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Journal of the Korean Physical Society

Ising antiferromagnet on a finite triangular lattice with free boundary conditions

Seung‐Yeon Kim

Published Nov 1, 2015• 8 citations
Publisher

Abstract

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.

DOI
10.3938/jkps.67.1517

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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.
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.
Exact analytical and numerical results for the site-diluted antiferromagnetic Ising model on the triangular lattice (AFIT) are presented. For infinitesimal dilution the change in the free energy of the system is related to the distribution of local fields, and it is shown that for a frustrated system such as the AFIT, dilution lowers the entropy per spin. For lattices of finite size and dilution the transfer matrix for the partition function is evaluated numerically. The entropy per spin shows a marked minimum near a concentration of spins x=0.70, in some disagreement with earlier transfer-matrix results.
The properties of the antiferromagnetic Ising model in the critical magnetic field are investigated by evaluating exactly its all possible ground states on L × L square lattice (L = 4∼16) with free boundary conditions for the first time. From the exact integer values for the number of ground states Ωc(L) on L × L square lattice, the exact ground-state entropy (in unit of the Boltzmann constant kB) per volume sc(L) = L−2lnΩc(L) and the exact ground-state magnetization mc(L;T = 0) = L−2〈Σiσi〉 are calculated in the critical magnetic field, where σi is the magnetic spin on a lattice site i. The values of the ground-state entropy sc(L) and the ground-state magnetization mc(L), in the limit L → ∞, are accurately estimated to be sc/kB = 0.40750±0.00108 and mc = 0.54696±0.00065. In addition, for the first time, we estimate the values of the scaling exponents ωs and ωm, controlling the finite-size scaling behaviors of the ground-state entropy and the ground-state magnetization as sc−sc(L)∼L−ωs and mc−mc(L)∼L−ωm respectively. Our results show clearly that ωs = ωm = 1.
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.