Replica theory of partition-function zeros in spin-glass systems

We study the phase transitions in spin-glass systems by analysing the partition-function zeros (Lee-Yang zeros) with respect to the complex temperature/field. For several models as the random energy and spherical models with many-body interactions, we extend the replica method and the procedure of the replica symmetry breaking ansatz to be applicable in the complex-parameter case. We derive the phase diagrams in the complex plane and calculate the density of zeros in each phase. We find that there is a replica symmetric phase having a large density near the imaginary axis away from the origin. In the spin-glass phase, the density is finite only when the chaos effect is present. This result indicates that the density of zeros is more closely connected to the chaos effect than the replica symmetry breaking. We also investigate the relevance of our result to the finite-dimensional systems by studying the renormalization group flow in the complex plane.