  Background: Spurious zero-energy Nambu-Goldstone (NG) modes appear when the
symmetry of a system is spontaneously broken. The Thouless-Valatin inertia, the
collective inertia of the NG mode, contains important information concerning
collective motion.
  Purpose: To establish an efficient and precise method for deriving the
collective inertia and the conjugate operator of a NG mode, we derive an
expression for the response function in terms of the coordinate-momentum
representation of the quasiparticle random-phase approximation which is valid
even if a symmetry-restoring zero-energy mode is present.
  Methods: We use the finite amplitude method for computing the response
function of superfluid nuclei with the nuclear density functional theory.
  Results: We derived analytically the collective inertia and the conjugate
coordinate operator of the NG mode from the zero-energy linear response with
the momentum operator of the NG mode. The formulation is tested in the cases of
translational and pairing rotational modes. Illustrative calculations are
performed for the neutron pairing rotation in Sn isotopes, the proton pairing
rotation in N = 82 isotones, and the neutron and proton pairing rotations
around the 130Xe nucleus.
  Conclusions: The proposed formulation allows us to compute the collective
inertia of the NG mode precisely and efficiently. The conjugate coordinate
operator can be utilized to remove spurious contributions to the strength
distribution in the finite amplitude method.
