A Theory of Cooperative Phenomena. III. Detailed Discussions of the Cluster Variation Method

Kurata, Michio; Kikuchi, Ryoichi; Watari, Tatsuro

doi:10.1063/1.1698926

Cited by 237 publications

(50 citation statements)

References 15 publications

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“…Therefore the betweenness centrality is smaller for vertices in more outer layers; more important vertices have smaller populations. Inhomogeneity has an influence on the type of phase transition; whereas the Ising model on a BL undergoes a second-order transition [10][11][12], that on a CT undergoes an infinite-order transition [13][14][15]. Similar behavior is observed in the bond-percolation transitions on a CT and a BL [3], which will be reported in detail in the following sections.…”

Section: Introductionmentioning

confidence: 67%

Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree

Nogawa¹,

Hasegawa²,

Nemoto³

2016

J. Stat. Mech.

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Abstract. We study bond percolation of the Cayley tree (CT) by focusing on the probability distribution function (PDF) of a local variable, namely, the size of the cluster including a selected vertex. Because the CT does not have a dominant bulk region, which is free from the boundary effect, even in the large-size limit, the phase of the system on it is not well defined. We herein show that local observation is useful to define the phase of such a system in association with the well-defined phase of the system on the Bethe lattice, that is, an infinite regular tree without boundary. Above the percolation threshold, the PDFs of the vertex at the center of the CT (the origin) and of the vertices near the boundary of the CT (the leaves) have different forms, which are also dissimilar to the PDF observed in the ordinary percolating phase of a Euclidean lattice. The PDF for the origin of the CT is bimodal: a decaying exponential function and a system-size-dependent asymmetric peak, which obeys a finite-size-scaling law with a fractal exponent. These modes are respectively related to the PDFs of the finite and infinite clusters in the nonuniqueness phase of the Bethe lattice. On the other hand, the PDF for the leaf of the CT is a decaying power function. This is similar to the PDF observed at a critical point of a Euclidean lattice but is attributed to the nesting structure of the CT around the boundary.

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Section: Introductionmentioning

confidence: 67%

Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree

Nogawa¹,

Hasegawa²,

Nemoto³

2016

J. Stat. Mech.

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“…Two new functions Q 1 and Q 2 entering the exact formula (19) for the sublattice magnetization of the Heisenberg spins are defined as follows can be readily obtained from a comparison of the effective coupling (9) with its critical value (16), which subsequently provides the following explicit form for the critical condition…”

Section: Model and Methodsmentioning

confidence: 99%

Presence or absence of order by disorder in a highly frustrated region of the spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices

Strečka

Ekiz

2015

Phys. Rev. E

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The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices is exactly solved by combining the generalized star-triangle transformation with the method of exact recursion relations. The ground-state and finite-temperature phase diagrams are rigorously calculated along with both sublattice magnetizations of the Ising and Heisenberg spins. It is evidenced that the Ising-Heisenberg model on triangulated Husimi lattices with two or three inter-connected triangles-in-triangles units displays in a highly frustrated region a quantum disorder irrespective of temperature, whereas the same model on triangulated Husimi lattices with a greater connectivity of triangles-in-triangles units exhibits at low enough temperatures an outstanding quantum order due to the order-by-disorder mechanism. The quantum reduction of both sublattice magnetizations in the peculiar quantum ordered state gradually diminishes with increasing the coordination number of underlying Husimi lattice.

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“…In the cluster variation method proposed by Kiku chi [2][3][4][5][6], the free energy of the compound being ordered and consisting of N atoms is expressed as (5) where are the probabilities of the ith configura tions of a basis cluster S; (n -1) is the number of over lap figures; o (S) and are the revaluation and multi plicity parameters, respectively; and ε i are the energies…”

Section: F E Ts -=mentioning

confidence: 99%

Internal energy and parameters of the order-disorder phase transition in titanium monoxide TiO y

Kostenko

Rempel

Lukoyanov

2013

J. Exp. Theor. Phys.

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Abstract-Quantum mechanical ab initio calculations are used to simulate the free energy functions for tita nium monoxide TiO y . The effect of the long range order of the Ti 5 O 5 type superstructure on the internal energy of the compound is studied by the supercell method. The dependences of the configuration entropy and free energy on the long range order parameter are determined. It is found that the order-disorder phase transition in titanium monoxide must occur in accordance with the mechanism of the first order phase tran sition with a critical value of the long range order parameter of 0.971. The calculated parameters of the phase transition are compared with the experimental data and the results obtained using the model of point charges and by calculating the Madelung energy. It is concluded that the short range order and the phonon entropy must be taken into account in calculating the equilibrium phase diagrams for strongly nonstoichiometric compounds.

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A Theory of Cooperative Phenomena. III. Detailed Discussions of the Cluster Variation Method

Cited by 237 publications

References 15 publications

Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree

Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree

Presence or absence of order by disorder in a highly frustrated region of the spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices

Internal energy and parameters of the order-disorder phase transition in titanium monoxide TiO y

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