Extended ensemble theory, spontaneous symmetry breaking, and phase transitions

Abstract

In this paper, as a personal review, we suppose a possible extension of Gibbs ensembletheory so that it can provide a reasonable description of phase transitions and spontaneoussymmetry breaking. The extension is founded on three hypotheses, and can beregarded as a microscopic edition of the Landau phenomenological theory of phasetransitions. Within its framework, the stable state of a system is determined by theevolution of order parameter with temperature according to such a principle thatthe entropy of the system will reach its minimum in this state. The evolution oforder parameter can cause a change in representation of the system Hamiltonian;different phases will realize different representations, respectively; a phase transitionamounts to a representation transformation. Physically, it turns out that phasetransitions originate from the automatic interference among matter waves as thetemperature is cooled down. Typical quantum many-body systems are studied with thisextended ensemble theory. We regain the Bardeen–Cooper–Schrieffer solution forthe weak-coupling superconductivity, and prove that it is stable. We find thatnegative-temperature and laser phases arise from the same mechanism as phase transitions,and that they are unstable. For the ideal Bose gas, we demonstrate that it willproduce Bose–Einstein condensation (BEC) in the thermodynamic limit, whichconfirms exactly Einstein’s deep physical insight. In contrast, there is no BEC eitherwithin the phonon gas in a black body or within the ideal photon gas in a solidbody. We prove that it is not admissible to quantize the Dirac field by usingBose–Einstein statistics. We show that a structural phase transition belongs physically tothe BEC happening in configuration space, and that a double-well anharmonicsystem will undergo a structural phase transition at a finite temperature. For theO(N)-symmetric vector model, we demonstrate that it will yield spontaneous symmetry breakingand produce Goldstone bosons; and if it is coupled with a gauge field, the gauge field willobtain a mass (Higgs mechanism). Also, we show that an interacting Bose gas is stable onlyif the interaction is repulsive. For the weak interaction case, we find that the BEC is a‘λ-transition’ and its transition temperature can be lowered by the repulsive interaction. In connection withliquid 4He, it is found that the specific heat at constant pressureCP willshow a T3 law at low temperatures, which is in agreement with the experiment.If the system is further cooled down, the theory predicts thatCP will vanish linearly as , which is anticipating experimental verifications.