Phase diagram of the Ising Model on a Cayley tree in the presence of competing interactions and magnetic field

Mariz, A.M.; Tsallis, Constantino; Albuquerque, E.L.

doi:10.1007/bf01017186

Cited by 60 publications

(78 citation statements)

References 14 publications

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“…Later Mariz et al [10] extended this results assuming existence also interaction J 3 of the one-level nearest-next-neighbours with k = 2. C.R.…”

Section: Introductionmentioning

confidence: 83%

“…In this paper we extend the model considered by Mariz et al [10] for a tree of arbitrary order using the procedures of Inawashiro, Thompson, and Honda [12] to write first-order recursion relations and give strong numerical evidence for the existence of chaotic phases associated with strange attractors. We consider the Vannimenus model on a Cayley tree of arbitrary order k with competing nearest-neighbor interactions J 1 , prolonged next-nearest-neighbor interactions J 2 and one-level k-tuple neighbours interaction J 3 in the presence of magnetic field h. Note that the inclusion of the k-tuple neighbours competing interaction J 3 is essential for the presence of different stable modulated phases at T = 0.…”

Section: Introductionmentioning

confidence: 99%

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Strange Attractors in the Vannimenus Model on an Arbitrary Order Cayley Tree

Ganikhodjaev

Akın

Temir

et al. 2013

J. Phys.: Conf. Ser.

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Abstract. We consider the Vannimenus model on a Cayley tree of arbitrary order k with competing nearest-neighbour interactions J1 and next-nearest-neighbour interactions J2 and J3 in the presence of an external magnetic field h. In this paper we study the phase diagram of the model using an iterative scheme for a renormalized effective nearest-neighbour coupling Kr and effective field per site Xr for spins on the rth level; it recovers, as particular cases, previous works by Vannimenus, Inawashiro et al, Mariz et al and Ganikhodjaev and Uguz. Each phase is characterized by a particular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. These changements can be either continuous or abrupt, respectively characterizing second-or first-order phase transitions. We present a few typical attractors and at finite temperatures, several interesting features (evolution of reentrances, separation of the modulated region into few disconnected pieces, etc) are exhibited for typical values of parameters. IntroductionThe anisotropic next-nearest-neighbour Ising (ANNNI) model, which consists of an Ising model with nearest-neighbour interactions augmented by competing next-nearest-neighbour couplings acting parallel to a single axis direction, is one of the simplest model displaying a rich phase diagram with a Lifshitz point and many spatially modulated phases [1]- [8]. There has been a considerable theoretical effort to obtain the structure of the global phase diagram of the ANNNI model in the T − p space, where T is temperature and p = −J 2 /J 1 is the ratio between the competing exchange interactions. On the basis of numerical mean-field calculations, Bak and von Boehm [7] suggested the existence of an infinite succession of commensurate phases, the so-called devil's staircase, at low temperatures. This mean-field picture has been supported by low-temperature series expansions performed by Fisher and Selke [8]. An Ising model with competing interactions on the Cayley tree has recently been studied extensively because of the appearance of nontrivial magnetic orderings (see [9] -[19] and references therein). The Cayley tree is not a realistic lattice; however, its amazing topology makes the exact calculation of various quantities possible. For many problems the solution on a tree is much simpler than on a regular lattice and is equivalent to the standard Bethe-Peierls theory [20].

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“…Later Mariz et al [10] extended this results assuming existence also interaction J 3 of the one-level nearest-next-neighbours with k = 2. C.R.…”

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Strange Attractors in the Vannimenus Model on an Arbitrary Order Cayley Tree

Ganikhodjaev

Akın

Temir

et al. 2013

J. Phys.: Conf. Ser.

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“…Inawashiro et al [5] independently of Vannimenus investigated the Ising model with nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree, but they allowed J p = J 0 , where J 0 is the one-level next-nearest-neighbor interaction on the Cayley tree of order two. Later Mariz et al [6] extended these results assuming also existence of binary interaction J 0 on the Cayley tree of order 2. Recently Ganikhodjaev et al have obtained a general result of the Vannimenus work on a Cayley tree of arbitrary finite order k [1].…”

Section: Introductionmentioning

confidence: 90%

“…In this case the phase diagram is very rich and there are at least four multicritical Lifshitz points, where two of them are at zero temperature and other two are at nonzero temperature. The main novelty lies in existing multicritical Lifshitz points that are at nonzero temperature, since in previous works [2] and [6] such point was at zero temperature. Let us note that in this case paramagnetic phase is absent.…”

Section: Phase Diagramsmentioning

confidence: 99%

“…In [6] the authors assumed that effects for k = 2 should not be very different in higher (but finite) order k. However from considerations above one can see that the role of k is rather significant. Secondary, for isomorphism between the system characterized by (J 1 , J t p ) and that characterized by (−J 1 , J t p ) stated in [6], there are small islands in the quadrant {β > 0, α < 0} where this isomorphism is broken, moreover the number of such islands increases with increasing k. To investigate the nature of such islands is rather a complete problem.…”

Section: Phase Diagramsmentioning

confidence: 99%

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Behaviors of Phase Diagrams of Ising Model with Competing Ternary and Binary Interactions on a Cayley Tree of Arbitrary Order

et al. 2012

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An Ising model with competing interactions has recently been studied extensively because of the appearance of nontrivial magnetic orderings. In this paper, we study the phase diagrams for the Ising model on a Cayley tree with competing nearest-neighbor interactions J and ternary prolonged interactions J tp on a Cayley tree of arbitrary order k and compare with the phase diagrams obtained in Uguz et al. and Vannimenus results for the Ising model on a Cayley tree with competing nearest-neighbor interactions J and ternary prolonged interactions J p . For some values of k, we obtain phase diagrams of the model. We clarify the role of order k of the Cayley tree. We also plot the variation of the wave vector with temperature.

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More Exact Results on a Binary Alloy with Next Nearest Neighbor Competing Interactions

Paraskevaidis

Haroni

2000

phys. stat. sol. (b)

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Phase diagram of the Ising Model on a Cayley tree in the presence of competing interactions and magnetic field

Cited by 60 publications

References 14 publications

Strange Attractors in the Vannimenus Model on an Arbitrary Order Cayley Tree

Strange Attractors in the Vannimenus Model on an Arbitrary Order Cayley Tree

Behaviors of Phase Diagrams of Ising Model with Competing Ternary and Binary Interactions on a Cayley Tree of Arbitrary Order

More Exact Results on a Binary Alloy with Next Nearest Neighbor Competing Interactions

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