Yang-Lee zeros and the critical behavior of the infinite-range two- and three-state Potts models

Abstract

The phase diagram of the two- and three-state Potts model with infinite-range interactions in the external field is analyzed by studying the partition function zeros in the complex field plane. The tricritical point of the three-state model is observed as the approach of the zeros to the real axis at the nonzero field value. Different regimes, involving several first- and second-order transitions of the complicated phase diagram of the three-state model, are identified from the scaling properties of the zeros closest to the real axis. The critical exponents related to the tricritical point and the Yang-Lee edge singularity are well reproduced. Calculations are extended to the negative fields, where the exact implicit expression for the transition line is derived.

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

“…While the former case have been already a subject of several studies (see e.g. [18,20]), the latter is addressed here for the first time.…”

“…We start from the analysis of the exact integral representation for the partition function at zero external field. First, we obtain the Fisher zeros by solving the system of equations Re Z(t, H) = 0 and Im Z(t, H) = 0 for the complex variables t = Re t + i Im t using function (18) at H = 0. In Fig.…”

“…Changing the integration variable √ Nx/T → x one gets [17]: where the temperature dependent prefactor again has been omitted. Being primarily interested in the properties of the partition function itself, we approximate (18) by its expansion at large N and small H. Keeping the leading order contributions in the linear in H term, we expand the exponent function for N → ∞ and get [17]:…”

“…In the remainder of this section we analyze the expressions for the exact and approximated partition functions of the Ising model on a complete graph, Eqs. (18) and (19).…”

Condensation of Lee-Yang zeros in scalar field theory

“…Along the red arcs, both the radial distance as well as the polar angle of the zeros scale with M −2/3 , verifying the proposed condensation law ρ = 1 + γM −2/3 obtained setting δ = 5 in Eq. (25). For the zeros located on the green line the radial distance scales in the same way with M (∼ M −2/3 ) while θ ρ = 2πn M with n ∈ N 0 .…”