The microcanonical transfer matrix is used to study the distribution of Yang-Lee zeros of the
Q-state Potts model in the complex magnetic field ( ${x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}e}^{\ensuremath{\beta}h}$) plane for the first time. Finite size scaling suggests that at (and below) the critical temperature the zeros lie close to, but not on, the unit circle with the two exceptions of the critical point
x=1 (
h=0) itself and the zeros in the limit
T=0.
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 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.
Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.
The microcanonical transfer matrix is used to evaluate the exact integer values for the number of states $\ensuremath{\Omega}(E,M)$, as a function of energy
E and magnetization
M, of the Ising model in an external magnetic field
H on $L\ifmmode\times\else\texttimes\fi{}2L$ honeycomb lattices
(L=6,8,…,14) and (due to memory limitations) the exact real values for $\overline{\ensuremath{\Omega}}(a,M)={\ensuremath{\sum}}_{E}\ensuremath{\Omega}(E,M){a}^{E}$
(L=16,…,22), where $a={e}^{2\ensuremath{\beta}J}$. Given $\ensuremath{\Omega}(E,M)$ or $\overline{\ensuremath{\Omega}}(a,M)$, the exact partition functions $Z(a,x)={\ensuremath{\sum}}_{E,M}\ensuremath{\Omega}(E,M){a}^{E}{x}^{M}={\ensuremath{\sum}}_{M}\overline{\ensuremath{\Omega}}(a,M){x}^{M}$, where $x={e}^{\ensuremath{-}2\ensuremath{\beta}H}$, are obtained. From the exact partition functions, the precise distributions of the partition function zeros in the complex
x plane of the honeycomb-lattice Ising antiferromagnet in a uniform external magnetic field, where the famous circle theorem for the partition function zeros of the Ising ferromagnet is not applicable, are determined. In addition, the critical points
xcp(a) and the magnetic scaling exponents
yh(a) of the honeycomb-lattice Ising antiferromagnet in a uniform magnetic field are estimated using its partition function zeros in the complex
x plane. Our results for the critical points are also compared to those of the two different closed-form approximations.
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)].