Numerical Calculations for the Surface on a Semi-Infinite Ising Model with a Random Surface Field

We study the critical behavior of the surface on a semi-infinite simple cubic lattice Ising model with a bimodal random surface field by large cell mean-field renormaliza tion group method and Monte Carlo simulations. Our results show that the surface ferromagnetic phase exists in the weak random field range above the bulk critical temperature. The surface. specific heat is not divergence and the susceptibility show a cusp singularity at the surface ferromagnetic-paramagnetic transition for a relatively large and om field.     

Universality of the ratio of the critical amplitudes of the magnetic susceptibility in a two-dimensional ising model with nonmagnetic impurities

The behavior of the magnetic susceptibility of a two-dimensional Ising model with nonmagnetic impurities is investigated numerically. A new method for determining the critical amplitudes and critical temperature is developed. The results of a numerical investigation of the ratio of the critical amplitudes of the magnetic susceptibility are presented. It is shown that the ratio of the critical amplitudes is universal right up to impurity concentrations q ≤ 0.25 (the percolation point of a square lattice is q _c = 0.407254). The behavior of the effective critical exponent γ(q) of the magnetic susceptibility is discussed. Apparently, a transition from Ising-type universal behavior to percolation behavior should occur in a quite narrow concentration range near the percolation point of the lattice.     

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.     

Critical and tricritical exponents of the Potts lattice gas

Critical and tricritical exponents for the two-dimensionalq-state Potts lattice gas are calculated using a Kadanoff lower-bound renormalization transformation with three variational parameters. The calculations support the picture proposed by Nienhuis et al. in which the critical and tricritical fixed points annihilate atq=4. For the most part, the exponents are in excellent agreement with the recent conjectures of den Nijs and of Nienhuis et al.     

A new type of decoupling approximation for ising model with spin 1/2

By introducing a new type of decoupling approximation within the framework of an effective field theory, we discuss the critical temperature of an isotropic square and simple cubic lattice in an Ising model. It is also derived the critical value _c above which the magnetism occurs in a free surface of a semi-infinite medium. The present formalism yields results quite superior to the other effective fields theories and in good agreement with series expansions and exact's one.     

High-temperature expansion of surface critical exponents for the classical n-vector model

The surface critical exponents of the classical n-vector model are investigated in the high-temperature series expansion. The first nine coefficients of the layer susceptibility for the semi-infinite simple cubic lattice are presented. The n-dependence of surface exponents is compared with the results of other theories.     

On Kadanoff's approximate renormalization group transformation

The fixed point structure resulting from the approximate renormalization group equations obtained by shifting bonds on the square Ising lattice is considered as a function of a free parameter h appearing in the definition of these equations. Next to the fixed point S considered by Kadanoff which is located in a symmetry plane two other “critical” fixed points A and B are found for h0.726. At the value h = 0.741, A crosses the fixed point S and vanishes together with the fixed point B at h = 0.726. Furthermore correction terms to the eigenvalues of the linearized renormalization group equations as obtained by Kadanoff are considered which arise if one chooses h to be optimal at all points of the coupling parameter space.     

Bulk and boundary critical behavior at Lifshitz points

Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard φ⁴ model. Analyzing these models systematically via modern field-theoretic renormalization group methods has been a long-standing challenge ever since their introduction in the middle of 1970s. We survey the recent progress made in this direction, discussing results obtained via dimensionality expansions, how they compare with Monte Carlo results, and open problems. These advances opened the way towards systematic studies of boundary critical behavior atm-axial Lifshitz points. The possible boundary critical behavior depends on whether the surface plane is perpendicular to one of them modulation axes or parallel to all of them. We show that the semi-infinite field theories representing the corresponding surface universality classes in these two cases of perpendicular and parallel surface orientation differ crucially in their Hamiltonian’s boundary terms and the implied boundary conditions, and explain recent results along with our current understanding of this matter.     

Site-bond percolation problems

We study a percolation process in which both sites and bonds are randomly blocked, independent of each other. In the Bethe lattice, the exact solution for the percolation threshold is found to be a hyperbola in thex-p plane, wherex andp are the respective probabilities of each site and bond being unblocked. Percolation threshold for a square and a simple cubic lattice is obtained by computer simulation. We also present a result obtained by a real-space renormalization group technique for the square lattice.     

