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.

• Received 1 December 1969

DOI:https://doi.org/10.1103/PhysRevB.1.3160