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).
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics