  By the topological argument that the identity matrix is surrounded by a set
of separable states follows the result that if a system is entangled at thermal
equilibrium for some temperature, then it presents a phase transition (PT)
where entanglement can be viewed as the order parameter. However, analyzing
different entanglement measures in the 2-qubit context, we see that different
entanglement quantifiers can indicate different orders for the same PT.
Examples are given for different Hamiltonians. Moving to the multipartite
context we show necessary and sufficient conditions for a family of
entanglement monotones to attest quantum phase transitions.
