Entanglement quantifiers, entanglement crossover and phase transitions

Entanglement has been widely used as a tool for the investigation of phase transitions (PTs). However, analysing several entanglement measures in the two-qubit context, we see that distinct entanglement quantifiers can indicate different orders for the same PT. Examples are given for different Hamiltonians. This leaves open the possibility of addressing different orders to the same PT if entanglement is used as an order parameter. Moving on to the multipartite context, we show necessary and sufficient conditions for a family of entanglement monotones to confirm quantum PTs