Local factorisation of the dynamics of quantum spin systems

Motivated by the study of area laws for the entanglement entropy of gapped ground states of quantum spin systems and their stability, we prove that the unitary cocycle generated by a local time-dependent Hamiltonian can be approximated, for any finite set $X$, by a tensor product of the corresponding unitaries in $X$ and its complement, multiplied by a dynamics strictly supported in the neighbourhood of the surface $\mathrm{\partial }X$. The error decays almost exponentially in the size of the neighbourhood and grows with the square of the area~$\mathrm{\mid }\mathrm{\partial }X{\mathrm{\mid }}^2$.