Symmetry Breaking and Order in the Age of Quasicrystals

TL;DR

This paper explores how quasicrystals challenge traditional notions of symmetry in condensed matter, examining their unique invariance properties, order parameters, and excitations like phonons and phasons.

Contribution

It provides a comprehensive review of symmetry breaking and order in quasicrystals, proposing a revised understanding of symmetry concepts in these aperiodic structures.

Findings

Quasicrystals lack traditional translational and rotational symmetries.

Introduction of new elementary excitations: phasons.

Discussion of dislocations and their role in quasicrystal order.

Abstract

The discovery of quasicrystals has changed our view of some of the most basic notions related to the condensed state of matter. Before the age of quasicrystals, it was believed that crystals break the continuous translation and rotation symmetries of the liquid-phase into a discrete lattice of translations, and a finite group of rotations. Quasicrystals, on the other hand, possess no such symmetries-there are no translations, nor, in general, are there any rotations, leaving them invariant. Does this imply that no symmetry is left, or that the meaning of symmetry should be revised? We review this and other questions related to the liquid-to-crystal symmetry-breaking transition using the notion of indistinguishability. We characterize the order-parameter space, describe the different elementary excitations, phonons and phasons, and discuss the nature of dislocations-keeping in mind that we are now living in the age of quasicrystals.