TY - JOUR

T1 - Q -plane zeros of the Potts partition function on diamond hierarchical graphs

AU - Chang, Shu Chiuan

AU - Roeder, Roland K.W.

AU - Shrock, Robert

N1 - Publisher Copyright:
© 2020 Author(s).

PY - 2020/7/1

Y1 - 2020/7/1

N2 - 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).

AB - 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).

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U2 - 10.1063/1.5127667

DO - 10.1063/1.5127667

M3 - Article

AN - SCOPUS:85088317838

VL - 61

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 7

M1 - 073301

ER -