The Yang-Lee zeros of the 1D Blume-Capel model on connected and non-connected rings

TL;DR

This paper analyzes the distribution of Yang-Lee zeros in the 1D Blume-Capel model on both connected and non-connected rings, revealing how zeros move off the unit circle with temperature and tend to overlap in the thermodynamic limit.

Contribution

It provides a combined numerical and analytical study of Yang-Lee zeros for the 1D Blume-Capel model on various lattice structures, including Feynman diagrams, and proves zeros lie on the unit circle for the 1D Ising case.

Findings

Zeros depart from the unit circle as temperature increases.

Zeros can bifurcate into two disjoint arcs, similar to 2D cases.

In the thermodynamic limit, zeros overlap for both models.

Abstract

We carry out a numerical and analytic analysis of the Yang-Lee zeros of the 1D Blume-Capel model with periodic boundary conditions and its generalization on Feynman diagrams for which we include sums over all connected and non-connected rings for a given number of spins. In both cases, for a specific range of the parameters, the zeros originally on the unit circle are shown to departure from it as we increase the temperature beyond some limit. The curve of zeros can bifurcate and become two disjoint arcs as in the 2D case. We also show that in the thermodynamic limit the zeros of both Blume-Capel models on the static (connected ring) and on the dynamical (Feynman diagrams) lattice tend to overlap. In the special case of the 1D Ising model on Feynman diagrams we can prove for arbitrary number of spins that the Yang-Lee zeros must be on the unit circle. The proof is based on a property of the zeros of Legendre Polynomials.