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Physical Review E

Density of the Fisher zeros for the three-state and four-state Potts models

Seung‐Yeon Kim

Published Jul 8, 2004• 11 citations
Publisher

Abstract

The numbers of states up to $L=12$ for the three-state Potts model and up to $L=10$ for the four-state Potts model on $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with self-dual boundary condition are enumerated using the microcanonical transfer matrix exploiting the permutation symmetry of the model. From these numbers of states, the densities of the Fisher zeros $g(\ensuremath{\theta})$ of the partition function in the complex $u=(1+(Q\ensuremath{-}1){e}^{\ensuremath{-}\ensuremath{\beta}J})∕\sqrt{Q}$ plane are determined for the zeros on the unit circle ${u}_{0}={e}^{i\ensuremath{\theta}}$. For small $\ensuremath{\theta}$ the density of zeros obeys the finite-size scaling, allowing us to estimate the order parameter and the thermal exponent.

DOI
10.1103/physreve.70.016110

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The numbers of states up to L=12L=12L=12 for the three-state Potts model and up to L=10L=10L=10 for the four-state Potts model on L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL square lattice with self-dual boundary condition are enumerated using the microcanonical transfer matrix exploiting the permutation symmetry of the model. From these numbers of states, the densities of the Fisher zeros g(\ensuremathθ)g(\ensuremath{\theta})g(\ensuremathθ) of the partition function in the complex u=(1+(Q\ensuremath−1)e\ensuremath−\ensuremathβJ)∕Qu=(1+(Q\ensuremath{-}1){e}^{\ensuremath{-}\ensuremath{\beta}J})∕\sqrt{Q}u=(1+(Q\ensuremath−1)e\ensuremath−\ensuremathβJ)∕Q​ plane are determined for the zeros on the unit circle u0=ei\ensuremathθ{u}_{0}={e}^{i\ensuremath{\theta}}u0​=ei\ensuremathθ. For small \ensuremathθ\ensuremath{\theta}\ensuremathθ the density of zeros obeys the finite-size scaling, allowing us to estimate the order parameter and the thermal exponent.
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.
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 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 ${\ensuremath{\alpha}}_{e}+2{\ensuremath{\beta}}_{e}+{\ensuremath{\gamma}}_{e}=2.$
We have evaluated numerically the zeros of the partition function of the qqq-state Potts model on the square lattice with reduced interactions KKK. On the basis of our numerical results, we conjecture that, both for finite planar self-dual lattices and for lattices with free or periodic boundary conditions in the thermodynamic limit, the zeros in the Re(x)>0\mathrm{Re}(x)>0Re(x)>0 region of the complex $x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({e}^{K}\ensuremath{-}1)/\sqrt{q}$ plane are located on the unit circle ∣x∣=1|x|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1∣x∣=1.
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.
The numerical transfer matrix for the partition function of discrete lattice models is generalized to allow the calculation of the density of states \ensuremath{\Omega}(E), and the restricted density of states \ensuremath{\Omega}(E,M). Given \ensuremath{\Omega}(E,M) the partition function is expressed as a polynomial in the variables x=${\mathit{e}}^{\mathrm{\ensuremath{\beta}}\mathit{h}}$ and y=${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$. These algorithms are illustrated with calculations for the Ising model on finite square lattices. The zeros of the partition function are examined in both the complex x and y planes. Finite size scaling analysis of the zeros leads to very accurate estimates for the critical temperature and critical exponents.