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 $Z(Q,a)\mathrm{}$ is a polynomial in Q and $v=a-1\hspace{0.5em}{(a=e}^{\beta J})$ and the coefficients of this polynomial, $\Phi \mathrm{}(b,c),$ 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 $\Phi \mathrm{}(b,c)\mathrm{}$ 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 $Z_{\mathrm{CP}}=Z\left({Q,a}_c\mathrm{}\right(Q\left)\mathrm{}\right),$ where $a_c\mathrm{}\left(Q\right)=1+\sqrt{Q}.$ We find a set of zeros in the complex $w=\sqrt{Q}$ plane that map to (or close to) the Beraha numbers for real positive Q. We also identify $Q_c\mathrm{}\left(L\right),$ the value of Q for a lattice of width L above which the locus of zeros in the complex $p=v/\sqrt{Q}$ plane lies on the unit circle. By finite-size scaling we find that $1/Q_c\mathrm{}\left(L\right)\mathrm{}\to 0$ as $\overrightarrow{L}\infty .$ We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine $Q_c\mathrm{}\left(a\right),$ 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 $Q_c^{\mathrm{AF}}\mathrm{}\left(a\right)=\left(1\mathrm{}-a\right)(a+3\mathrm{}).$ We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that $Q_c^{\mathrm{FM}}\mathrm{}\left(a\right)=(a-{1)}^2.$ We show that the edge singularity in the complex Q plane approaches $Q_c$ as $Q_c\mathrm{}\left(L\right)\mathrm{}\sim Q_c{+AL}^{-y_q},$ 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 $y_t{=(1+Au+Bu}^2)/(C+Du)\mathrm{}$ where $u=-(2/\pi ){\cos }^{-1}(\sqrt{Q}/2).\mathrm{}$ For $Q=3\mathrm{}$ we find $y_t\simeq 0.6;$ however if we include lattices up to $L=12\mathrm{}$ we find $y_t\simeq 0.50\left(8\right)\mathrm{}$ in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. 96, 461 (1999)].

• Received 18 May 2000

DOI:https://doi.org/10.1103/PhysRevE.63.066107