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Physical Review E

Partition function zeros of the honeycomb-lattice Ising antiferromagnet in the complex magnetic-field plane

Seung‐Yeon Kim

Published Oct 11, 2010• 7 citations
Publisher

Abstract

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 $(L=6,8,\dots{},14)$ and (due to memory limitations) the exact real values for $\overline{\ensuremath{\Omega}}(a,M)={\ensuremath{\sum}}_{E}\ensuremath{\Omega}(E,M){a}^{E}$ $(L=16,\dots{},22)$, 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 ${x}_{cp}(a)$ and the magnetic scaling exponents ${y}_{h}(a)$ 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.

DOI
10.1103/physreve.82.041107

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The microcanonical transfer matrix is used to evaluate the exact integer values for the number of states \ensuremathΩ(E,M)\ensuremath{\Omega}(E,M)\ensuremathΩ(E,M), as a function of energy EEE and magnetization MMM, of the Ising model in an external magnetic field HHH on L\ifmmode×\else\texttimes\fi2LL\ifmmode\times\else\texttimes\fi{}2LL\ifmmode×\else\texttimes\fi2L honeycomb lattices (L=6,8,…,14)(L=6,8,\dots{},14)(L=6,8,…,14) and (due to memory limitations) the exact real values for \ensuremathΩ‾(a,M)=\ensuremath∑E\ensuremathΩ(E,M)aE\overline{\ensuremath{\Omega}}(a,M)={\ensuremath{\sum}}_{E}\ensuremath{\Omega}(E,M){a}^{E}\ensuremathΩ​(a,M)=\ensuremath∑E​\ensuremathΩ(E,M)aE (L=16,…,22)(L=16,\dots{},22)(L=16,…,22), where a=e2\ensuremathβJa={e}^{2\ensuremath{\beta}J}a=e2\ensuremathβJ. Given \ensuremathΩ(E,M)\ensuremath{\Omega}(E,M)\ensuremathΩ(E,M) or \ensuremathΩ‾(a,M)\overline{\ensuremath{\Omega}}(a,M)\ensuremathΩ​(a,M), the exact partition functions Z(a,x)=\ensuremath∑E,M\ensuremathΩ(E,M)aExM=\ensuremath∑M\ensuremathΩ‾(a,M)xMZ(a,x)={\ensuremath{\sum}}_{E,M}\ensuremath{\Omega}(E,M){a}^{E}{x}^{M}={\ensuremath{\sum}}_{M}\overline{\ensuremath{\Omega}}(a,M){x}^{M}Z(a,x)=\ensuremath∑E,M​\ensuremathΩ(E,M)aExM=\ensuremath∑M​\ensuremathΩ​(a,M)xM, where x=e\ensuremath−2\ensuremathβHx={e}^{\ensuremath{-}2\ensuremath{\beta}H}x=e\ensuremath−2\ensuremathβH, are obtained. From the exact partition functions, the precise distributions of the partition function zeros in the complex xxx 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){x}_{cp}(a)xcp​(a) and the magnetic scaling exponents yh(a){y}_{h}(a)yh​(a) of the honeycomb-lattice Ising antiferromagnet in a uniform magnetic field are estimated using its partition function zeros in the complex xxx plane. Our results for the critical points are also compared to those of the two different closed-form approximations.
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.
From the exact partition functions of the Ising model on L x L square lattices (up to L = 14) in an arbitrary nonzero external magnetic field H, the precise distributions of the Fisher zeros in the complex temperature (a = e(2betaJ)) plane are obtained. Our results indicate that in the limit L-->infinity the Fisher zeros for H not = 0 do cut the positive real axis at the antiferromagnetic critical point ac(H); these values are compared with the results of closed-form approximations for the antiferromagnetic critical line. From the Fisher zeros the thermal scaling exponent yt = 1 is also obtained along the critical line, indicating the logarithmic singularity of the specific heat for the antiferromagnetic Ising model even in a strong uniform magnetic field.
The exact partition functions of the Ising model on L × L square lattices with free boundary conditions are evaluated up to L = 22 for the first time by using a microcanonical transfer matrix and counting all possible spin states. The total number of states is 2 22×22 = 2 484 ≈ 5.0 × 10 145 for L = 22. Using the partition function zeros in the complex temperature plane of the Ising model on L × L square lattices with free boundary conditions, we analyze the critical behavior of the square-lattice Ising antiferromagnet and compare them with the results for periodic boundary conditions (the most popular boundary conditions). Our results show clearly that free boundary conditions are much more efficient than periodic boundary conditions in the study of an antiferromagnet.
The microcanonical transfer matrix is used to study the distribution of the Fisher zeros of the Q>2Q>2Q>2 Potts models in the complex temperature plane with nonzero external magnetic field Hq.{H}_{q}.Hq​. Unlike the Ising model for ${H}_{q}\ensuremath{\ne}0,$ which has only a nonphysical critical point (the Fisher edge singularity), the Q>2Q>2Q>2 Potts models have physical critical points for Hq<0{H}_{q}<0Hq​<0 as well as the Fisher edge singularities for Hq>0.{H}_{q}>0.Hq​>0. For Hq<0,{H}_{q}<0,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,{H}_{q}>0,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< 6,3<~Q<~6,3< 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 Ising model in an external magnetic field is one of the most outstanding and intriguing unsolved problems. One of the important properties of the Ising model in an external magnetic field is the Yang-Lee edge singularity. The zeros in the complex magnetic-field plane of the grand partition function of the Ising model on a honeycomb lattice in an external magnetic field are exactly calculated by using the exact integer values for the density of states on L × 2L honeycomb lattices (up to L = 14). The critical line of the Yang-Lee edge singularity of the Ising model on a honeycomb lattice in an external magnetic field is accurately investigated from its Yang-Lee zeros.
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.