  We present an example where Spontaneous Symmetry Breaking may effect not only
the behavior of the entanglement at Quantum Phase Transitions, but also the
origin of the non-analyticity. In particular, in the XXZ model, we study the
non analyticities in the concurrence between two spins, which was claimed to be
accidental, since it had its origin in the optimization involved in the
concurrence definition. We show that when one takes in account the effect of
the Spontaneous Symmetry Breaking, even tough the values of the entanglement
measure does not change, the origin the the non-analytical behavior changes: it
is not due to the optimization process anymore and in this sense it is a
"natural" non-analyticity. This is a much more subtle influence of the
Spontaneous Symmetry Breaking not noticed before. However the non-analytical
behavior still suggests a second order quantum phase transition and not the
first order that occurs and we explain why. We also show that the value of
entanglement between one site and the rest of the chain does change when taking
into account the Spontaneous Symmetry Breaking.
