Zeros of partition function for continuous phase transitions using cumulants

Abstract

This paper explores the use of a cumulant method to determine the zeros of partition functions for continuous phase transitions going up to the second zero. Unlike a first-order transition, with a uniform density of zeros near the transition point, a continuous transition shows a power law dependence of the density with a nontrivial slope for the line of zeros. Different types of models and methods of generating cumulants are used as testing grounds for the method. These include exactly solvable DNA melting problem on hierarchical lattices, heterogeneous DNA melting with randomness in sequence, Monte Carlo simulations for the well-known square lattice Ising model. The method is applicable for zeros near the imaginary axis which control dynamical quantum phase transitions. In all cases, the method is found to provide the basic information about the transition, and most importantly, avoids root finding methods.