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Physical review. B, Condensed matter

Critical properties of the spin-1 Ashkin-Teller model from the differential operator method

Mohamed Loulidi

Published May 1, 1997• 11 citations
Publisher

Abstract

In order to study the effect of the anisotropy parameter on the two coupled Ising models, we propose an Ashkin-Teller model (ATM) with spin variable S=1. Using an effective-field theory, we show that near the partially ordered phase ${\mathrm{PO}}_{1}$ that appears in spin-1/2 ATM, this model presents a new partially ordered phase ${\mathrm{PO}}_{2}$. The anisotropy parameter introduces a first-order phase transition between different phases. However, the model exhibits a rich variety of phase diagrams and higher-order critical points.

DOI
10.1103/physrevb.55.11611

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Using Monte Carlo simulations, we have studied the critical behaviours and phase diagrams of the spin-1/2 and spin-1 Ashkin Teller model ferromagnetic thin films. In the first part: The effect of the interaction with the exchange coupling 𝐽4 is studied. The phase diagrams present different ratios of the surface exchange interaction at the special point, in which all the films thickness have the same critical temperature. First and second order transitions appears in the system. Moreover, at the special point descriptions, we found another partially ordered phase for certain values of the parameter’s interaction. Finally, the critical exponents of the surface magnetization 𝛽1𝑚 are estimated. In the second part, the effects of the crystal field 𝐷 𝐽 ⁄ 2𝑏 and four-spins coupling (𝐽4𝑠 𝐽 ⁄ 2𝑏, 𝐽4𝑏 𝐽 ⁄ 2𝑏) has been studied in details. The phase diagrams in the (𝑘𝐵𝑇𝑐 𝐽 ⁄ 2𝑏 ,𝐽2𝑠 𝐽 ⁄ 2𝑏) plane exhibits the special point, for different values of 𝐷 𝐽 ⁄ 2𝑏 and (𝐽4𝑠 𝐽 ⁄ 2𝑏, 𝐽4𝑏 𝐽 ⁄ 2𝑏). Then, we found rich phase diagrams with firstand second-order phase transitions that meet at tricritical points, that are depended on the film thickness N. Finally, this model presents a new partially ordered phase < 𝜎𝑆 >, between the paramagnetic phase and the Baxter phase, which are separated by a second-order phase transition, for positive values of 𝐷 𝐽 ⁄ 2𝑏 and for each film thickness N.

