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Physical review. B, Solid state

Condensation of the Rotating Two-Dimensional Ideal Bose Gas

John J. Rehr, N. David Mermin

Published Apr 1, 1970• 32 citations

Abstract

The two-dimensional ideal Bose gas in a circular container rotating at fixed rim velocity has some unexpected and instructive properties. In the thermodynamic limit, the system undergoes a phase transition, as argued by Widom. We find that the transition is a type of Bose-Einstein condensation, in which, however, infinitely many one-particle states are macroscopically occupied. The system evades general theorems forbidding Bose-Einstein condensation in two dimensions, not merely because it is inhomogeneous, as Widom suggests, but because the equilibrium density is unbounded near the circumference in the thermodynamic limit. We point out that, independently of the particular model, Bose-Einstein condensation into any single-particle state (or states), spatially uniform or nonuniform, is forbidden in one and two dimensions provided only that (a) the Hamiltonian is real, and (b) the equilibrium density is everywhere bounded in the thermodynamic limit.

DOI: 10.1103/physrevb.1.3160

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The two-dimensional ideal Bose gas in a circular container rotating at fixed rim velocity has some unexpected and instructive properties. In the thermodynamic limit, the system undergoes a phase transition, as argued by Widom. We find that the transition is a type of Bose-Einstein condensation, in which, however, infinitely many one-particle states are macroscopically occupied. The system evades general theorems forbidding Bose-Einstein condensation in two dimensions, not merely because it is inhomogeneous, as Widom suggests, but because the equilibrium density is unbounded near the circumference in the thermodynamic limit. We point out that, independently of the particular model, Bose-Einstein condensation into any single-particle state (or states), spatially uniform or nonuniform, is forbidden in one and two dimensions provided only that (a) the Hamiltonian is real, and (b) the equilibrium density is everywhere bounded in the thermodynamic limit.

A recently proved theorem of statistical physics states that a one- or two-dimensional Bose liquid in homogeneous thermodynamic equilibrium cannot undergo Bose-Einstein condensation. The theorem does not apply to inhomogeneous liquids. This is shown by considering the ideal Bose liquid (i) in the presence of a gravitational field and (ii) in rotational motion. The existence of well-defined one- and two-dimensional phase transitions with Bose-Einstein condensation is proved by these explicit examples.

Evading the Mermin–Wagner–Hohenberg no-go theorem and revisiting with rigor the ideal Bose gas confined in a square box, we explore a discrete phase transition in two spatial dimensions. Through both analytic and numerical methods, we verify that thermodynamic instability emerges if the number of particles is sufficiently yet finitely large: specifically, N ⩾ 35131 . The instability implies that the isobar of the gas zigzags on the temperature–volume plane, featuring supercooling and superheating phenomena. The Bose–Einstein condensation can then persist from absolute zero to the superheating temperature. Without necessarily taking the large N limit, under a constant pressure condition, the condensation takes place discretely both in the momentum and in the position spaces. Our result is applicable to a harmonic trap. We assert that experimentally observed Bose–Einstein condensations of harmonically trapped atomic gases are a first-order phase transition that involves a discrete change of the density at the center of the trap.

Bose-Einstein condensation (BEC) of an ideal gas is investigated for a finite number of particles. In three dimensions, we find a transition temperature which is lower than in the thermodynamic limit. Lowering the dimension increases the transition temperature and is therefore favorable for BEC. This is in contrast to the standard result obtained in the thermodynamic limit which states that BEC is not possible in, e.g., a one-dimensional (1D) harmonic potential. As a result, 1D atom traps, such as radially tightly confining magnetic traps or optical dipole traps, are promising for studying BEC. \textcopyright{} 1996 The American Physical Society.

We study the Bose-Einstein condensation (BEC) of a free Bose gas under rigid rotation. The aim is to explore the impact of rotation on the thermodynamic quantities associated with BEC, including the Bose-Einstein (BE) transition temperature and condensate fraction. We begin by introducing the rotation in the Lagrangian density of free charged Klein-Gordon fields and determine the corresponding grand canonical partition function at finite temperature, chemical potential, and finite angular velocity. Assuming slow rotation, we derive analytical expressions for the pressure, energy, number, and angular momentum densities of a free Bose gas in nonrelativistic and ultrarelativistic limits in terms of the corresponding fugacities. We then focus on the phenomenon of BEC. We calculate the critical temperature of BEC transition and the condensate fraction in a slowly rotating Bose gas including only particles. Our findings indicate that the critical exponent associated with the BE transition in a rotating gas is lower compared to that in a nonrotating Bose gas. We also determine the fugacity in a rotating Bose gas in the aforementioned limits and examine how rotation affects its temperature dependence, both below and above the critical temperature. By analyzing the behavior of heat capacity at these temperatures, we demonstrate that in a nonrelativistic Bose gas, the rotation transforms the nature of the BE phase transition from a continuous to a discontinuous transition. In general, we find that a nonrelativistic Bose gas under rotation behaves similarly to a nonrotating Bose gas in ultrarelativistic limit.