  First order quantum phase transitions (1QPTs) are signaled, in the
thermodynamic limit, by discontinuous changes in the ground state properties.
These discontinuities affect expectation values of observables, including
spatial correlations. When a 1QPT is crossed in the vicinity of a second order
one (2QPT), due to the correlation length divergence of the latter, the
corresponding ground state is modified and it becomes increasingly difficult to
determine the order of the transition when the size of the system is finite.
Here we show that, in such situations, it is possible to apply finite size
scaling to entanglement measures, as it has recently been done for the order
parameters and the energy gap, in order to recover the correct thermodynamic
limit. Such a finite size scaling can unambigously discriminate between first
and second order phase transitions in the vicinity of multricritical points
even when the singularities displayed by entanglement measures lead to
controversial results.
