Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model

Abstract

The complex zeros of the canonical (fixed walk-length) partition function are calculated for both the self-avoiding trails model and the vertex-interacting self-avoiding walk model, both in bulk and in the presence of an attractive surface. The finite-size behavior of the zeros is used to estimate the location of phase transitions: the collapse transition in the bulk and the adsorption transition in the presence of a surface. The bulk and surface cross-over exponents, ϕ and ϕ S , are estimated from the scaling behavior of the leading partition function zeros.

Critical behavior of magnetic polymers in two and three dimensions

“…Another useful tool to use for estimating critical temperatures and, in the case of polymers, the exponent φ is to look at the behaviour of the dominant complex zero of the partition function [9,[32][33][34]. The dominant zero is the one which is closest to the real axis in the complex temperature plane, and will pinch the real axis at the transition temperature in the thermodynamic limit.…”

“…Canonically, if the interactions are short-ranged and the walks studied in their infinite length limit, there should exist a universality where the critical behaviour is insensitive to details of the model [2][3][4]. An early indication that things are not so simple arose in the study of the two-dimensional O(n → 0) model introduced by Blöte and Nienhuis [5], more recently known under the title of Vertex Interacting Self-Avoiding Walk [6][7][8][9]. They found that in this model, despite having short-ranged interactions, corresponding to non-crossing doubly-visited sites, the critical exponents were not the same as those found previously for the interacting self-avoiding walk (ISAW).…”