•May 22, 2021•0 citations
In order to study the effect of the anisotropy parameter on the two coupled Ising models, we propose an Ashkin-Teller model (ATM) with spin variable S=1. Using an effective-field theory, we show that near the partially ordered phase PO1{\mathrm{PO}}_{1}PO1​ that appears in spin-1/2 ATM, this model presents a new partially ordered phase PO2{\mathrm{PO}}_{2}PO2​. The anisotropy parameter introduces a first-order phase transition between different phases. However, the model exhibits a rich variety of phase diagrams and higher-order critical points.
Two nonperturbative methods by means of the transfer-matrix finite-size-scaling and Monte Carlo simulations are used to investigate the spin-1 Ashkin-Teller model (ATM). We have obtained rich phase diagrams with first- and second-order phase transitions with several multicritical points of higher order. Also this model exhibits a partially ordered phase PO2{\mathrm{PO}}_{2}PO2​ which does not exist in the spin-1/2 Ashkin-Teller model (ATM). Finally, the critical behavior of this model is discussed.
The study of the Ashkin-Teller model (ATM) of spin-3/2 on a hypercubic lattice is undertaken via Monte Carlo simulation.The phase diagrams are displayed and discussed in the physical parameter space.Rich physical properties are recovered, namely the second order transition and multicritical points.The phase diagrams have been obtained by varying the strength describing the four spin interaction and the single ion potential.This model shows a new high temperature partially ordered phase, called 〈S〉 and a new Baxter 3/2 ground state which do not exist either in the spin-1/2 ATM or in the spin-1 ATM.
In order to study the nature of phase transitions in magnetic systems, the aim of our research is to explore the S=1 and σ=3/2 states where the ferromagnetism as well as the influence of different crystal fields will be studied by the Monte Carlo method of The Ashken-Teller model which is a coupling of two Ising models with a four-component parameter. We have given some theoretical definitions related to the phase transition as well as the types of magnetism also the spin models and the simulation method. First, we identify the most stable phases in the phase diagrams at temperature T=0 by energy minimization. Then For higher temperatures, we find phase diagrams rich in ordered phases: a 3/2 Baxter phase and a 1/2 Baxter phase as well as a phase that appears neither in spin 1 nor in spin 3/2 ATM, and finally, a <σ> = 1/2 phases with 1st and 2nd order.
We study a model Hamiltonian consisting of NNN Ising models coupled pairwise through a four-spin interaction K4{K}_{4}K4​. When N=2N=2N=2 this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of K4=0{K}_{4}=0K4​=0. To see if this behavior persists for $N\ensuremath{\ne}2$ we perform a first-order-perturbation expansion around the decoupling point in two dimensions. As an aid in interpreting the results of this perturbation expansion we have determined the phase diagram of the system through mean-field theory and Monte Carlo studies in both two and three dimensions for N=3N=3N=3. The results show that N=2N=2N=2 is special because the coupling between Ising models is marginal over a range of values of K4{K}_{4}K4​. We discuss the effect of the coupling K4{K}_{4}K4​ for $N\ensuremath{\ne}2$.
Magnetic properties and phase diagrams of a spin-1/2 anisotropic Ashkin-Teller model in a transverse field are studied using the mean field theory. On the one hand, it is found that there exists a critical value of the transverse field Ωc/J1 above which the partially ordered phase 〈σS〉 appears at very low temperature and sufficiently large values of the four-spin interaction coefficient J4/J1. Such result is not obtained in the Ω/J1 = 0 case. On the other hand, the multicritical behaviour is also studied as a function of the transverse field Ω/J1 for selected values of the spin interaction coefficient J2/J1. However, depending on the transverse field value, and a fixed J2/J1 value, four varieties of phase diagrams are found. General formulae are established for an arbitrary coordination number q. The phase diagrams and the behaviour of the transverse and longitudinal magnetisations are given for q = 4.
Using Monte Carlo simulations, we have studied the critical behaviours and phase diagrams of the spin-1/2 and spin-1 Ashkin Teller model ferromagnetic thin films. In the first part: The effect of the interaction with the exchange coupling 𝐽4 is studied. The phase diagrams present different ratios of the surface exchange interaction at the special point, in which all the films thickness have the same critical temperature. First and second order transitions appears in the system. Moreover, at the special point descriptions, we found another partially ordered phase for certain values of the parameter’s interaction. Finally, the critical exponents of the surface magnetization 𝛽1𝑚 are estimated. In the second part, the effects of the crystal field 𝐷 𝐽 ⁄ 2𝑏 and four-spins coupling (𝐽4𝑠 𝐽 ⁄ 2𝑏, 𝐽4𝑏 𝐽 ⁄ 2𝑏) has been studied in details. The phase diagrams in the (𝑘𝐵𝑇𝑐 𝐽 ⁄ 2𝑏 ,𝐽2𝑠 𝐽 ⁄ 2𝑏) plane exhibits the special point, for different values of 𝐷 𝐽 ⁄ 2𝑏 and (𝐽4𝑠 𝐽 ⁄ 2𝑏, 𝐽4𝑏 𝐽 ⁄ 2𝑏). Then, we found rich phase diagrams with firstand second-order phase transitions that meet at tricritical points, that are depended on the film thickness N. Finally, this model presents a new partially ordered phase < 𝜎𝑆 >, between the paramagnetic phase and the Baxter phase, which are separated by a second-order phase transition, for positive values of 𝐷 𝐽 ⁄ 2𝑏 and for each film thickness N.