  We study the decoherence dynamics of a quantum Ising lattice of finite size
with a transverse dissipative interaction, namely the coupling with the bath is
assumed perpendicular to the direction of the spins interaction and parallel to
the external transverse magnetic field. In the limit of small transverse field,
the eigenstates and spectrum are obtained by a strong coupling expansion, from
which we derive the Lindblad equation in the Markovian limit. At temperature
lower than the energy gap and for weak dissipation, the decoherence dynamics
can be restricted to take only the two degenerate ground states and the first
excited subspace into account. The latter is formed by pairs of topological
excitations (domain walls or kinks), which are quantum delocalized along the
chain due to the small magnetic field. We find that some of these excited
states form a relaxation-free subspace, i.e. they do not decay to the ground
states. We also discuss the decoherence dynamics for an initial state formed by
a quantum superposition of the two degenerate ground states corresponding to
the orthogonal classical, ferromagnetic states.
