  We establish a general scaling law for the entanglement of a large class of
ground states and dynamically evolving states of quantum spin chains: we show
that the geometric entropy of a distinguished block saturates, and hence
follows an entanglement-boundary law. These results apply to any ground state
of a gapped model resulting from dynamics generated by a local hamiltonian, as
well as, dually, to states that are generated via a sudden quench of an
interaction as recently studied in the case of dynamics of quantum phase
transitions. We achieve these results by exploiting ideas from quantum
information theory and making use of the powerful tools provided by
Lieb-Robinson bounds. We also show that there exist noncritical fermionic
systems and equivalent spin chains with rapidly decaying interactions whose
geometric entropy scales logarithmically with block length. Implications for
the classical simulatability are outlined.
