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Physical review. B, Condensed matter

N -color Ashkin-Teller model

Gary S. Grest, Michael Widom

Published Dec 1, 1981• 75 citations
Publisher

Abstract

We study a model Hamiltonian consisting of $N$ Ising models coupled pairwise through a four-spin interaction ${K}_{4}$. When $N=2$ this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of ${K}_{4}=0$. 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 $N=3$. The results show that $N=2$ is special because the coupling between Ising models is marginal over a range of values of ${K}_{4}$. We discuss the effect of the coupling ${K}_{4}$ for $N\ensuremath{\ne}2$.

DOI
10.1103/physrevb.24.6508

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Physical review. B, Condensed matter•Mar 1, 1999•15 citations
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≠2N\ensuremath{\ne}2N\ensuremath=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≠2N\ensuremath{\ne}2N\ensuremath=2.
High- and low-temperature series, Monte Carlo simulations, mean-field calculations, and renormalization-group ideas are used to analyze the behavior of the Ashkin-Teller model in three dimensions. The data generated by series and Monte Carlo methods in the neighborhood of the decoupling point are consistent with the renormalization-group conclusion that this system cannot show a line of continuously varying exponents. Thus the phase diagram for the d=3d=3d=3 system is quite different from that in d=2d=2d=2. Indeed a new phase (not seen in d=2d=2d=2) is observed, one in which the system spontaneously breaks its natural symmetry between its two kinds of Ising spins. A variety of continuous transitions are observed and explained in terms of an analysis which is fully consonant with renormalization-group concepts.
The NNN-color Ashkin-Teller model is solved exactly in the large-NNN limit in two dimensions. The phase diagram is found. It is shown that for positive four-spin coupling the transition is first order while for negative four-spin coupling the transition is continuous and Ising type. The specific heat near the second-order transition is calculated and it is found to be finite at the transition because of large corrections to scaling. The latent heat and correlation length at the first-order transition are also calculated.
An intriguing result of statistical mechanics is that a first-order phase transition can be rounded by disorder coupled to energylike variables. In fact, even more intriguing is that the rounding may manifest itself as a critical point, quantum or classical. In general, it is not known, however, what universality classes, if any, such criticalities belong to. In order to shed light on this question we examine in detail the disordered three-color Ashkin-Teller model by Monte Carlo methods. Extensive analyses indicate that the critical exponents define a new universality class. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as criticality. However, the magnetization critical exponent, β, and the correlation length critical exponent, ν, are found to vary with disorder and the four-spin coupling strength, and we conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0{J}_{1}<0J1​<0 (nearest neighbor, ferromagnetic) and J2>0{J}_{2}>0J2​>0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4{Z}_{4}Z4​ symmetry and is stabilized at low temperatures for g=J2/∣J1∣>1/2g={J}_{2}/|{J}_{1}|>1/2g=J2​/∣J1​∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1{J}_{1}J1​-J2{J}_{2}J2​ model has a line of very weak first-order phase transitions in the whole region 1/2<g<g∗1/2<g<{g}^{*}1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from g=g∗g={g}^{*}g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g∗{g}^{*}g∗ in the J1{J}_{1}J1​-J2{J}_{2}J2​ model. The continuous transitions near g∗{g}^{*}g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2g>1/2g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
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.