  A two-dimensional (2D) electron gas formed in a modulation-doped GaAs/AlGaAs
single quantum well undergoes a first-order transition when the first excited
subband is occupied with electrons, as the Fermi level is tuned into resonance
with the excited subband by applying a dc voltage. Direct evidence for this
effect is obtained from low-temperature photoluminescence spectra which display
the sudden renormalization of the intersubband energy $E_{01}$ upon the abrupt
occupation of the first excited subband. Calculations within density-functional
theory, which treat the 2D exchange potential {\it exactly}, show that this
thermodynamical instability of the electron system is mainly driven by {\it
intersubband} terms of the exchange Coulomb interaction. From
temperature-dependent measurements the existence of a critical point at $T_c =
35\pm 5$ K is inferred.
