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International Journal of Modern Physics B

GRAPHICAL SOLUTION OF THE ISING MODEL ON HONEYCOMB LATTICE

S. A. Jafari

Published Jan 30, 2009• 2 citations
Publisher

Abstract

In this work, we present a detailed graphical solution for the Ising model on the honeycomb lattice. In view of the mapping between the calculation of the partition function for generalizations of the Ising model and the calculus of resonating valence bond (RVB) states at zero temperature, our calculation may be of relevance to the RVB physics on the honeycomb lattice.

DOI
10.1142/s0217979209049620

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In this work, we present a detailed graphical solution for the Ising model on the honeycomb lattice. In view of the mapping between the calculation of the partition function for generalizations of the Ising model and the calculus of resonating valence bond (RVB) states at zero temperature, our calculation may be of relevance to the RVB physics on the honeycomb lattice.
A general technique is presented which allows site-bond disorder problems in Ising systems to be mapped onto a certain class of decorated lattice problems. The specific case of site-bond disorder on the honeycomb lattice is studied in detail through the use of annealed model solutions of the corresponding decoration problems. The critical temperature and magnetisation as functions of bond and site concentrations are calculated. The zero-temperature magnetisation is used to obtain an approximation to the percolocation probability of the system.
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 Ising model on the honeycomb lattice with spins of magnitudes 1/2 and s with random exchange constant and random uniaxial anisotropy is considered. The two cases are studied separately, and in both cases binary annealed distributions of the random parameters are considered. An exact solution is obtained by mapping the model onto the uniform spin 1/2 Ising model on the triangular lattice, and the transition temperature is determined as a function of the parameters for arbitrary s. Explicit results are presented for s=1 and s=3/2, and it is shown that some critical parameters, surprisingly, are not universal.
We study bond and spin correlations of the nearest-neighbor resonating valence bond (RVB) wave function for a SU(2) symmetric S=1/2S=1/2S=1/2 antiferromagnet on the honeycomb lattice. We find that spin correlations in this wave function are short ranged, while the bond energy correlation function takes on an oscillatory power-law form $D(\stackrel{P\vec}{r})\ensuremath{\sim}cos(\mathbf{Q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{r})/{|\stackrel{P\vec}{r}|}^{{\ensuremath{\eta}}_{w}(2)}$, where $\mathbf{Q}=(2\ensuremath{\pi}/3,\ensuremath{-}2\ensuremath{\pi}/3)$ is the wave vector corresponding to ``columnar'' valence-bond solid order on the honeycomb lattice, and ${\ensuremath{\eta}}_{w}(2)\ensuremath{\approx}1.49(3)$. We use a recently introduced large-ggg expansion approach to relate bond-energy correlators of the SU(g)\mathrm{SU}(g)SU(g) wave function to dimer correlations of an interacting fully packed dimer model with a three-dimer interaction of strength $V(g)=\ensuremath{-}ln(1+1/{g}^{2})$. Putting g=2g=2g=2, we find numerically that the dimer correlation function ${D}^{d}(\stackrel{P\vec}{r})$ of this dimer model has power-law behavior ${D}^{d}(\stackrel{P\vec}{r})\ensuremath{\sim}cos(\mathbf{Q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{r})/{|\stackrel{P\vec}{r}|}^{{\ensuremath{\eta}}_{d}(2)}$ with ${\ensuremath{\eta}}_{d}(2)\ensuremath{\approx}1.520(15)$, in rather good agreement with the wave function results. We also study the same quantities for g=3,4,10g=3,4,10g=3,4,10 and find that the bond-energy correlations in the SU(g)\mathrm{SU}(g)SU(g) wave function are consistently well reproduced by the corresponding dimer correlations in the interacting dimer model.
The mapping which produces a path between the p-state symmetric vertex model and the Ising model with spin [Formula: see text] on the honeycomb lattice is constructed. The mapping construction is based on a combination of decoration and gauge transformations. The formulation enables one to find the manifold in the vertex weights parameter space on which the mapping can be performed and to derive dual relations among the model parameters as well as the free energies of related symmetric vertex and Ising models. The extension of the mapping to arbitrary lattice coordination is outlined.