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Physica A Statistical Mechanics and its Applications

Star–triangle approach to boundary behavior in the two-dimensional Ising model

Theodore W. Burkhardt, Ihnsouk Guim

Published Mar 1, 1998• 2 citations
Publisher
DOI
10.1016/s0378-4371(97)00591-8

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We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance $m$ from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount ${\mathrm{Am}}^{\ensuremath{-}y}$ for large $m$ is examined in detail. For $y<1$ the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For $y=1$, $A>{A}_{c}$ (${A}_{c}$ being a positive critical value), and for $y<1$, $A>0$, there is a spontaneous surface magnetization at the bulk critical temperature.

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The solvable sl(n)sl(n)sl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model on the body centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly.
We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance mmm from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount ${\mathrm{Am}}^{\ensuremath{-}y}$ for large mmm is examined in detail. For y<1y<1y<1 the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For y=1y=1y=1, A>AcA>{A}_{c}A>Ac​ (Ac{A}_{c}Ac​ being a positive critical value), and for y<1y<1y<1, A>0A>0A>0, there is a spontaneous surface magnetization at the bulk critical temperature.