Scaling relationship between effective critical exponents throughout the crossover region in thin Ising films

The Monte Carlo (MC) approach is used to check the validity of the scaling relationship for the effective critical exponents in thin Ising films. We investigate this relationship not just in the critical region but throughout the crossover to the expected two-dimensional behavior. Our results indicate that this scaling relationship is very well-fulfilled throughout the entire crossover temperature region, as predicted by a previous renormalization group analysis. The two-dimensional universality class of Ising films is confirmed by means of data collapsing plots for plates with increasing L, up to L=100. The evolution of the maximum value of the effective critical exponents with film thickness is discussed. Received 22 April 1999     

Renormalization group for matrix models with branching interactions

We develop a method to obtain the large-N renormalization group flows for matrix models of two-dimensional gravity plus branched polymers. This method gives precise results for the critical points and exponents for one-matrix models. We show that it can be generalized to two-matrix models and we recover the Ising critical points.     

The classical Ising model: A quantum renormalization group approach

Proceeding from the equivalence between the d-dimentional classical Ising model and the (d?1)-dimentional quantum mechanical Ising model in a transverse magnetic field, we study the critical properties of the classical model via the quantum mechanical model. Quantum renormalization group transformations based on the truncation method and the ground state projection operator method are used to calculate the critical exponents. They are found to agree well with the “exact” values.     

Spatial evolutionary game with bond dilution

In this paper, we study numerically the prisoner’s dilemma game (PDG) and snowdrift game (SG) on a two-dimensional square lattice with both quenched and annealed bond dilution. For quenched bond dilution, the system undergoes a dynamical transition at the critical occupation probability q^∗, which is higher than the bond percolation transition point for a square lattice. In the critical region, the defined order parameter has a scaling form as P_e∼(q−q^∗)^β for q<q^∗ with the critical exponents β=1.42 for PDG and β=1.52 for SG, which differ from those with quenched site dilution. For annealed bond dilution, the system exhibits a distinct cooperative behavior. We find that the cooperation is much enhanced in the range of small payoff parameters on a lattice with slightly annealed bond dilution.     

EXACT DECIMATION APPROACH WITH MEAN-FIELD APPROXIMATION FOR RANDOMLY DILUTED ISING MODELS

A new renormalization approach developed by authors previously is generalized to disordered systems. Calculations of the critical temperature, concentration, exponents, and of limiting slope of critical curve for the random bond-diluted Ising model on the square lattice, even in the lowest approximation, are in very satisfactory agreement with all known exact or series results.     

An introduction to the fundamentals of the renormalization group in critical phenomena

The fundamental concepts underlying the application of the renormalization group and related techniques to critical phenomena are reviewed at an elementary level. Topics discussed include: the definition of the renormalization group as a functional integral over high momentum components of the spin field, the behaviour of the renormalization group near the fixed point and the derivation of scaling, Wilson's approximate recursion relation, trivial and non-trivial fixed points of isotropic spin systems near d = 4, Feynman graph expansions for critical exponents, ? = 4 ? d and 1/n-expansions, the derivation of exact recursion relations and co-ordinate space transformations for d = 2 Ising systems     

Phase transition in a two-dimensional Ising ferromagnet based on the generalized zero-temperature Glauber dynamics

At zero temperature, based on the Ising model, the phase transition in a two-dimensional square lattice is studied using the generalized zero-temperature Glauber dynamics. Using Monte Carlo （MC） renormalization group methods, the static critical exponents and the dynamic exponent are studied; the type of phase transition is found to be of the first order.     

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

A study is made of a two-dimensional Ising model with staggered three-spin interactions in one direction and two-spin interactions in the other. The phase diagram of the model and its critical behavior are explored by conventional finite-size scaling and by exploiting relations between mass gap amplitudes and critical exponents predicted by conformal invariance. The model is found to exhibit a line of continuously varying critical exponents, which bifurcates into two Ising critical lines. This similarity of the model with the Ashkin-Teller model leads to a conjecture for the exact critical indices along the nonuniversal critical curve. Earlier contradictions about the universality class of the uniform (isotropic) case of the model are clarified.     

