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Journal of Physics A Mathematical and General

An exactly solvable two-dimensional Ising model with magnetic field

P. Azaria, Héctor Giacomini

Published Oct 7, 1988• 14 citations
Publisher

Abstract

An Ising model formulated on a Kagome lattice with anisotropic ferromagnetic and antiferromagnetic interactions, and a magnetic field, is found to be exactly solvable for arbitrary values of temperature. The magnetic field acts on two of the three sublattices of the Kagome lattice. Explicit expressions for the partition function and the critical variety of the model are given.

DOI
10.1088/0305-4470/21/19/003

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An Ising model formulated on a Kagome lattice with anisotropic ferromagnetic and antiferromagnetic interactions, and a magnetic field, is found to be exactly solvable for arbitrary values of temperature. The magnetic field acts on two of the three sublattices of the Kagome lattice. Explicit expressions for the partition function and the critical variety of the model are given.
A two-dimensional Ising model on a particular topological structure, a ``triangles-in-triangles'' Kagom\'e structure, which is called a triangular-Kagom\'e lattice, is investigated. In this model, we consider two interaction parameters Jaa{J}_{aa}Jaa​ and Jab{J}_{ab}Jab​, corresponding to the interactions among spins on the inner sites and between spins on the inner and vertex sites of the triangular in the Kagom\'e lattice, respectively. By summing over all spins at inner sites in the partition function, we arrive at the partition function of the Kagom\'e lattice. The effective interaction of the corresponding Kagom\'e lattice is always ferromagnetic, even for antiferromagnetic Jaa{J}_{aa}Jaa​. The critical properties of the system depend only on ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$. When ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}>\ensuremath{-}1$, the system has long range order at low temperature. However, when ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$, the partition function of the triangular-Kagom\'e lattice can be related to that of the Kagom\'e lattice with effective interaction at a temperature higher than its critical temperature. Therefore, the system will be in paramagnetic phase at all temperatures for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$. The phase diagram for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$ is given exactly.
The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.
The interface method is used to derive the order-disorder critical temperature of the two-dimensional Ising model on a rectangular lattice with ferromagnetic interactions in the x direction and antiferromagnetic interactions in the y direction. Employing this approximation in two different ways gives different critical temperatures Tc{T}_{c}Tc​ for nonzero uniform magnetic fields H. However, both of the phase boundaries Tc{T}_{c}Tc​(H) reduce to the exact results available at H=0 and T=0. .AE
We investigate a generalized inhomogeneous two-dimensional Ising model on the kagomé lattice with two alternating couplings J i and J i ′ , i = 1 , 2 , 3 , along the three lattice directions. The local Boltzmann weights are mapped onto a non-symmetric eight-vertex R -matrix satisfying the free-fermion condition for arbitrary values of the six couplings. We analyze the Yang–Baxter structure for the cases J i ′ = J i and J i ′ = − J i and derive the corresponding one-dimensional quantum spin chains in the anisotropic limit. The first case yields a transverse-field Ising-type Hamiltonian, while the second leads to a modified chain with a graded permutation structure and shifted partition-function zeros. For a partially anisotropic model, we also obtain the free energy and specific heat.