We report some results on the complex-temperature (CT) singularities of
q-state Potts models on the square lattice. We concentrate on the problematic region
Re(a)<0 (where
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< 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,
ae∗ of arcs protruding into the (complex-temperature extension of the) ferromagnetic phase. Exponents for these singularities are determined; e.g., for
q=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=5, our finding is consistent with the exact value
ae, ${a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)$. Further discussions of complex-temperature phase diagrams are given.
The spin-one Ising model on the honeycomb lattice has never been solved exactly in spite of its simplicity. Even its exact critical temperature is not known. The exact integer values for the density of states of the spin-one Ising model on the L×2L honeycomb lattice are enumerated up to L=14. The partition function zeros in the complex temperature plane of the spin-one Ising model on the L×2L honeycomb lattice are exactly obtained, using the density of states. The properties of the partition function zeros in the complex temperature plane are related to the behaviors of various thermodynamic functions, in particular, their singular behaviors. The unknown properties of the spin-one Ising model on the honeycomb lattice are investigated, based on its partition function zeros in the complex temperature plane.
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.
We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field H, i.e., for 0\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}\ensuremath{\infty}, where \ensuremath{\mu}=${\mathit{e}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\beta}}\mathit{H}}$. We also carry out a similar analysis for -\ensuremath{\infty}\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}0. The results for the interval -1\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1 provide a way of continuously connecting the two known exact solutions of this model, viz., at \ensuremath{\mu}=1 (Onsager, Yang) and \ensuremath{\mu}=-1 (Lee and Yang). Our methods include numerical calculations of complex-temperature zeros of the partition function and an analysis of low-temperature series expansions. For real nonzero H, the inner branch of a lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For \ensuremath{\mu}0, there are two line segments of singularities on the negative and positive u axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at \ensuremath{\mu}=-1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams. \textcopyright{} 1996 The American Physical Society.
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.