the rest of the lattice was first considered in lattice field theories. It was found that the entanglement entropy of *noncritical* chains with short-ranged interactions reaches a saturation level, while the entanglement entropy diverges logarithmically with $L$ in the field limit, i.e. in the case of *critical* chains. This approach was subsequently pursued numerically [@VidalLatorre] in the particular context of the one-dimensional XY model and, in more generality, in an analytical treatment based on random matrix theory [@Keating], applicable to any spin chain Hamiltonian that can be cast into a quadratic form of fermionic operators.

At the same time, progress has been hampered by the realization that *bi-partite* entanglement, whether of blocks of spins or otherwise, at best only gives us an incomplete, local picture of the entanglement exhibited by generic many-body ground states. It is something of a truism, of course, to note that a truly global, multi-partite approach is indispensable if one wishes to capture the entanglement that pervades critical many-body ground states simultaneously at all length scales (for recent work on multi-partite entanglement in quantum spin chains see for example Refs. [@Bruss; @Costantini]).

Unfortunately, the theory of multi-partite entanglement is still very much in its infancy. It 