  We study the entanglement wedge cross-section (EWCS) in holographic massive
gravity theory, in which a first and second-order phase transition can occur.
We find that the mixed state entanglement measures, the EWCS and mutual
information (MI) can characterize the phase transitions. The interesting
phenomenon is that the EWCS and MI show exactly the opposite behavior in the
critical region, which suggests that the EWCS captures distinct degrees of
freedom from that of the MI. Meanwhile, the holographic entanglement entropy
(HEE) exhibits more complicated behaviors, totally distinct from that of the
AdS-RN black holes. Moreover, EWCS, MI and HEE all show obvious critical
scaling law near the critical point of the second-order phase transition and
have the same critical exponent. We offer an analytical understanding on this
and point out that this critical phenomenon is distinct from that of the
superconductivity phase transition.
