There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).
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 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.
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 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.
We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable
u=e−4K, where
K=J/(kBT), for nonzero external magnetic field,
H. Exact results are given for the phase diagram in the
u plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real
h=H/(kBT), these results provide new insights into features of our earlier study of this case. We also consider complex
h=H/(kBT) and
μ=e−2h. Calculations of complex-
u zeros of the partition function on sections of the square lattice are presented. For the case of imaginary
h, i.e.,
μ=eiθ, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the
u plane. We find that in this case, the phase boundary
Bu contains a real line segment extending through part of the physical ferromagnetic interval
0≤u≤1, with a right-hand endpoint
urhe at the temperature for which the Yang-Lee edge singularity occurs at
μ=e±iθ. Conformal field theory arguments are used to relate the singularities at
urhe and the Yang-Lee edge.