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arXiv:cond-mat/0102090 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Density of states, Potts zeros, and Fisher zeros of theQ-state Potts model for continuousQ

Seung‐Yeon Kim, Richard J. Creswick

Published May 17, 2001• 26 citations
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Abstract

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)].

DOI
10.1103/physreve.63.066107

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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 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 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 $L\ifmmode\times\else\texttimes\fi{}L$ 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.
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 ${\ensuremath{\alpha}}_{e}+2{\ensuremath{\beta}}_{e}+{\ensuremath{\gamma}}_{e}=2.$
A new algorithm is presented, which allows us to calculate numerically the partition function Z for systems, which can be described by arbitrary interaction graphs and lattices, e.g., Ising models or Potts models (for arbitrary values q>0), including random or diluted models. The new approach is suitable for large systems. The basic idea is to measure the distribution of the number of connected components in the corresponding Fortuin-Kasteleyn representation and to compare with the case of zero degrees of freedom, where the exact result Z=1 is known. As an application, d=2 and d=3 dimensional ferromagnetic Potts models are studied, and the critical values qc, where the transition changes from second to first order, are determined. Large systems of sizes N=1000(2) and N=100(3) are treated. The critical value qc(d=2)=4 is confirmed and qc(d=3)=2.35(5) is found.
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.
We present exact solutions for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial P)P)P) on tube sections of the simple cubic lattice of fixed transverse size ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ and arbitrarily great length Lz,{L}_{z},Lz​, for sizes ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}=2\ifmmode\times\else\texttimes\fi{}3$ and $2\ifmmode\times\else\texttimes\fi{}4$ and boundary conditions (a) (FBCx,FBCy,FBCz)({\mathrm{FBC}}_{x},{\mathrm{FBC}}_{y},{\mathrm{FBC}}_{z})(FBCx​,FBCy​,FBCz​) and (b) (PBCx,FBCy,FBCz),({\mathrm{PBC}}_{x},{\mathrm{FBC}}_{y},{\mathrm{FBC}}_{z}),(PBCx​,FBCy​,FBCz​), where FBC (PBC) denote free (periodic) boundary conditions. In the limit of infinite length, ${L}_{z}\ensuremath{\rightarrow}\ensuremath{\infty},$ we calculate the resultant ground-state degeneracy per site W (=exponent of the ground-state entropy). Generalizing q from Z+{Z}_{+}Z+​ to C,C,C, we determine the analytic structure of W and the related singular locus B\mathcal{B}B which is the continuous accumulation set of zeros of the chromatic polynomial. For the ${L}_{z}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of a given family of lattice sections, W is analytic for real q down to a value qc.{q}_{c}.qc​. We determine the values of qc{q}_{c}qc​ for the lattice sections considered and address the question of the value of qc{q}_{c}qc​ for a d-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph Km,m.{K}_{m,m}.Km,m​.