Symmetry, quasisymmetry, and critical phenomena (1,2).(Technical report)

Abstract: This presentation is an analysis of the role of symmetry in second-order quantum phase transitions. It seeks to explain why transitions between phases of systems, associated with different symmetries, exhibit critical phenomena. It transpires that a system in one phase, tends to hold onto the symmetry associated with that phase until a breaking point is reached at which a rapid transition occurs to a new phase associated with a different symmetry. Understanding what is happening presents the challenge of explaining how a system "holds on" to a symmetry in the face of strong symmetry-breaking interactions. The explanation is of fundamental interest in physics, for understanding why models with built in symmetries often work extremely well even when the models ignore large symmetry-breaking interactions that are known to be present. When this phenomenon occurs, we say that the system has a quasidynamical symmetry. This concept is of interest in mathematics because it turns out that quasidyndamical symmetries are the physical realizations of a new concept in group theory, which we refer to as an embedded representation.

PACS Nos.: 21.60.Fw, 21.60.Ev, 64.6-.Ht, 68.18.Jk

Resume : Nous etudions le role la symetrie dans des transitions de phase du deuxieme ordre. Nous cherchons a expliquer pourquoi des transitions entre les phases d'un systeme associees a differentes symetries, ont un comportement critique. Il ressort qu'un systeme dans une phase cherche a maintenir la symetrie associee a cette phase jusqu'a ce qu'il atteigne un point de bris, auquel moment une transition rapide se produit vers une nouvelle phase avec une symetrie differente. Comprendre ce qui se passe presente le defi de comprendre comment un systeme << s'accroche >> a une symetrie lorsqu'il fait face a une forte interaction de bris de symetrie. L'explication a un interet fondamental en physique, pour comprendre pourquoi des modeles avec une symetrie interne fonctionnent si bien, meme lorsque de fortes interactions de bris de symetrie sont presentes. Lorsque cala se produit, nous disons que le systeme a une symetrie quasi-dynamique. Ce concept est interessant en mathematiques, parce que les symetries quasi-dynamiques sont des realisations physiques d'un nouveau concept en theorie des groupes appele representation incluse.

[Traduit par la Redaction]

1. Introduction

It is often said that nature loves symmetry. Indeed, it often takes considerable energy to shake a system out of a symmetric state. This hardly needs explaining at a classical level but understanding howit comes about at a quantal level proves to be insightful. It turns out that there is a natural expression of how and why symmetries are effectively preserved, when there are strong symmetry-breaking interactions, in terms of quasidynamical symmetry [1] (or effective symmetry as it is also called [2]).A system is said to exhibit a quasidynamical symmetry if a subset of its states exhibit properties that would result if the system had a dynamical symmetry which, in fact, may be badly broken.

An example of quasidynamical symmetry at a macroscopic classical level is given by considering an array of molecules of a particular type, characterized by a parameter a, which might be a parameter in the Hamiltonian signifying, for example, the magnitude of an applied field, or it might be the temperature or pressure. Suppose that the array forms a crystal structure with a particular symmetry for a small value of a and a different symmetry for a large value of a. One can imagine that, if a could be increased in steps from a small value to a larger value, the initial symmetry might be essentially preserved but the positions of the molecules might begin to exhibit increasing displacements from their equilibrium positions and, at some critical …