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Entanglement and violation of Bell inequalities are aspects of quantumnonlocality that have been often confused in the past. It is now known thatthis equivalence is only true for pure states. Even though almost all thestudies of quantum correlations at quantum phase transitions deal only withentanglement, we here argue that Bell inequalities can also reveal a generalquantum phase transition. This is also shown for a particular case of twospin-1/2 particles in an infinite one-dimensional chain described by the XXZmodel. In this case, the Bell inequality is able to signal not only thefirst-order phase transition, but also the infinite-order Kosterlitz-Thoulessquantum phase transition, which cannot be revealed either by the energy of thesystem nor by the bipartite entanglement. We also show that although thenearest-neighbor spins are entangled, they, unexpectedly, never violate theBell inequality. This indicates that the type of entanglement which is relevantfor quantum phase transition is not trivial, i.e., it cannot be revealed by theBell inequality.

Bell inequalities and entanglement at quantum phase transition in the XXZ Model