  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 $N\geq 35131$. The
instability implies that the isobar of the gas zigzags on the
temperature-volume plane, featuring supercooling and superheating phenomena.
The Bose-Einstein condensation then can persist from absolute zero to the
superheating temperature. Without necessarily taking the large $N$ limit, under
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 which
involves a discrete change of the density at the center of the trap.