The effects of surfaces on dynamic critical behavior

The field theoretic renormalization-group approach for the study of critical behavior near free surfaces is generalized to dynamic properties. Time-dependent Ginzburg-Landau models with nonconserved or conserved order parameters — semi-infinite generalizations of the so-called modelsA andB — are considered. The asymptotic behavior of response and correlation functions is analyzed at the ordinary and special transitions in 4-? dimensions, and dynamic scaling laws for surface quantities are obtained. It is shown that the critical exponents can be expressed entirely in terms of static bulk and surface exponents andz, the dynamic bulk exponent. The critical exponents for the leading frequency, temperature and momentum singularities of the surface two-spin correlation function at the ordinary transition differ appreciably from the corresponding bulk analogues. In addition, the shape function which describes its frequency dependence differs qualitatively from the one of the bulk correlation function.     

Fractal structure of zeros in hierarchical models

We consider an Ising and aq-state Potts model on a diamond hierarchical lattice. We give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq. These zeros are just the Julia set corresponding to the renormalization group transformation.     

Surface Transitions of the Semi-Infinite Potts Model II: The Low Bulk Temperature Regime

We consider the semi-infinite q–state Potts model. We prove, for large q, the existence of a first order surface phase transition between the ordered phase and the the so-called “new low temperature phase” predicted in,Li in which the bulk is ordered whereas the surface is disordered.     

Investigation of the critical behavior of the three-dimensional weakly diluted Potts model

The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v.     

Classical spin systems in the presence of a wall: Multicomponent spins

In this paper we investigate classical spin systems on a semi-infinite lattice. We establish detailed properties of such systems near the surface layer. For the Ising- and the classicalXY models on a semi-infinite lattice we study the phase diagram, the critical properties and the decay of spin-spin correlations near the surface layer.     

Semi-infinite Ising model

For the semi-infinite Ising model in two or more dimensions, we prove analyticity properties of the surface free energy and map out the phase diagram in the absence of an external magnetic field. We prove that this phase diagram contains critical lines where the parallel and/or the transverse correlation lengths diverge. The critical exponent,v _⊥, of the transverse correlation length is shown to be equal to the exponentv of the Ising model on an infinite lattice. In a second paper, these results will be used to analyze the wetting transition.     

Effective medium approach to real space renormalization

We use the ideas of effective medium theory to present a self-consistent decimation which yields the exact fixed point for the q-component Potts model in a square lattice and also generates the exact critical surface for anisotropic interactions.     

Critical wetting in the square Ising model with a boundary field

The Ising square lattice with nearest-neighbor exchangeJ>0 and a free surface at which a boundary magnetic fieldH ₁ acts has a second-order wetting transition. We study the surface excess magnetization and the susceptibility ofL×M lattices by Monte Carlo simulation and probe the critical behavior of this wetting transition, applying finite-size scaling methods. For the cases studied, the results are not consistent with the presumably exactly known values of the critical exponents, because the asymptotic critical region has not yet been reached. Implication of our results for critical wetting in three dimensions and for the application of the present model to adsorbed wetting layers at surface steps are briefly discussed.Alexander von Humboldt-Fellow     

The Kadanoff lowerbound renormalization transformation for the q-state Potts model

The Kadanoff lowerbound renormalization transformation when applied to the 2-dimensional q-state Potts model, is found to show a bifurcation phenomenon at q = 4, that might be considered as signalling the onset to the first-order transition. At the value of the free parameter where the bifurcation is found, the specific heat exponent takes almost the value predicted by weak universality $\text{α(4) =}\text{2}\text{3}$, while the cross-over exponent in the Potts-lattice gas direction becomes marginal. The cross-over exponent in the cubic direction is found already to be irrelevant for q > 3.3. Further a duality relation for a class of models obtained by a partial breaking of the Potts symmetry in the hamiltonian, including the cubic model is derived.     

Critical behavior of the one-dimensional commensurate-incommensurate transition

We study the low-temperature critical behavior of one-dimensional charge-density-waves coupled to an underlying lattice, using the McMillan free energy. For weak coupling, the incommensurate CDW orders at T = 0 as a lattice of phase slip solutions with XY critical behavior. For strong coupling, the commensurate CDW orders at T = 0 with Ising critical behavior. Analytic expressions for the low-temperature inverse correlation length and average phase change are obtained for all values of the coupling to the lattice.     

On the electrostatic potential due to a slab shaped and a semi-infinite NaCl-type lattice

Consider N layers of a NaCl-type ionic lattice such that in every layer one has an infinite square lattice of positive and negative unit point charges. We present formulae in which the electrostatic potential in an arbitrary field point is expressed as a sum of two rapidly converging lattice sums. For N→∞ we obtain formulae applicable for a semi-infinite lattice.