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Journal of the Korean Physical Society

Partition Function Zeros and Shift Exponent of the Ising Model on a Square Lattice with Self-dual Boundary Conditions

Seung‐Yeon Kim

Published Mar 1, 2018• 3 citations
Publisher

Abstract

The exact integer values for the density of states of the Ising model on an L×L square lattice with self-dual boundary conditions are evaluated up to L = 22 for the first time by counting all possible spin states (enormous 2 485 ≈ 10 146 states for L = 22). The exact partition function zeros in the complex temperature plane, very sensitive indicators for phase transitions and critical phenomena, are obtained by using the density of states. From the behavior of the partition function zeros, the shift exponent λ of the square-lattice Ising model with self-dual boundary conditions is accurately and precisely estimated to be λ = 2.0000000(2), clearly indicating λ = 2.

DOI
10.3938/jkps.72.646

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The exact integer values for the density of states of the Ising model on an L×L square lattice with self-dual boundary conditions are evaluated up to L = 22 for the first time by counting all possible spin states (enormous 2 485 ≈ 10 146 states for L = 22). The exact partition function zeros in the complex temperature plane, very sensitive indicators for phase transitions and critical phenomena, are obtained by using the density of states. From the behavior of the partition function zeros, the shift exponent λ of the square-lattice Ising model with self-dual boundary conditions is accurately and precisely estimated to be λ = 2.0000000(2), clearly indicating λ = 2.
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 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.
The exact integer values for the density of states of the Ising model on an equilateral triangular lattice with free boundary conditions are evaluated up to L = 24 spins on a side for the first time by using the microcanonical transfer matrix. The total number of states is 2 N s = 2300 ≈ 2.037 × 1090 for L = 24, where N s = L(L+1)/2 is the number of spins. Classifying all 2300 spin states according to their energy values is an enormous work. From the density of states, the exact partition function zeros in the complex temperature plane of the triangular-lattice Ising model are evaluated. Using the density of states and the partition function zeros, we investigate the properties of the triangularlattice Ising antiferromagnet. The scaling behavior of the ground-state entropy and the form of the correlation length at T = 0 are studied for the triangular-lattice Ising antiferromagnet with free boundary conditions. Also, the scaling behavior of the Fisher edge singularity is investigated.
We have evaluated numerically the zeros of the partition function of the qqq-state Potts model on the square lattice with reduced interactions KKK. On the basis of our numerical results, we conjecture that, both for finite planar self-dual lattices and for lattices with free or periodic boundary conditions in the thermodynamic limit, the zeros in the Re(x)>0\mathrm{Re}(x)>0Re(x)>0 region of the complex $x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({e}^{K}\ensuremath{-}1)/\sqrt{q}$ plane are located on the unit circle ∣x∣=1|x|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1∣x∣=1.