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We present an example where Spontaneous Symmetry Breaking may effect not onlythe behavior of the entanglement at Quantum Phase Transitions, but also theorigin of the non-analyticity. In particular, in the XXZ model, we study thenon analyticities in the concurrence between two spins, which was claimed to beaccidental, since it had its origin in the optimization involved in theconcurrence definition. We show that when one takes in account the effect ofthe Spontaneous Symmetry Breaking, even tough the values of the entanglementmeasure does not change, the origin the the non-analytical behavior changes: itis 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 theSpontaneous Symmetry Breaking not noticed before. However the non-analyticalbehavior still suggests a second order quantum phase transition and not thefirst order that occurs and we explain why. We also show that the value ofentanglement between one site and the rest of the chain does change when takinginto account the Spontaneous Symmetry Breaking.

Effects of spontaneous symmetry break in the origin of non-analytic behavior of entanglement at quantum phase transitions