A Critical Point on the Hairy Black Hole Phase Boundary in the Improved Holographic Einstein-Maxwell-Dilaton Theory

In this work, we investigate the hairy black hole solutions and their dual phase diagram in the improved holographic Einstein-Maxwell-Dilaton (EMD) model.From the gravitational perspective, the rich phase structures observed in the dual boundary field theory originate from the intricate interplay between the scalar field formalism and the Maxwell field coupling mechanism. Two distinct types of hairy black hole solutions are found in this framework. Type-I hairy black holes are predominantly governed by scalar potential dynamics, whereas Type-II solutions emerge through nonminimal coupling to the $U(1)$ gauge field. We map out the phase distribution in the $(μ_B,T)$ parameter plane and delineate the boundary separating these two hairy phases. The phase diagram exhibits a first-order phase transition line consistent with previous findings, accompanied by a subtle third-order phase transition line that terminates at a critical point positioned at the turning point of the entire phase boundary curve. Our results complement existing research on holographic EMD theory by offering a comprehensive characterization of phase distributions, transition boundaries, and their gravitational sector interpretations. These insights will enable more effective engineering of specific phase structures for simulating strongly coupled systems through targeted modifications to the EMD model.