Interpolating relativistic and nonrelativistic Nambu-Goldstone and Higgs modes

When a continuous symmetry is spontaneously broken in nonrelativistic theories, there appear Nambu-Goldstone (NG) modes, the dispersion relations of which are either linear (type I) or quadratic (type II). We give a general framework to interpolate between relativistic and nonrelativistic NG modes, revealing a nature of type-I and -II NG modes in nonrelativistic theories. The interpolating Lagrangians have the nonlinear Lorentz invariance which reduces to the Galilei or Schrödinger invariance in the nonrelativistic limit. We find that type-I and type-II NG modes in the interpolating region are accompanied with a Higgs mode and a chiral NG partner, respectively, both of which are gapful. In the ultrarelativistic limit, a set of a type-I NG mode and its Higgs partner remains, while a set of a type-II NG mode and its gapful NG partner turns to a set of two type-I NG modes. In the nonrelativistic limit, the both types of accompanied gapful modes become infinitely massive, disappearing from the spectrum. The examples contain a phonon in Bose-Einstein condensates or helium superfluids, a phonon and magnon in spinor Bose-Einstein condensates, a magnon in ferromagnets, and a kelvon and dilaton-magnon localized around a Skyrmion line in ferromagnets.