Entanglement wedge minimum cross-section in holographic massive gravity theory

We study the entanglement wedge cross-section (EWCS) in holographic massive gravity theory, which a first and second-order phase transition can occur. find that mixed state measures, EWCS mutual information (MI) characterize transitions. The MI show exactly opposite behavior critical region, suggests captures distinct degrees of freedom from MI. More importantly, EWCS, HEE all same scaling region. give an analytical understanding this phenomenon. By comparing quantum thermodynamic superconductors, we analyze relationship difference between them, provide two mechanisms transition.