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Journal of Mathematical Physics

Broken Symmetry and Generalized Bose Condensation in Restricted Geometries

Marvin D. Girardeau

Published Jun 1, 1969• 6 citations
Publisher

Abstract

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.

DOI
10.1063/1.1664945

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Soviet Journal of Low Temperature Physics•Oct 1, 1983•1 citation
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.
Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode (k=0)(\mathbf{k}=0)(k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.
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.
The theory of ultracold, dilute Bose gases is the subject of intensive studies, driven by new experimental applications, which also motivate the study of Bose-Einstein condensation (BEC) in low dimensions. From the theoretical point of view there are few, quite special, models in which we are able to prove BEC for interacting bosons. With the aim of studying the condensation problem I considered a simplified model for a three and two dimensional system of bosons interacting with a repulsive short range potential, obtained by introducing an ultraviolet momentum cutoff. I proved that the interacting theory is well defined at all orders in terms of one (in three dimensions) or two (in two dimensions) effective parameters, related to the intensity of the interaction. Moreover the correlations do not exhibit anomalous dimensions, i.e. the model is in the same universality class of the exactly soluble Bogoliubov model. The result is obtained by using Rigorous Renormalization Group techniques and implementing local Ward identities (WI), which reduce the number of independent running couplings, this fact being crucial for the control of the two dimensional theory, where the effective interactions are relevant. Since the ultraviolet momentum cutoff breaks local gauge invariance, the corrections to the formal WI have been considered.
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.
We have developed a microscopic theory of superfluidity in a two-dimensional, slightly nonideal Bose gas. The theory is rooted in the fact that a number of bosons are concentrated in the small momentum region at low temperatures when the interaction is weak, in spite of the fact that there is no Bose condensate in the two-dimensional case when T≠0. This fact lets us reduce the description of the system from one employing creation ak+ and annihilation ak operators for bosons to one based on combinations of these operators ρk and φk(ρk = S−1Σqaq+aq+k, φk· = −(i/ns) Σ(kq/k2 + l)aq+aq+k, where n is the boson density, and S is the area of the system). The description in terms of ρk and φk is simple because the hamiltonian is quadratic with respect to ρk and φk. The operators ρk and φk can also be used to express the occupation number Nk = ak+ak and current operators. We find the elementary excitation spectrum (which is identical to the Bogolyubov spectrum) and the momentum distribution of the bosons. We calculate the normal density due to thermal oscillations in the Bose gas and to interactions with impurity centers in the substrate.