We report our Monte Carlo results on the critical and multicritical behavior of the ± J Ising model [with a random-exchange probability P ( J xy ) = p δ( J xy - J )+(1- p )δ( J xy + J )], in two and three dimensions. We study the transition line between the paramagnetic and ferromagnetic phase, which extends from p = 1 to a multicritical (Nishimori) point. By a finite-size scaling analysis, we provide strong numerical evidence that in three dimensions the critical behavior along this line belongs to the same universality class as that of the critical transition in the randomly dilute Ising model. In two dimensions we confirm that the critical behavior is controlled by the pure Ising fixed point and that disorder is marginally irrelevant, giving rise to universal logarithmic corrections. In both two and three dimensions, we also determine the location of the multicritical Nishimori point, as well as the renormalization-group dimensions of the operators that control the renormalization-group flow close to it.We report our Monte Carlo results on the critical and multicritical behavior of the ± J Ising model [with a random-exchange probability P ( J xy ) = p δ( J xy - J )+(1- p )δ( J xy + J )], in two and three dimensions. We study the transition line between the paramagnetic and ferromagnetic phase, which extends from p = 1 to a multicritical (Nishimori) point. By a finite-size scaling analysis, we provide strong numerical evidence that in three dimensions the critical behavior along this line belongs to the same universality class as that of the critical transition in the randomly dilute Ising model. In two dimensions we confirm that the critical behavior is controlled by the pure Ising fixed point and that disorder is marginally irrelevant, giving rise to universal logarithmic corrections. In both two and three dimensions, we also determine the location of the multicritical Nishimori point, as well as the renormalization-group dimensions of the operators that control the renormalization-group flow close to it.

arXiv:0810.0685Journal of Physics Conference SeriesJan 1, 20094 citations

Crossover Critical Behavior in the Three-Dimensional Ising Model

Young C. Kim, Mikhail A. Anisimov, Jan V. Sengers +1 more

arXiv:cond-mat/0203190Journal of Statistical PhysicsMar 1, 200349 citations

Interdimensional Scaling Laws

Y. Imry, G. Deutscher, David J. Bergman +1 more

We consider in detail the critical behavior of a $d$ -dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: $d$ -dimensional mean field, $d$ -dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in $d$ and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.We consider in detail the critical behavior of a $d$-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: $d$-dimensional mean field, $d$-dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in $d$ and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.

Physical Review AFeb 1, 197323 citations

Critical behavior of weakly disordered anisotropic systems in two dimensions

Giancarlo Jug, B. N. Shalaev

The critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the so-called generalized Ashkin-Teller model (GATM) is studied. In the critical region this model is shown to be described by a multifermion field theory similar to the Gross-Neveu model with a few independent quartic coupling constants. Renormalization group calculations are used to obtain the temperature dependence near the critical point of some thermodynamic quantities and the large-distance behavior of the two-spin correlation function. The equation of state at criticality is also obtained in this framework. We find that random models described by the GATM belong to the same universality class as that of the two-dimensional Ising model. The critical exponent \ensuremath{\nu} of the correlation length for the three- and four-state random-bond Potts models is also calculated in a three-loop approximation. We show that this exponent is given by an apparently convergent series in \ensuremath{\epsilon}=c-1/2 (with c the central charge of the Potts model) and that the numerical values of \ensuremath{\nu} are very close to that of the 2D Ising model. This work therefore supports the conjecture (valid only approximately for the three- and four-state Potts models) of a superuniversality for the 2D disordered models with discrete symmetries. \textcopyright{} 1996 The American Physical Society.The critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the so-called generalized Ashkin-Teller model (GATM) is studied. In the critical region this model is shown to be described by a multifermion field theory similar to the Gross-Neveu model with a few independent quartic coupling constants. Renormalization group calculations are used to obtain the temperature dependence near the critical point of some thermodynamic quantities and the large-distance behavior of the two-spin correlation function. The equation of state at criticality is also obtained in this framework. We find that random models described by the GATM belong to the same universality class as that of the two-dimensional Ising model. The critical exponent \ensuremath{\nu} of the correlation length for the three- and four-state random-bond Potts models is also calculated in a three-loop approximation. We show that this exponent is given by an apparently convergent series in \ensuremath{\epsilon}=c-1/2 (with c the central charge of the Potts model) and that the numerical values of \ensuremath{\nu} are very close to that of the 2D Ising model. This work therefore supports the conjecture (valid only approximately for the three- and four-state Potts models) of a superuniversality for the 2D disordered models with discrete symmetries. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9607201Physical review. B, Condensed matterAug 1, 199628 citations

We consider in detail the critical behavior of a

d

-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior:

d

-dimensional mean field,

d

-dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in

d

and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.