Yang-Lee Zeros of the Two- and Three-State Potts Model Defined on $\phi^3$ Feynman Diagrams

TL;DR

This paper investigates the distribution of Yang-Lee zeros for the Ising and 3-state Potts models on $\,\phi^3$ Feynman diagrams, revealing zeros on the unit circle for Ising and outside it for Potts, supported by analytic and numerical methods.

Contribution

It provides the first nonperturbative proof for the Ising model's zeros on the unit circle on random graphs and extends understanding of phase transitions in these models.

Findings

Ising model zeros are on the unit circle, confirmed analytically and numerically.

Potts model zeros are outside the unit circle, based on perturbative analysis.

Analytic proofs are supported by finite lattice numerical calculations.

Abstract

We present both analytic and numerical results on the position of the partition function zeros on the complex magnetic field plane of the $q=2$ (Ising) and $q=3$ states Potts model defined on $\phi^3 $ Feynman diagrams (thin random graphs). Our analytic results are based on the ideas of destructive interference of coexisting phases and low temperature expansions. For the case of the Ising model an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. For the $q=3$ states Potts model our perturbative results indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic results are confirmed by finite lattice numerical calculations.