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Communications in Theoretical Physics

The Critical Lines of the Yang–Lee Edge Singularities of the Anisotropic Ising Ferromagnets on Square, Triangular and Honeycomb Lattices

Xianzhi Wang

Published Mar 15, 2001• 2 citations
Publisher

Abstract

Using the numerical results and some results from the renormalization group theory, we extend our previous approach of determining the Yang–Lee edge singularities of Ising ferromagnets on square, triangular and honeycomb lattices (Phys. Rev. Lett. 78 (1997) 413; Phys. Rev. E56 (1998) 2793; E57 (1998) 5013) and obtain accurate closed-form approximations of the critical lines of anisotropic Ising ferromagnets on these lattices.

DOI
10.1088/0253-6102/35/3/365

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Using the numerical results and some results from the renormalization group theory, we extend our previous approach of determining the Yang–Lee edge singularities of Ising ferromagnets on square, triangular and honeycomb lattices (Phys. Rev. Lett. 78 (1997) 413; Phys. Rev. E56 (1998) 2793; E57 (1998) 5013) and obtain accurate closed-form approximations of the critical lines of anisotropic Ising ferromagnets on these lattices.
We extend our previous approach to determine the critical lines of the Yang-Lee edge singularity of Ising ferromagnets on square, triangular, and honeycomb lattices by considering the zeros of the Ising partition function on elementary cycles of these lattices. It is found that the critical lines have the properties ${h}_{0}\ensuremath{\rightarrow}{t}^{15/8}$ as $\stackrel{\ensuremath{\rightarrow}}{T}{T}_{c}+$ and $\ensuremath{\beta}{h}_{0}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ as $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty}$. Using the asymptotic formulas valid in the high-temperature limit and the zero-field limit we obtain the functional form of the critical line.
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 Ising ferromagnet, consisting of magnetic spins, is the simplest system showing phase transitions and critical phenomena at finite temperatures. The Ising ferromagnet has played a central role in our understanding of phase transitions and critical phenomena. Also, the Ising ferromagnet explains the gas-liquid phase transitions accurately. In particular, the Ising ferromagnet in a nonzero magnetic field has been one of the most intriguing and outstanding unsolved problems. We study analytically the partition function zeros in the complex magnetic-field plane and the Yang-Lee edge singularity of the infinite-range Ising ferromagnet in an external magnetic field. In addition, we compare the Yang-Lee edge singularity of the infinite-range Ising ferromagnet with that of the square-lattice Ising ferromagnet in an external magnetic field.
We have studied numerically the Lee-Yang singularities of the four dimensional Ising model at criticality, which is believed to be in the same universality class as the φ44 scalar field theory. We have focused in the numerical characterization of the logarithmic corrections to the scaling of the zeros of the partition function and its cumulative probability distribution, finding a very good agreement with the predictions of the renormalization group computation on the φ44 scalar field theory. To obtain these results, we have extended a previous study [R. Kenna, C. B. Lang, Nucl. Phys., 1993, B393, 461] in which there were computed numerically the first two zeros for L≤24 lattices, to the computation of the first four zeros for L≤64 lattices.