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Computer Physics Communications

Exact enumeration of the ground states of a triangular Ising antiferromagnet

Nicholas D. Blakeley, Rick P. Millane

Published Nov 12, 2005• 4 citations
Publisher
DOI
10.1016/j.cpc.2005.10.004

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We evaluate the density of states g ( M , E ) as a function of energy E and magnetization M of Ising models on square and triangular lattices, using the exact enumeration method for small systems and the Wang–Landau method for larger systems. From the density of states the average magnetization per spin, m ( T , h ), of the antiferromagnets has been obtained for any values of temperature T and uniform magnetic field h . Also, based on g ( M , E ), the behaviour of m ( T , h ) is understood microcanonically. The microcanonical approach reveals the differences between the unfrustrated model (on the square lattice) and the frustrated one (on the triangular lattice).
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.