  The interplay between dissipation and internal interactions in quantum
many-body systems gives rise to a wealth of novel phenomena. Here we
investigate spin-1/2 chains with uniform local couplings to a Markovian
environment using the time-dependent density matrix renormalization group
(tDMRG). For the open XXZ model, we discover that the decoherence time diverges
in the thermodynamic limit. The coherence decay is then algebraic instead of
exponential. This is due to a vanishing gap in the spectrum of the
corresponding Liouville superoperator and can be explained on the basis of a
perturbative treatment. In contrast, decoherence in the open transverse-field
Ising model is found to be always exponential. In this case, the internal
interactions can both facilitate and impede the environment-induced
decoherence.
