Phase structure of the Born-Infeld-anti-de Sitter black holes probed by   non-local observables
  With the non-local observables such as two point correlation function and holographic entanglement entropy, we probe the phase structure of the Born-Infeld-anti-de Sitter black holes. We find for the case $bQ>0.5$, the phase structure is similar to that of the Reissner-Nordstr\"om-AdS black hole, namely the black hole undergoes a Hawking-Page phase transition, a first order phase transition, and a second order phase transition. While for the case $bQ<0.5$, we find there is a new branch for the infinitesimally small black hole so that a pseudo phase transition emerges besides the original first order phase transition. For the first order phase transition and the pseudo phase transition, the equal area law is checked, and for the second order phase transition, the critical exponent of the analogous heat capacity is obtained in the neighborhood of the critical points. All the results show that the phase structure of the non-local observables is the same as that of the thermal entropy regardless of the size of the boundary region in the field theory. 
Xiao-Xiong Zeng, Xian-Ming Liu, Li-Fang Li
hep-th gr-qc