Multiple levels of symmetry breaking

Abstract

In the previous paper in this volume we have studied the p-spin interaction model just below the critical temperature, and we have rigorously proved several aspects of the physicist's prediction that this model exhibits "one level of symmetry breaking". In the present paper we show how to construct systems that exhibit an arbitrarily large, but finite number of "levels of symmetry-breaking". As the temperature decreases, such systems exhibit many phase transitions, as the structure of the overlaps gains complexity. This phenomenon does not seem to have been described previously, even in the physics literature.