is observed. The excitations are described in a self-consistent way within the Bogoliubov-Popov approximation, and the interactions between thermal atoms are described by means of the Hartree-Fock method.
Taking into account the possibility of particle-antiparticle pair production, we have investigated the onset of Bose-Einstein condensation in an ideal relativistic Bose gas confined to restricted geometries. Through an extensive use of the Poisson summation formula, we have carried out an explicit evaluation of the summations over states appearing in the problem, which enables us to make a rigorous analysis of the temperature dependence of the thermogeometric parameter yyy of the system in the case of a cubical enclosure under periodic boundary conditions. This, in turn, leads us to determine the growth of the condensate fraction $\frac{{\ensuremath{\rho}}_{0}}{\ensuremath{\rho}}$ as a smooth function of temperature from $T\ensuremath{\gtrsim}{T}_{c}$ down to T=0T=0T=0 K. Finite-size corrections to the standard bulk results are obtained in explicit terms and are shown to be in complete agreement with the Fisher-Barber scaling theory for such effects. In the end, special geometries, such as narrow channels and thin films, are also examined.
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.