  We study the decoherence and the relaxation dynamics of topological states in
an extended class of quantum Ising chains which can present a manyfold ground
state subspace. The leading interaction of the spins with the environment is
assumed to be the local fluctuations of the transverse magnetic field. By
deriving the Lindblad equation using the many-body states, we investigate the
relation between decoherence, energy relaxation and topology. In particular, in
the topological phase and at low temperature, we analyze the dephasing rates
between the different degenerate ground states.
