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

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.

Figure 1

Solutions of the equations $\text{Re}{\mathcal{Z}}_N=0$ [blue (dark gray) lines] and $\text{Im}{\mathcal{Z}}_N=0$ [orange (light gray) lines, including the $h_1=0$ axis] for $N=1000000$ particles at $K=2$. For distant points errors can be observed as deviations of intersections from the imaginary axis.

Figure 2

Phase diagram: Ising transition ($h<0$), open circles; KMS transition ($h>0$), thick full line. At low temperatures, $K>K_0\left(3\right)$, Ising and KMS lines are merged and a transition exists only for $h=0$, broken line.

Figure 3

The free-energy density $-f(x,S)$ at (a) $(K,h)=(3.5,0)$ and (b) $(K,h)=(2.79,0)$.

Figure 4

The free energy density $-f(x,S)$ at inverse temperature $K=K_0\left(3\right)=4\ln 2$ and $h=0$.

Figure 5

The free-energy density $-f(x,S)$ at $K=2.70$ and two values of the field: (a) $h_{\text{Ising}}=-0.04169621964961\cdots$ and (b) $h_{\text{KMS}}=0.0181471805599452\cdots .$

Figure 6

The free-energy density $-f(x,S)$ at inverse temperature $K=K_{\text{tr}}^{\text{KMS}}=8/3=2.\dot{6}$ and external field: (a) $h=-0.06333826060\cdots$ and (b) $h=h_{\text{tr}}^{\text{KMS}}=0.02648051389\cdots .$

Figure 7

$q=3$ and $N=3000$. The positions of the part of the Yang-Lee zeros closest to the real field axis are given as the intersections of the blue (dark gray) and orange (light gray) lines: (a) $K=2.6$ and (b) $K=2.7$. The broken black line is the KMS value of the real field predicted by relation (32).

Figure 8

$q=3$ and $N=10000$. The positions of the part of the Yang-Lee zeros closest to the real field axis are given as the intersections of the blue (dark gray) and orange (light gray) lines: (a) $K=K_0\left(3\right)=2.772...$ and (b) $K=3.5$.