When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu–Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, the number of resulting NGMs can be lower than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear.

In this talk, I will briefly review spontaneous symmetry breaking and Nambu-Goldstone theorem at first. Then I will introduce the rules that govern the number and energy dispersion of NGMs associated with broken internal symmetry in nonrelativistic systems. An overview of the derivation of these rules will be given next, based on the effective Lagrangian approach. At last, I will discuss the properties of NGMs originating from space-time symmetry breaking.

References:

[1] H.Watanabe. Annu. Rev. Condens.Matter Phys. 2020. 11:169–87.

[2] H.Watanabe and H. Murayama. Phys. Rev. X 4, 031057 (2014).