12. Re-Discussion on Symmetry Breaking and Elementary Excitations

The chapter delves into the fundamental concepts of symmetry breaking and elementary excitations, with a particular focus on the Landau theory. It introduces and discusses phonons, phasons, and fluid phonons, which are essential excitations in quasicrystals. The author explores the interactions and implications of these excitations in both solid and soft-matter quasicrystals, highlighting the crucial role they play in understanding the dynamics and behavior of these complex structures. By revisiting the pioneering work of scientists like Planck, Einstein, Debye, and Landau, the chapter provides a historical context and builds on their theories to offer a contemporary perspective on quasicrystals. The discussion includes the classification of two-dimensional quasicrystals and the coupling or decoupling of phonons and phasons, emphasizing their significant impact on the motion of matter. Additionally, the chapter underscores the importance of fluid phonons in soft-matter quasicrystals, demonstrating their agreement with classical fluid dynamics. This comprehensive analysis not only validates the theoretical framework but also opens avenues for further research in the field of quasicrystals.