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Physical Review Letters

The Critical Line of an Ising Antiferromagnet on Square and Honeycomb Lattices

Xianzhi Wang, Jai Sam Kim

Published Jan 20, 1997• 44 citations
Publisher

Abstract

We show that the singularity of the free energy of Ising models in the absence of a magnetic field on the triangular, square, and honeycomb lattices is related to zeros of the pseudopartition function on an elementary cycle. Using the Griffiths' smoothness postulate, we extend these results to the case in a magnetic field and derive a formula of the critical line of an Ising antiferromagnet, which is in good agreement with the numerical results.

DOI
10.1103/physrevlett.78.413

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Journal of the Korean Physical Society•Aug 1, 2020•4 citations
We show that the singularity of the free energy of Ising models in the absence of a magnetic field on the triangular, square, and honeycomb lattices is related to zeros of the pseudopartition function on an elementary cycle. Using the Griffiths' smoothness postulate, we extend these results to the case in a magnetic field and derive a formula of the critical line of an Ising antiferromagnet, which is in good agreement with the numerical results.
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.
For the two-dimensional ferromagnetic Ising critical point, I show that the known values of the critical exponents imply the absence of logarithms of the reduced temperature in the leading contributions to any field derivative of the free energy at zero magnetic field. For the square-lattice Ising antiferromagnet in a weak magnetic field, I compute the critical line Tc{T}_{c}Tc​(H)=Tc0{T}_{c}^{0}Tc0​(1-0.038 023 259H2{H}^{2}H2) and the leading contribution to the susceptibility \ensuremath{\chi}=0.014 718 006 6H2{H}^{2}H2ln(1/\ensuremath{\Vert}t\ensuremath{\Vert}), where t is the reduced temperature.
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.