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Journal of Physics A Mathematical and General

Decorated Ising model in a magnetic field

L.L. Gonçalves, Tomoya Horiguchi

Published Jun 1, 1986• 10 citations
Publisher

Abstract

A study of the two-dimensional Ising model on a rectangular lattice decorated with v-dimensional classical vector spins or Ising spins of magnitude s in an external field is presented. Exact expressions are obtained for the magnetisation and correlation functions, and the phase diagram is also determined exactly. It is shown that when there is no direct interaction between the spins of the host lattice there is a critical field h c , above which the system does not present long range order, and that this h c does not depend on v or s. The isothermal susceptibility related to the decorating spins is also obtained exactly, and it is shown that it diverges logarithmically at the transition temperature only for non-vanishing external field. A discussion is presented on the solution of equivalent models, and as a by-product it is shown how to map the model onto the two-dimensional eight-vertex model with temperature dependent parameters in an arbitrary staggered field.

DOI
10.1088/0305-4470/19/8/025

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A study of the two-dimensional Ising model on a rectangular lattice decorated with v-dimensional classical vector spins or Ising spins of magnitude s in an external field is presented. Exact expressions are obtained for the magnetisation and correlation functions, and the phase diagram is also determined exactly. It is shown that when there is no direct interaction between the spins of the host lattice there is a critical field h c , above which the system does not present long range order, and that this h c does not depend on v or s. The isothermal susceptibility related to the decorating spins is also obtained exactly, and it is shown that it diverges logarithmically at the transition temperature only for non-vanishing external field. A discussion is presented on the solution of equivalent models, and as a by-product it is shown how to map the model onto the two-dimensional eight-vertex model with temperature dependent parameters in an arbitrary staggered field.
The critical behavior of the two-dimensional Ising model (square lattice, exchange constant J) in an uniform field, and in an annealed random field is considered. The random field is generated by decorating the horizontal and vertical bonds of the lattice, and it satisfies an arbitrary distribution which is imposed by introducing a pseudo-chemical potential. By decimating the decorating variables the model can be mapped onto a homogeneous Ising model with effective exchange constant J′ and effective external field h′, dependent on the temperature. These parameters, which satisfy a set of coupled equations, depend on the spin average and nearest-neighbor two-spin correlation, and are obtained numerically. For the symmetric field distribution [Formula: see text] the mapping of the critical frontier on the (K′=βJ′,H′=βh′) plane onto the (K=β J,H=βh) plane is determined and, as in the model introduced by Essam and Place, there is a region on the (K, H) plane which cannot be reached from any real values of (K′, H′). The critical exponents are determined numerically, and it is shown that they do not satisfy renormalization relations obtained for their model.
Abstract The Kramers–Wannier transfer matrix method is generalized to an arbitrary decoration number of the Ising chain. An exact analytical expression is obtained for the largest eigenvalue of the transfer matrix of a decorated Ising chain in the presence of an external magnetic field. Frustration points and the values of frustration magnetic fields are found that depend on the magnitudes and signs of exchange interactions. Exact expressions are obtained for zero-temperature entropies and zero-temperature magnetizations of the model under consideration. Magnetic phase diagrams of the ground state of the system are constructed for decoration values of d = 1 and d = 2, including those in the absence of a magnetic field. A comparison is made with a decorated square lattice not only in the absence but also in the presence of a magnetic field.
We study a two-dimensional XXZ -Ising model on a square-hexagon (denoted for simplicity by 4-6) lattice with spin 1/2. The phase diagram at zero temperature is discussed, where five states are found, two types of ferrimagnetic states, two types of antiferromagnetic states, and one ferromagnetic state. To solve this model, we have mapped onto the eight-vertex model with union Jack interaction term, and it was verified that the model cannot be completely mapped onto eight-vertex model. However, by imposing an exact solution condition, we have found the region where the XXZ -Ising model on 4-6 lattice is exactly soluble with one free parameter, particularly for the case of symmetric eight-vertex model condition. In this manner we have explored the properties of the system and have analyzed the interacting competition parameters which preserve the region where there is an exact solution. Unfortunately the present model does not satisfy the free fermion condition of the eight-vertex model, unless for a trivial solution. Even so, we are able to discuss the critical point region, beyond the region of exact resolvability.
The effect of a discrete distribution of annealed random fields on the properties of the two-dimensional Ising model is studied. The random field is allowed to assume the values 0,\ifmmode\pm\else\textpm\fi{}h with the probabilities p0{p}_{0}p0​,${p}_{\ifmmode\pm\else\textpm\fi{}}$, and a parameter \ensuremath{\mu} is introduced which is determined by specifying p0{p}_{0}p0​. Exact results are obtained by mapping the model onto an effective Ising model on the rectangular lattice without field, and the probabilities p0{p}_{0}p0​,${p}_{\ifmmode\pm\else\textpm\fi{}}$ are expressed in terms of the magnetization and two-spin correlations of the effective model. It is shown that the long-range order is not destroyed by the random field and the transition temperature increases as p0{p}_{0}p0​ decreases. For a given p0{p}_{0}p0​ the critical temperature increases with the field strength and these combined effects show that the annealed random field favors the ordering of the system, therefore presenting behavior opposite to that expected for the system in a quenched random field.
. A two-dimensional Ising model with nearest neighbour exchange interactions among electrons, in the field of a finite intensity, is investigated. By including interactions along a given direction into the structure of the transfer matrix, we pass over from the direct lattice space to the pseudo-spin space. A field dependent unitary transformation in the pseudo-spin space is applied in order to evaluate the partition function exactly. There are two important consequences of the general field dependence of the partition function: (a) The order parameter undergoes a finite discontinuity at the transition temperature; and (b) The induced susceptibility discovers a logarithmic singularity just below the transition temperature. Neither consequence seems to conform with an existing theory of a two-dimensional ferromagnet.
A hybrid lattice-statistical model of doubly decorated two-dimensional lattices, which have localized Ising spins at its nodal sites and itinerant electrons delocalized over decorating sites, is exactly solved with the help of a generalized decoration-iteration transformation. Under the assumption of a quarter filling of each couple of the decorating sites, the ground state constitutes either spontaneously long-range ordered ferromagnetic or ferrimagnetic phase in dependence on whether the ferromagnetic or antiferromagnetic interaction between the localized Ising spins and itinerant electrons is considered. The critical temperature of the spontaneously long-range ordered phases monotonically increases upon strengthening the ratio between the kinetic term and the Ising-type exchange interaction.