  This note contains discussions on the entanglement entropy and mutual
information of a strongly coupled field theory with a critical point which has
a holographic dual. We investigate analytically, in the specific regimes of
parameters, how these non-local operators behave near the critical point.
Interestingly, we observe that although the mutual information is constant at
the critical point, its slope shows a power-law divergence in the vicinity of
the critical point. We show that the leading behavior of mutual information at
and near the critical point could yield a set of critical exponents if we
regard it as an order parameter. Our result for this set of static critical
exponents is (1/2,1/2,1/2,2) which is identical to the one calculated via the
thermodynamic quantities. Hence it suggests that beside the numerous merits of
mutual information, this quantity also captures the critical behavior of the
underlying field theory and it could be used as a proper measure to probe the
phase structure associated with the strongly coupled systems.
