q-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs

Abstract

We report exact results concerning the zeros of the partition function of the Potts model in the complex q-plane, as a function of a temperature-like Boltzmann variable v, for the m-th iterate graphs Dm of the diamond hierarchical lattice, including the limit m → ∞. In this limit, we denote the continuous accumulation locus of zeros in the q-planes at fixed v = v0 as ℬ𝑞(𝑣0). We apply theorems from complex dynamics to establish the properties of ℬ𝑞(𝑣0). For v = −1 (the zero-temperature Potts antiferromagnet or, equivalently, chromatic polynomial), we prove that ℬ𝑞(−1) crosses the real q-axis at (i) a minimal point q = 0, (ii) a maximal point q = 3, (iii) q = 32/27, (iv) a cubic root that we give, with the value q = q1 = 1.638 896 9…, and (v) an infinite number of points smaller than q1, converging to 32/27 from above. Similar results hold for ℬ𝑞(𝑣0) for any −1 < v < 0 (Potts antiferromagnet at nonzero temperature). The locus ℬ𝑞(𝑣0) crosses the real q-axis at only two points for any v > 0 (Potts ferromagnet). We also provide the computer-generated plots of ℬ𝑞(𝑣0) at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of Z(D4, q, v0).