Singularities in the complex temperature plane at the first order phase transitions and critical points

Abstract

In the early 50’s, Lee and Yang developed the theory of phase transitions based on the singular nature of partition function in the complex parameter space. Ever since there have been a long history of extensive study of the behavior of the partition function in the complex temperature plane near the phase transition point as well as the critical point. Recent study reveals that the so-called Lee-Yang circle theorem is a special case of a more general mathematical relation in the probability theory in the case of the first order phase transition. On the other hand the finite size scaling theory shows that the complex temperature behavior of the partition function at the critical point has a unique scaling behavior. Instead of zeros of the partition function there is a flat region where the complex partion function vanishes as one moves away from the critical point. We will discuss these recent findings.