An extended approach for computing the critical properties in the two-and three-dimensional lattices within the effective-field renormalization group method

In this letter we employing the effective-field renormalization group (EFRG) to study the Ising model with nearest neighbors to obtain the reduced critical temperature and exponents ν for bi- and three-dimensional lattices by increasing cluster scheme by extending recent works. The technique follows up the same strategy of the mean field renormalization group (MFRG) by introducing an alternative way for constructing classical effective-field equations of state takes on rigorous Ising spin identities.     

Critical Behavior of Gaussian Model on X Fractal Lattices in External Magnetic Fields

Using the renormalization group method, the critical behavior of Gaussian model is studied in external magnetic fields on X fractal lattices embedded in two-dimensional and d-dimensional (d ＞ 2) Euclidean spaces, respectively. Critical points and exponents are calculated. It is found that there is long-range order at finite temperature for this model, and that the critical points do not change with the space dimensionality d (or the fractal dimensionality dr). It is also found that the critical exponents are very different from results of Ising model on the same lattices, and that the exponents on X lattices are different from the exact results on translationally symmetric lattices.     

Critical behaviour of the hard-core lattice gas on the honeycomb lattice

The hard triangle lattice-gas model (lattice-gas on the honeycomb lattice with first neighbour exclusion) is studied by the phenomenological renormalization method. The critical activity is found to be z = 7.85 and the critical exponents suggest that this model belongs to the 2-D Ising universality class.     

Eight-parameter renormalization group for Penrose lattices

The Ising model and the bond percolation model are set up with eight parameters on two-dimensional Penrose lattices. The behavior of their phase transition is studied by the use of a real-space renormalization group method. The resulting critical indices suggest that they belong to the universality class of two-dimensional periodic lattices.     

Double-Chain Mean-Field Renormalizat ion Group Approach for the Ising Model

A double-chain mean-field renormalization group approach for the Ising model is formulated. With the approach, transition temperatures and critical exponents for several numbers of nearest-neighbor chains are obtained.     

Determination of critical points and flow diagrams by Monte Carlo renormalization group methods

A general method for calculating block renormalized coupling constants within the framework of the Monte Carlo renormalization group is presented. The method is applicable for any values of the couplings and in particular for those far from the critical point. A new technique for evaluating separately the derivatives of the block renormalized couplings is also discussed. The utility of these methods is demonstrated on the two-dimensional Ising model, where knowledge of the exact critical point in the multiparameter space of coupling constants results in improved values of the critical exponents.     

Ising model in an external field on a hierarchical lattice

The two-dimensional renormalization map of the diamond-hierarchical Ising model in an external field is given, and pictures of the distribution of zeros of the partition function in the complex plane of temperature for varying values of coupling constant and external field are shown. Critical exponents of the model are found, and results are different from those of the Ising model on a two- or three-dimensional regular lattice.     

Operator oriented optimization of block spin transformations

We propose a generally applicable method to optimize block spin transformations. Working in the sector of even operators the renormalization transformation is adjusted such that the resulting renormalized trajectory is close to the trajectory of a simple few-parameter hamiltonian. The idea is tested within the d = 2 Ising model and leads to clear improvements in the determination of the critical exponents.     

Supersymmetry and its spontaneous breaking in the random field Ising model

We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) ? 5.1 and leads to a breakdown of the "dimensional reduction" property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.     

Block spin techniques applied to the ising model

A detailed study of the critical behaviour of the Ising model in 1+1 and 2+1 dimensions is made using an approximate real space renormalization transformation which involves block spins. The critical indices α, β, η, and ν are calculated and compared with previous results, as is the critical couplingy _c. The method is shown to respect one of the scaling relations, and in 1+1D some exact results are reproduced (y _c=1, ν=1).     

Dimensional reduction by Monte Carlo

A Monte Carlo method for mapping a field theoretical or statistical system to a new theory embedded in a space-time of lesser dimensionality is presented. Typically, the critical properties of the dimensionally reduced system depend upon the details of the mapping. As an example, the two-dimensional Ising model is mapped to a one-dimensional Ising model with long-range forces and a phase transition. Systems with long-range interactions and known exponents can thus be constructed with this procedure.