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Journal of Physics C Solid State Physics

Crossed Ising chains: a special decorated two-dimensional Ising model showing a transition to first-order behaviour

K Jungling, Gustav M. Obermair

Published Mar 7, 1975• 2 citations
Publisher

Abstract

A system of crossed Ising chains that can be interpreted as a multiply decorated square Ising lattice is considered and the local spontaneous magnetization which 'hangs through' between the crossing points is discussed. With growing chain distance the coupling constant along the decorated horizontal bonds is allowed to increase such that the critical temperature of the second-order transition remains constant. A narrowing of the critical region with increasing chain distance is observed; in the limit of infinite distance spontaneous magnetization, specific heat and susceptibility exhibit a first-order transition. Thus the model serves as an example for the continuous variation of the character of a phase transition from second to first kind as a function of the 'external parameter' chain distance.

DOI
10.1088/0022-3719/8/5/006

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A system of crossed Ising chains that can be interpreted as a multiply decorated square Ising lattice is considered and the local spontaneous magnetization which 'hangs through' between the crossing points is discussed. With growing chain distance the coupling constant along the decorated horizontal bonds is allowed to increase such that the critical temperature of the second-order transition remains constant. A narrowing of the critical region with increasing chain distance is observed; in the limit of infinite distance spontaneous magnetization, specific heat and susceptibility exhibit a first-order transition. Thus the model serves as an example for the continuous variation of the character of a phase transition from second to first kind as a function of the 'external parameter' chain distance.
The two-dimensional Ising model defined on square lattices with diamond-type bond-decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems. The model is studied analytically under the bond-renormalization scheme. For an n-level decorated lattice, the long-range ordering occurs at the critical temperature given by the fitting function (kBTc/J)n=1.6410+(0.6281)exp[−(0.5857)n], and the local ordering inside n-level decorated bonds occurs at the temperature given by the fitting function (kBTm/J)n=1.6410−(0.8063)exp[−(0.7144)n]. The critical amplitude Asing(n) of the logarithmic singularity in specific heat characterizes the width of the critical region, and it varies with the decoration-level n as Asing(n)=(0.2473)exp[−(0.3018)n], obtained by fitting the numerical results. The cross over from a finite-decorated system to an infinite-decorated system is not a smooth continuation. For the case of infinite decorations, the critical specific heat becomes a cusp with the height c(n)=0.639852. The results are compared with those obtained in the cell-decorated Ising model.
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 Ising linear chain with nearest-neighbor interactions of strength J s and infinite long-range ferromagnetic exponential interactions of strength J l was solved exactly in the presence of the external field. The spontaneous magnetization and the susceptibility are obtained. The transition temperature depends on J s and J l . For J s >0, the model shows the second order phase transion. The specific heat shows both a broad peak and a sharp peak. The value of sharp peak of the specific heat depends on J s and J l . For J s <0, the properties of the transition agree with those of Nagle model with long-range equivalent-neighbor ferromagnetic interactions. The exact free energy and the long-range order of our model are identical with those of the variational theory of a quasi-one-dimensional model.
The thermal phase transitions of a spin-1/2 Ising-Heisenberg model on the diamond-decorated square lattice in a magnetic field are investigated using a decoration-iteration transformation and classical Monte Carlo simulations. A generalized decoration-iteration transformation maps this model exactly onto an effective classical Ising model on the square lattice with temperature-dependent effective nearest-neighbor interactions and magnetic field strength. The effective field vanishes along a ground-state phase boundary of the original model, separating a ferrimagnetic and a quantum monomer-dimer phase. At finite temperatures this phase boundary gives rise to an exactly solvable surface of discontinuous (first-order) phase transitions, which terminates in a line of Ising critical points. The existence of discontinuous reentrant phase transitions within a narrow parameter regime is reported and explained in terms of the low-energy excitations from both phases. These exact results, obtained from the mapping to the zero-field effective Ising model, are corroborated by classical Monte Carlo simulations of the effective model.
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.
We explore the phase diagram of Ising spins on one-dimensional chains that criss-cross in two perpendicular directions and that are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian that has been proposed as a model of magnetic behavior in Nb_{12}O_{29} and also, conceptually, as a geometry that is intermediate between one and two dimensions. Using mean-field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit, we demonstrate nontrivial critical behavior, including tricritical points that separate first- and second-order phase transitions. Finally, we present evidence that this model belongs to the two-dimensional Ising universality class.