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We consider the time evolution of a system described by a
1+1-dimensional conformal field theory (CFT) from states with short-range
correlations and entanglement which are more general than that considered
in Calabrese-Cardy [2006]. We show that, while the correlation functions
in any finite region still become stationary at late times, they do not
have a thermal form. Instead, the effective temperature as measured by
correlations depends on the observable. We argue that this is consistent
with the stationary distribution being given by a generalised Gibbs
ensemble, which may, however, depend on non-local parafermionic charges.
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