Ashkin-Teller Research Articles

Study of the Critical Behavior and the Multi-Layer Transition in a Spin-1 Ashkin–Teller Model under the Effect of the RKKY Interaction: Finite-Size-Scaling and Monte Carlo Studies

By using a Monte Carlo simulation, we have studied the effects on the multilayer transition and magnetic properties of the crystal field [Formula: see text], the four-spin coupling [Formula: see text] and RKKY interaction in the spin-1 Ashkin–Teller (AT) model of a system composed of two magnetic multi-layer materials, of various thicknesses, separated by a nonmagnetic spacer of thickness [Formula: see text]. By varying the strength of [Formula: see text], we have obtained a new partially ordered phase [Formula: see text], for different values of [Formula: see text]. We have established a rich phase diagram with first and second-order phase transitions with tricritical points. It is shown that the transition temperature of both magnetic blocks depends strongly on the thickness of the magnetic layers [Formula: see text], as well as [Formula: see text] and [Formula: see text]. There exists a critical thickness of the magnetic layers [Formula: see text] above which the multilayer blocks undergo a transition separately at different temperatures, for a high value of [Formula: see text] and [Formula: see text]. Besides, the critical exponents are computed.

Cluster Monte Carlo method for the 3D Ashkin–Teller model

This paper presents a cluster Monte Carlo method suitable for both, the first- and the second-order phase transitions in the 3D Ashkin–Teller (AT) model. A cluster algorithm is necessary to verify correctness of the results obtained so far using Metropolis algorithm exhibiting significant critical slowing down. Moreover, metastable states have recently been investigated in this model which can affect the results, especially obtained using the Metropolis algorithm. Our Wolff type algorithm is described and its dynamic critical behavior is demonstrated. Our computer experiments exploit the properties of Binder and Challa cumulants and additionally the one proposed by Lee and Kosterlitz, the last two adapted by us to give clear results for the AT model. The energy distribution histogram method is also independently applied for the first time for the 3D AT model using the Wolff type algorithm. For validation of the previous results and of our algorithm, it is demonstrated that the results of our computations along the line between Ising and Potts points, which are rescaled to their thermodynamic limits, are consistent with those obtained using the Metropolis algorithm. It is also shown that the presented cluster algorithm of the Wolff type significantly reduces the problem of critical slowing down for the 3D AT model, and the dynamic critical exponent reaches values close to zero. As the best strategy, it is suggested to use the cluster algorithm in the critical region and the Metropolis one beyond.

Gapless Coulomb State Emerging from a Self-Dual Topological Tensor-Network State.

In the tensor network representation, a deformed Z_{2} topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the Ashkin-Teller model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or the confining phase.

Wide crossover in the 3D Ashkin–Teller model

The phase transition line, for which the possibility of the non-universal behavior studied by Monte Carlo (MC) simulations had been announced, in the standard 3D Ashkin–Teller (AT) model phase diagram is carefully investigated. The results of our MC experiments for transitions between paramagnetic, mixed, and Baxter phases for negative coupling between neighboring spin pairs, the most complex and least recognized region in the phase diagram, are presented. Binder, Challa, and also for the first time of Lee–Kosterlitz cumulants, the last two adapted to the AT model, are exploited. The uncommon coexistence of continuous and first order phase transitions along the same phase boundary on different components of the order parameter are reported. The paper is closed with the conclusion of a wide crossover effect that covers the whole line between tricritical points, where earlier papers indicate the possibility of non-universal behavior with the critical exponent values varying continuously.

Ashkin-Teller model and diverse opinion phase transitions on multiplex networks.

Multiplex networks (MNs) have become a platform of recent research in network sciences because networks in many real-world systems interact and function together. One of the main scientific issues in MNs is how the interdependence changes the emerging patterns or phase transitions. Until now, studies of such an issue have concentrated on cluster-breakdown phenomena, aiming to understand the resilience of the system under random failures of edges. These studies have revealed that various phase transition (PT) types emerge in MNs. However, such studies are rather limited to percolation-related problems, i.e., the limit q→1 of the q-state Potts model. Thus, a systematic study of opinion formation in social networks with the effect of interdependence between different social communities, which may be seen as the study of the emerging pattern of the Ising model on MNs, is needed. Here we study a well-known spin model called the Ashkin-Teller (AT) model in scale-free networks. The AT model can be regarded as a model for interacting systems between two species of Ising spins placed on respective layers in double-layer networks. Our study shows that, depending on the interlayer coupling strength and a network topology, unconventional PT patterns can also emerge in interaction-based phenomena: continuous, discontinuous, successive, and mixed-order PTs and a continuous PT not satisfying the scaling relation. The origins of such rich PT patterns are elucidated in the framework of Landau-Ginzburg theory.

