  We study a quantum antiferromagnetic Heisenberg model on a hypercubic lattice
in three or higher dimensions $d\ge 3$. When a phase transition occurs with the
continuous symmetry breaking, the nonvanishing spontaneous magnetization which
is obtained by applying the infinitesimally weak symmetry breaking field is
equal to the maximum spontaneous magnetization at zero or non-zero low
temperatures. In addition, the transverse correlation in the infinite-volume
limit exhibits a Nambu-Goldstone-type slow decay. In this paper, we assume that
the transverse correlation decays by power law with distance. Under this
assumption, we prove that the power is equal to $2-d$ at non-zero low
temperatures, while it is equal to $1-d$ at zero temperature. The method is
applied also to a quantum XY model and a classical Heisenberg model at non-zero
low temperatures in three or higher dimensions. The resulting power is given by
the same $2-d$ at non-zero low temperatures.
