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

High Energy Physics - Theory;Materials Science;High Energy Physics - Phenomenology