Title: On The Patterns of The Fisher's Zeros Maps to Classify Phase Transition

Abstract: Phase transition is one of the most interesting natural phenomena and until nowadays several techniques are being developed to study it. One of the main concerns in the topic is how to classify a specific transition as being of first, second, or even of a higher order, according to the Ehrenfest classification. The partition function provides all the thermodynamic information about the physical system, and a phase transition can be identified by the complex temperature where it is equal to zero. In addition, the pattern of the zeros on the complex temperature plan can provide evidences of the order of the transition. In this manuscript, we present an analytical and simulational study connecting the microcanonical analysis of the unstable region of the entropy to the canonical partition function zeros. We show that for the first-order transition the zeros accumulate uniformly in a vertical line on the complex inverse temperature plane as discussed in previous works. We illustrated our calculation using the $147$ particles Lennard-Jones cluster. In the second-order case the zeros can assume different slops, besides that the inverse distance between them following a power law with exponent $0<\alpha<1$. For higher order phase transitions $\alpha > 1$ is expected. We studied the 2D square lattice Ising model where we found $\alpha \approx 1.4$ which is inconsistent with the expected transition for this model.