  We investigate analytically a star network of spins, in which all spins
interact exclusively with a central spin through Heisenberg XX couplings of
equal strength. We find that the central spin correlates and entangles the
other spins at zero temperature to a degree that depends on the total number of
spins. Surprisingly, the entanglement depends on the evenness or oddness of
this number and some correlations are substantial even for an infinite
collection of spins. We show how symmetric multi-party states for optimal
sharing and splitting of entanglement can be obtained in this system using a
magnetic field.
