We consider the Ashkin–Teller model on the square lattice, which is represented by two Ising models (σ and τ) having a four-spin coupling of strength, , between them. We introduce an asymmetric defect line in the system along which the couplings in the σ Ising model are modified. In the Hamiltonian version of the model we study the scaling behavior of the critical magnetization at the defect, both for σ and for τ spins, by means of density matrix renormalization. For > 0 we observe identical scaling for σ and τ spins, whereas for < 0 one model becomes locally ordered and the other locally disordered. This is different from the critical behavior of the uncoupled model ( = 0) and is in contradiction with the results of recent field-theoretical calculations.We consider the Ashkin–Teller model on the square lattice, which is represented by two Ising models (σ and τ) having a four-spin coupling of strength, , between them. We introduce an asymmetric defect line in the system along which the couplings in the σ Ising model are modified. In the Hamiltonian version of the model we study the scaling behavior of the critical magnetization at the defect, both for σ and for τ spins, by means of density matrix renormalization. For > 0 we observe identical scaling for σ and τ spins, whereas for < 0 one model becomes locally ordered and the other locally disordered. This is different from the critical behavior of the uncoupled model ( = 0) and is in contradiction with the results of recent field-theoretical calculations.

arXiv:1105.1687Journal of Statistical Mechanics Theory and ExperimentMay 25, 20112 citations

Critical behavior of the N -component Ashkin-Teller model with bond impurities

Ganpathy N. Murthy

The $N$ -component Ashkin-Teller model is a system of $N$ Ising models coupled by a four-spin interaction. The continuum limit of this model is the $\mathrm{O}(N)$ Gross-Neveu model. In this report the critical behavior of the model with weak bond impurities is derived from a renormalization-group analysis. It is found that quenched randomness converts a first-order transition into a second-order transition.The $N$-component Ashkin-Teller model is a system of $N$ Ising models coupled by a four-spin interaction. The continuum limit of this model is the $\mathrm{O}(N)$ Gross-Neveu model. In this report the critical behavior of the model with weak bond impurities is derived from a renormalization-group analysis. It is found that quenched randomness converts a first-order transition into a second-order transition.

Physical review. B, Condensed matterNov 1, 198713 citations

On the critical behavior of the Ising model with mixed two- and three-spin interactions

Francisco Castilho Alcaraz, Michael N. Barber

Journal of Statistical PhysicsFeb 1, 19879 citations

Universality and tricritical behavior of three-dimensional Ising models with two- and four-spin interactions

Ole G. Mouritsen, Svend J. Knak Jensen, B. Frank

The Monte Carlo technique is applied to a study of the phase transitions and the critical behavior of the spin-\textonehalf{} Ising model on an fcc lattice with mixtures of two- ( ${J}_{2}$ ) and four - ( ${J}_{4}$ ) spin interactions. In the limit ${J}_{2}=0$ the model exhibits a first-order transition. The transition remains of first order for $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\gtrsim}\frac{1}{2}$, but a crossover to continuous transitions is found around $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\approx}\frac{1}{4}\ensuremath{-}\frac{1}{2}$ indicating that the model exhibits tricritical behavior. A modified mean-field theory is presented leading to an approximate description of the tricritical behavior in agreement with the Monte Carlo calculations. In the region of continuous transitions. $0<~\frac{{J}_{4}}{{J}_{2}}\ensuremath{\lesssim}\frac{1}{4}$, the critical exponent $\ensuremath{\beta}$ pertaining to the order parameter derived from the Monte Carlo data retains the Ising value, in accordance with the universality hypothesis. Our findings show that the four-spin interactions do not lead to nonuniversal critical behavior, contrary to the conclusions made by Griffiths and Wood from a series analysis.The Monte Carlo technique is applied to a study of the phase transitions and the critical behavior of the spin-\textonehalf{} Ising model on an fcc lattice with mixtures of two- (${J}_{2}$) and four - (${J}_{4}$) spin interactions. In the limit ${J}_{2}=0$ the model exhibits a first-order transition. The transition remains of first order for $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\gtrsim}\frac{1}{2}$, but a crossover to continuous transitions is found around $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\approx}\frac{1}{4}\ensuremath{-}\frac{1}{2}$ indicating that the model exhibits tricritical behavior. A modified mean-field theory is presented leading to an approximate description of the tricritical behavior in agreement with the Monte Carlo calculations. In the region of continuous transitions. $0<~\frac{{J}_{4}}{{J}_{2}}\ensuremath{\lesssim}\frac{1}{4}$, the critical exponent $\ensuremath{\beta}$ pertaining to the order parameter derived from the Monte Carlo data retains the Ising value, in accordance with the universality hypothesis. Our findings show that the four-spin interactions do not lead to nonuniversal critical behavior, contrary to the conclusions made by Griffiths and Wood from a series analysis.

Physical review. B, Condensed matterJul 1, 198127 citations

A discussion of critical behaviour in a mixed-spin Ising model

A. Bobák, Marian Jurčisin

Physica A Statistical Mechanics and its ApplicationsJun 1, 199758 citations

The

N

-component Ashkin-Teller model is a system of

N

Ising models coupled by a four-spin interaction. The continuum limit of this model is the

\mathrm{O}(N)

Gross-Neveu model. In this report the critical behavior of the model with weak bond impurities is derived from a renormalization-group analysis. It is found that quenched randomness converts a first-order transition into a second-order transition.

The Monte Carlo technique is applied to a study of the phase transitions and the critical behavior of the spin-\textonehalf{} Ising model on an fcc lattice with mixtures of two- (

{J}_{2}

) and four - (

{J}_{4}

) spin interactions. In the limit

{J}_{2}=0

the model exhibits a first-order transition. The transition remains of first order for $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\gtrsim}\frac{1}{2}$, but a crossover to continuous transitions is found around $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\approx}\frac{1}{4}\ensuremath{-}\frac{1}{2}$ indicating that the model exhibits tricritical behavior. A modified mean-field theory is presented leading to an approximate description of the tricritical behavior in agreement with the Monte Carlo calculations. In the region of continuous transitions. $0<~\frac{{J}_{4}}{{J}_{2}}\ensuremath{\lesssim}\frac{1}{4}$, the critical exponent $\ensuremath{\beta}$ pertaining to the order parameter derived from the Monte Carlo data retains the Ising value, in accordance with the universality hypothesis. Our findings show that the four-spin interactions do not lead to nonuniversal critical behavior, contrary to the conclusions made by Griffiths and Wood from a series analysis.