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)].
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
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∣=1.
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.
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
Hq, 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>2 Potts models in the complex temperature plane with nonzero external magnetic field
Hq. Unlike the Ising model for ${H}_{q}\ensuremath{\ne}0,$ which has only a nonphysical critical point (the Fisher edge singularity), the
Q>2 Potts models have physical critical points for
Hq<0 as well as the Fisher edge singularities for
Hq>0. For
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, 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, 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 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.