  We study the dynamics caused by transport of transverse magnetization in one
dimensional transverse Ising chain at zero temperature. We observe that a class
of initial states having product structure in fermionic momentum-space and
satisfying certain criteria, produce spatial variation in transverse
magnetization. Starting from such a state, we obtain the transverse
magnetization analytically and then observe its dynamics in presence of a
homogeneous constant field $\Gamma$. In contradiction with general expectation,
whatever be the strength of the field, the magnetization of the system does not
become homogeneous even after infinite time. At each site, the dynamics is
associated with oscillations having two different timescales. The envelope of
the larger timescale oscillation decays algebraically with an exponent which is
invariant for all such special initial states. The frequency of this
oscillation varies differently with external field in ordered and disordered
phases. The local magnetization after infinite time also characterizes the
quantum phase transition.
