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arXiv:cond-mat/0307010

Fisher zeros and Potts zeros of the Q-state Potts model for nonzero external magnetic field

Seung‐Yeon Kim

Published Jul 1, 2003• 0 citations
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Abstract

The properties of the partition function zeros in the complex temperature plane (Fisher zeros) and in the complex $Q$ plane (Potts zeros) are investigated for the $Q$-state Potts model in an arbitrary nonzero external magnetic field $H_q$, using the exact partition function of the one-dimensional model.

DOI
10.48550/arxiv.cond-mat/0307010

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The properties of the partition function zeros in the complex temperature plane (Fisher zeros) and in the complex QQQ plane (Potts zeros) are investigated for the QQQ-state Potts model in an arbitrary nonzero external magnetic field HqH_qHq​, using the exact partition function of the one-dimensional model.
The microcanonical transfer matrix is used to study the distribution of the Fisher zeros of the Q>2Q>2Q>2 Potts models in the complex temperature plane with nonzero external magnetic field Hq.{H}_{q}.Hq​. Unlike the Ising model for ${H}_{q}\ensuremath{\ne}0,$ which has only a nonphysical critical point (the Fisher edge singularity), the Q>2Q>2Q>2 Potts models have physical critical points for Hq<0{H}_{q}<0Hq​<0 as well as the Fisher edge singularities for Hq>0.{H}_{q}>0.Hq​>0. For Hq<0,{H}_{q}<0,Hq​<0, the crossover of the Fisher zeros of the Q-state Potts model into those of the $(Q\ensuremath{-}1)$-state Potts model is discussed, and the critical line of the three-state Potts ferromagnet is determined. For Hq>0,{H}_{q}>0,Hq​>0, we investigate the edge singularity for finite lattices and compare our results with the high-field, low-temperature series expansion of Enting. For 3< Q< 6,3<~Q<~6,3< Q< 6, we find that the specific heat, magnetization, susceptibility, and the density of zeros diverge at the Fisher edge singularity with exponents ${\ensuremath{\alpha}}_{e},$ ${\ensuremath{\beta}}_{e},$ and ${\ensuremath{\gamma}}_{e},$ which satisfy the scaling law ${\ensuremath{\alpha}}_{e}+2{\ensuremath{\beta}}_{e}+{\ensuremath{\gamma}}_{e}=2.$
The distribution of the zeros of the partition function in the complex temperature plane (Fisher zeros) of the two-dimensional Q-state Potts model is studied for non-integer Q. On L×L self-dual lattices studied (L⩽8), no Fisher zero lies on the unit circle p0=eiθ in the complex p=(eβJ−1)/Q plane for Q<1, while some of the Fisher zeros lie on the unit circle for Q>1 and the number of such zeros increases with increasing Q. The ferromagnetic and antiferromagnetic properties of the Potts model are investigated using the distribution of the Fisher zeros. For the Potts ferromagnet we verify the den Nijs formula for the thermal exponent yt. For the Potts antiferromagnet we also verify the Baxter conjecture for the critical temperature and present new results for the thermal exponents in the range 0<Q<3.
From the exact partition functions of the Ising model on L x L square lattices (up to L = 14) in an arbitrary nonzero external magnetic field H, the precise distributions of the Fisher zeros in the complex temperature (a = e(2betaJ)) plane are obtained. Our results indicate that in the limit L-->infinity the Fisher zeros for H not = 0 do cut the positive real axis at the antiferromagnetic critical point ac(H); these values are compared with the results of closed-form approximations for the antiferromagnetic critical line. From the Fisher zeros the thermal scaling exponent yt = 1 is also obtained along the critical line, indicating the logarithmic singularity of the specific heat for the antiferromagnetic Ising model even in a strong uniform magnetic field.
The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a-1 (a=e(betaJ)) and the coefficients of this polynomial, Phi(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute Phi(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z(CP)=Z(Q,a(c)(Q)), where a(c)(Q)=1+square root[Q]. We find a set of zeros in the complex w=square root[Q] plane that map to (or close to) the Beraha numbers for real positive Q. We also identify Q(c)(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/square root[Q] plane lies on the unit circle. By finite-size scaling we find that 1/Q(c)(L)-->0 as L-->infinity. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q(c)(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Baxter's conjecture Q(AF)(c)(a)=(1-a)(a+3). We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that Q(FM)(c)(a)=(a-1)(2). We show that the edge singularity in the complex Q plane approaches Q(c) as Q(c)(L) approximately Q(c)+AL(-y(q)), and determine the scaling exponent y(q) for several values of Q. Finally, by finite-size scaling of the Fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y(t) as a function of Q in the range 2</=Q</=3. Using data for lattices of size 3</=L</=8 we find that y(t) is a smooth function of Q and is well fitted by y(t)=(1+Au+Bu2)/(C+Du) where u=-(2/pi)cos(-1)(squareroot[Q]/2). For Q=3 we find y(t) approximately 0.6; however if we include lattices up to L=12 we find y(t) approximately 0.50(8) in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. 96, 461 (1999)].
The microcanonical transfer matrix and its extensions offer a new way of obtaining exact partition functions on finite two dimensional lattices. We show the density of the partition function zeros in the complex x- plane for the Ising model at y=ycy=y_cy=yc​ and y=0.5ycy=0.5y_cy=0.5yc​, and the density of the partition function zeros in the complex y-plane for the three-state Potts model.