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Physical review. B, Solid state

Decay of Order in Isotropic Systems of Restricted Dimensionality. I. Bose Superfluids

David Jasnow, Michael Ellis Fisher

Published Feb 1, 1971• 35 citations
Publisher

Abstract

The off-diagonal ordering of one- and two-dimensional Bose particle systems of finite thickness and cross-section is considered in the presence and absence of a symmetry-breaking field. It is shown rigorously, by applying Bogoliubov's inequality to a subdomain of the system, that no spontaneous ordering can occur; several definitions of long-range order are discussed. Explicit bounds on the order-order correlation function, integrated over a subdomain, are obtained which indicate how the short-range order decays with distance. Conditions under which information on the pointwise behavior of the correlation function can be inferred are also discussed. The inequalities are assessed numerically for real situations.

DOI
10.1103/physrevb.3.895

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The off-diagonal ordering of one- and two-dimensional Bose particle systems of finite thickness and cross-section is considered in the presence and absence of a symmetry-breaking field. It is shown rigorously, by applying Bogoliubov's inequality to a subdomain of the system, that no spontaneous ordering can occur; several definitions of long-range order are discussed. Explicit bounds on the order-order correlation function, integrated over a subdomain, are obtained which indicate how the short-range order decays with distance. Conditions under which information on the pointwise behavior of the correlation function can be inferred are also discussed. The inequalities are assessed numerically for real situations.
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At any finite temperature, the correlation function $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\mathcal{r})\ensuremath{\psi}(0)〉$ of a two-dimensional Bose system is known to approach zero as the separation r\mathcal{r}r approaches infinity. This paper is devoted to supporting the point of view that superfluid phenomena may still persist in such a system. To describe the low-temperature behavior of a condensed two-dimensional Bose system, we use a coherent-state representation of the low-lying phonon excitations. The expectation of the particle field operator $〈\ensuremath{\psi}(\mathcal{r})〉$ in a coherent phonon state of the system is shown to be a well-defined function. This function has only small fluctuations in its magnitude and, therefore, it is possible to make the usual arguments for superfluid behavior based on the existence of the condensate wave function. The large phase fluctuations of the function, when the coherentphonon states are collected into the thermal ensemble, give rise to the spatial decay of the correlation function or reduced density matrix $〈{\ensuremath{\psi}}^{+}(\mathcal{r})\ensuremath{\psi}(0)〉$. It is concluded that two-dimensional Bose systems are superfluid at sufficiently low temperatures, even in the inifinite-volume limit. An analogous argument would indicate the possibility of superconductivity in a two-dimensional Fermi system.
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