Worm-type Monte Carlo simulation of the Ashkin-Teller model on the triangular lattice

We investigate the symmetric Ashkin-Teller (AT) model on the triangular lattice in the antiferromagnetic two-spin coupling region (J<0). In the J→-∞ limit, we map the AT model onto a fully packed loop-dimer model on the honeycomb lattice. On the basis of this exact transformation and the low-temperature expansion, we formulate a variant of worm-type algorithms for the AT model, which significantly suppress the critical slowing down. We analyze the Monte Carlo data by finite-size scaling, and locate a line of critical points of the Ising universality class in the region J<0 and K>0, with K the four-spin interaction. Further, we find that, in the J→-∞ limit, the critical line terminates at the decoupled point K=0. From the numerical results and the exact mapping, we conjecture that this "tricritical" point (J→-∞,K=0) is Berezinsky-Kosterlitz-Thouless-like and the logarithmic correction is absent. The dynamic critical exponent of the worm algorithm is estimated as z=0.28(1) near (J→-∞,K=0).

Critical interfaces of the Ashkin–Teller model at the parafermionic point

We present an extensive study of interfaces defined in theZ4 spin lattice representation of the Ashkin–Teller (AT) model. In particular, wenumerically compute the fractal dimensions of boundary and bulk interfaces at theFateev–Zamolodchikov point. This point is a special point on the self-dual criticalline of the AT model and it is described in the continuum limit by the parafermionic theory. Extending on previous analytical and numerical studies,we point out the existence of three different values of fractal dimensions whichcharacterize different kind of interfaces. We argue that this result may berelated to the classification of primary operators of the parafermionic algebra.The scenario emerging from the studies presented here is expected to unveilgeneral aspects of geometrical objects of the critical AT model, and thus ofc = 1 critical theories in general.

Anomalous Universality in the Anisotropic Ashkin–Teller Model

The Ashkin–Teller (AT) model is a generalization of Ising 2–d to a four states spin model; it can be written in the form of two Ising layers (in general with different couplings) interacting via a four–spin interaction. It was conjectured long ago (by Kadanoff and Wegner, Wu and Lin, Baxter and others) that AT has in general two critical points, and that universality holds, in the sense that the critical exponents are the same as in the Ising model, except when the couplings of the two Ising layers are equal (isotropic case). We obtain an explicit expression for the specific heat from which we prove this conjecture in the weakly interacting case and we locate precisely the critical points. We find the somewhat unexpected feature that, despite universality, holds for the specific heat, nevertheless nonuniversal critical indexes appear: for instance the distance between the critical points rescale with an anomalous exponent as we let the couplings of the two Ising layers coincide (isotropic limit); and so does the constant in front of the logarithm in the specific heat. Our result also explains how the crossover from universal to nonuniversal behaviour is realized.

THE BOSONIZATION FORM AND NEW CRITICAL REGION OF THE QUANTUM ASHKIN-TELLER CHAIN

In the continuum limit, the critical phenomenon of the quantum Ashkin-Teller(A-T) chain is discussed by the bosonization method. A new critical region of the A-T chain besides the Ising decoupling point has been found.

Quantum Ashkin-Teller Model and Conformal Theory11The project supported by Science Foundation of National Education Committee.

A simple derivation of the field theory of quantum Ashkin-Teller (AT) model shows that the AT model is equivalent to a complex fermion field. At the critical point, the field equation is conformally covariant. Thus the critical point condition and the normalizing factor are given.

Cluster method for the Ashkin-Teller model.

A cluster Monte Carlo algorithm for the Ashkin-Teller (AT) model is constructed according to the guidelines of a general scheme for such algorithms. Its dynamical behavior is tested for the square-lattice AT model. We perform simulations on the line of critical points along which the exponents vary continuously, and find that critical slowing down is significantly reduced. We find continuous variation of the dynamical exponent z along the line, following the variation of the ratio of specific-heat and correlation-length exponents \ensuremath{\alpha}/\ensuremath{\nu}, in a manner which satisfies the Li-Sokal bound ${\mathit{z}}_{\mathrm{cluster}}$\ensuremath{\ge}\ensuremath{\alpha}/\ensuremath{\nu}, that was so far proved only for Potts models.

Non-universal exponents and marginal operators via a Monte Carlo renormalisation group

The authors study the critical exponents, eigenoperators and the phase diagram of the N=2 Ashkin-Teller (AT) model in two dimensions using a Monte Carlo renormalisation group. They pay special attention to the marginal operator responsible for the non-universal behaviour. The crossover exponent is measured and used to numerically calculate the thermal exponent of the eight-vertex model and of the AT model near the Potts-four-point, giving better results than a direct determination, since the crossover operator is not affected by the additional AT marginal operators.

Two-layered Ising model with short-range two-spin interactions showing the critical behaviour of the Ashkin-Teller model

A two-layered Ising model is presented which maps to the Ashkin-Teller (AT) model (1943) for certain ratios of the two-spin couplings. Thus the model is able to exhibit the same critical exponents as the AT model, which are in general different from the usual Ising exponents; even continuous dependence of the critical exponents on the energy parameters is to be expected.