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Physical Review Letters

Partition Function Zeros of the Square Lattice Potts Model

Chi-Ning Chen, Chin‐Kun Hu, Fa Yueh Wu

Published Jan 8, 1996• 81 citations
Publisher

Abstract

We have evaluated numerically the zeros of the partition function of the $q$-state Potts model on the square lattice with reduced interactions $K$. 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 $\mathrm{Re}(x)>0$ 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 $|x|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.

DOI
10.1103/physrevlett.76.169

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arXiv:cond-mat/0102090•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 17, 2001•26 citations
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=(eK\ensuremath−1)/qx\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({e}^{K}\ensuremath{-}1)/\sqrt{q}x=(eK\ensuremath−1)/q​ plane are located on the unit circle ∣x∣=1|x|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1∣x∣=1.
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 exact integer values for the density of states of the Ising model on an L×L square lattice with self-dual boundary conditions are evaluated up to L = 22 for the first time by counting all possible spin states (enormous 2 485 ≈ 10 146 states for L = 22). The exact partition function zeros in the complex temperature plane, very sensitive indicators for phase transitions and critical phenomena, are obtained by using the density of states. From the behavior of the partition function zeros, the shift exponent λ of the square-lattice Ising model with self-dual boundary conditions is accurately and precisely estimated to be λ = 2.0000000(2), clearly indicating λ = 2.
The distributions of the partition function zeros in the complex a=e2betaJ1 plane of the square-lattice Ising model with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions are investigated as a function of R=J2/J1. Starting from the well-known two-circle distribution of the zeros a=+/-1+sqrt[2]eitheta for R=0 , finally the partition function zeros lie on the unit circle a=eitheta for R=infinity . Between these two ends, the changes in the zero distributions are described. Using the partition function zeros, the critical point ac(R) and the thermal scaling exponent yt(R) are estimated for the Ising ferromagnet (equivalently, antiferromagnet) and superantiferromagnet. For the special case of R=1/2 , the possible implications of the zero distributions are also discussed.
We report some results on the complex-temperature (CT) singularities of qqq-state Potts models on the square lattice. We concentrate on the problematic region Re(a)<0\mathrm{Re}(a)<0Re(a)<0 (where a=eKa={e}^{K}a=eK) in which CT zeros of the partition function are sensitive to finite lattice artifacts. From analyses of low-temperature series expansions for 3< q< 83<~q<~83< q< 8, we establish the existence, in this region, of complex-conjugate CT singularities at which the magnetization and susceptibility diverge. From calculations of zeros of the partition function, we obtain evidence consistent with the inference that these singularities occur at endpoints ae{a}_{e}ae​,ae∗{a}_{e}^{*}ae∗​ of arcs protruding into the (complex-temperature extension of the) ferromagnetic phase. Exponents for these singularities are determined; e.g., for q=3q=3q=3, we find ${\ensuremath{\beta}}_{e}=\ensuremath{-}0.125(1)$, consistent with ${\ensuremath{\beta}}_{e}=\ensuremath{-}\frac{1}{8}$. By duality, these results also imply associated arcs extending into the (CT extension of the) symmetric paramagnetic phase. Analytic expressions are suggested for the positions of some of these singularities; e.g., for q=5q=5q=5, our finding is consistent with the exact value ae{a}_{e}ae​, ${a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)$. Further discussions of complex-temperature phase diagrams are given.
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)].