  Spatial modulation has been studied for a long time in condensed matter,
nuclear matter and quark matter, so far in non-relativistic field theories. In
this paper, spatially modulated vacua at zero temperature and zero density are
studied in relativistic field theories. We first propose an adaptation of the
Nambu-Goldstone theorem to higher derivative theories under the assumption of
the absence of ghosts: when a global symmetry is spontaneously broken due to
vacuum expectation values of space-time derivatives of fields, a
Nambu-Goldstone (NG) boson appears without a canonical kinetic (quadratic
derivative) term with a quartic derivative term in the modulated direction
while a Higgs boson appears with a canonical kinetic term. We demonstrate this
in a simple model allowing (meta)stable modulated vacuum of a phase modulation
(Fulde-Ferrell state), where an NG mode associated with spontaneously broken
translational and $U(1)$ symmetries appears.
