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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical nature of ideal Bose-Einstein condensation: Similarity with Yang-Lee theory of phase transition

Xianzhi Wang, Jai Sam Kim

Published Jan 1, 1999• 6 citations
Publisher

Abstract

It is found that the singularity of the thermodynamic potential of an ideal Bose gas is connected with the physical root ${z}_{0}=1$ of the inverse of the grand partition function. The critical nature is determined solely by the behavior of the root distribution of the inverse of the grand partition function near ${z}_{0}=1$. This is quite similar to the situation of phase transition described by Yang-Lee theory.

DOI
10.1103/physreve.59.1242

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It is found that the singularity of the thermodynamic potential of an ideal Bose gas is connected with the physical root z0=1{z}_{0}=1z0​=1 of the inverse of the grand partition function. The critical nature is determined solely by the behavior of the root distribution of the inverse of the grand partition function near z0=1{z}_{0}=1z0​=1. This is quite similar to the situation of phase transition described by Yang-Lee theory.
Ideal Bose-Einstein condensation (BEC) remains a paradigmatic example of a continuous phase transition and a cornerstone for understanding quantum degenerate bosonic matter. We demonstrate that critical behavior of the ideal Bose gas near the BEC phase transition falls into three distinct classes, determined exclusively by the low-energy scaling of the density of states. Depending on its scaling exponent, which is controlled by dimensionality and confinement, the transition displays either the usual algebraic divergences of thermodynamic susceptibilities, divergent behavior with marginal logarithmic corrections, or a more subtle form of criticality, where only the correlation length diverges. Our work provides a unified framework for criticality in noninteracting bosonic systems. This classification applies broadly to atomic, photonic, polaritonic, and magnonic condensates, where dimensionality, confinement, and spectral engineering can strongly reshape the density of states.
The analytical theory of Bose–Einstein condensation of an ideal gas in mesoscopic systems has been briefly reviewed in application to traps with arbitrary shapes and dimension. This theory describes the phases of the classical gas and the formed Bose–Einstein condensate, as well as the entire vicinity of the phase transition point. The statistics and thermodynamics of Bose–Einstein condensation have been studied in detail, including their self-similar structure in the critical region, transition to the thermodynamic limit, effect of boundary conditions on the properties of a system, and nonequivalence of the description of Bose–Einstein condensation in different statistical ensembles. The complete classification of universality classes of Bose–Einstein condensation has been given.
Gas-liquid condensation (GLC) in the attractive uniform Bose gas is studied by applying the idea of the Yang-Lee zeros to its grand partition function. The followings are proved: (i) When the temperature decreases and the density increases, the GLC occurs prior to the Bose-Einstein condensation (BEC), (ii) an explosive growth of the Bose-statistical coherence to a macroscopic scale occurs simultaneously with the GLC, which are triggered by bosons with a zero momentum, and (iii) the GLC in the Bose gas is not only a condensation in coordinate space, but also in momentum space. On the basis of these results, a comparative study of the BEC and the GLC is developed. Further, we discuss its implication to the trapped atomic gas.