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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical lines of the Yang-Lee edge singularity of Ising ferromagnets on square, triangular, and honeycomb lattices

Xianzhi Wang, Jai Sam Kim

Published May 1, 1998• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.57.5013

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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 h0\ensuremath→t15/8{h}_{0}\ensuremath{\rightarrow}{t}^{15/8}h0​\ensuremath→t15/8 as T\ensuremath→Tc+\stackrel{\ensuremath{\rightarrow}}{T}{T}_{c}+T\ensuremath→​Tc​+ and \ensuremathβh0\ensuremath→\ensuremathπ/2\ensuremath{\beta}{h}_{0}\ensuremath{\rightarrow}\ensuremath{\pi}/2\ensuremathβh0​\ensuremath→\ensuremathπ/2 as T\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty}T\ensuremath→​\ensuremath∞. 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 densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of 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.