Finite-size scaling for the ising model on the Möbius strip and the klein bottle

We study the finite-size scaling properties of the Ising model on the M?bius strip and the Klein bottle. The results are compared with those of the Ising model under different boundary conditions, that is, the free, cylindrical, and toroidal boundary conditions. The difference in the magnetization distribution function p(m) for various boundary conditions is discussed in terms of the number of the percolating clusters and the cluster size. We also find interesting aspect-ratio dependence of the value of the Binder parameter at T = T(c) for various boundary conditions. We discuss the relation to the finite-size correction calculations for the dimer statistics.     

Superscaling of percolation on rectangular domains

For percolation on (RL)xL two-dimensional rectangular domains with a width L and aspect ratio R, we propose that the existence probability of the percolating cluster E(p)(L,epsilon,R) as a function of L, R, and deviation from the critical point epsilon can be expressed as F(epsilonL(y(t))R(a)), where y(t) identical with1/nu is the thermal scaling power, a is a new exponent, and F is a scaling function. We use Monte Carlo simulation of bond percolation on square lattices to test our proposal and find that it is well satisfied with a=0.14(1) for R>2. We also propose superscaling for other critical quantities.     

Neél order in square and triangular lattice Heisenberg models

We show that the density matrix renormalization group can be used to study magnetic ordering in two-dimensional spin models. Local quantities should be extrapolated with the truncation error, not with its square root. We introduce sequences of clusters, using cylindrical boundary conditions with pinning fields, which provide for rapidly converging finite-size scaling. We determine the magnetization for both the square and triangular Heisenberg lattices with errors comparable to the best alternative approaches.     

Percolation processes in monomer-polyatomic mixtures

In this paper, the percolation of mixtures of monomers and polyatomic species (k$k$-mers) on a square lattice is studied. By means of a finite-size scaling analysis, the critical exponents and the scaling collapsing of the fraction of percolating lattices are found. A phase diagram separating a percolating from a non-percolating region is determined. The main features of the phase diagram are discussed in order to predict its evolution for larger k$k$-mer sizes.     

Monte carlo simulation of the two-spin facilitated model above the thermodynamic transition temperature

Results of a Monte Carlo simulation of the two-spin facilitated model proposed by Fredrickson and Andersen above the thermodynamic transition temperature without external field on the square lattice are presented. The model is a kinetic Ising model with a special constraint to its kinetic process and was designed to simulate viscous fluid. A time correlation function for uniform magnetization is analyzed using the idea of finite-size scaling and percolation length. For the system size that was simulated, our data fit into a scaling plot adopting the percolation length as a length scale, and we obtain the dynamical critical exponent z ∼ 4, which is different from the usual value z ∼ 2.     

Yang-Lee zeros of the antiferromagnetic Ising model

There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a相似文献   

Multicritical point in a diluted bilayer Heisenberg quantum antiferromagnet

The S=1/2 Heisenberg bilayer antiferromagnet with randomly removed interlayer dimers is studied using quantum Monte Carlo simulations. A zero-temperature multicritical point (p(*),g(*)) at the classical percolation density p=p(*) and interlayer coupling g(*) approximately equal 0.16 is demonstrated. The quantum critical exponents of the percolating cluster are determined using finite-size scaling. It is argued that the associated finite-temperature quantum critical regime extends to zero interlayer coupling and could be relevant for antiferromagnetic cuprates doped with nonmagnetic impurities.     

Entropic rigidity of randomly diluted two- and three-dimensional networks

In recent work, we presented evidence that site-diluted triangular central-force networks, at finite temperatures, have a nonzero shear modulus for all concentrations of particles above the geometric percolation concentration p(c). This is in contrast to the zero-temperature case where the (energetic) shear modulus vanishes at a concentration of particles p(r)>p(c). In the present paper we report on analogous simulations of bond-diluted triangular lattices, site-diluted square lattices, and site-diluted simple-cubic lattices. We again find that these systems are rigid for all p>p(c) and that near p(c) the shear modulus mu approximately (p-p(c))(f), where the exponent f approximately 1.3 for two-dimensional lattices and f approximately 2 for the simple-cubic case. These results support the conjecture of de Gennes that the diluted central-force network is in the same universality class as the random resistor network. We present approximate renormalization group calculations that also lead to this conclusion.     

Scaling of the Distribution of Shortest Paths in Percolation

We present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for (i) the effect of the finite size of the system and (ii) the dependence of this distribution on the site occupancy probability p. We test the hypothesis for the case of two-dimensional percolation.     

Numerical results for the three-state critical Potts model on finite rectangular lattices

Partition functions for the three-state critical Potts model on finite square lattices and for a variety of boundary conditions are presented. The distribution of their zeros in the complex plane of the spectral variable is examined and is compared to the expected infinite-lattice result. The partition functions are then used to test the finite-size scaling predictions of conformal and modular invariance.     

Linking numbers for self-avoiding loops and percolation: application to the spin quantum hall transition

Nonlocal twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which count the numbers of self-avoiding loops (respectively, percolation clusters) surrounding a given point. Their scaling dimensions are computed exactly. This yields many results: for example, the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. Its mean behaves as (1/3sqrt[3] pi) |ln( p(c)-p)| as p-->p(c)-. As an application we compute the exact value sqrt[3]/2 for the conductivity at the spin Hall transition, as well as the shape dependence of the mean conductance in an arbitrary simply connected geometry with two extended edge contacts.     

Percolation on patchwise heterogeneous surfaces under equilibrium conditions

The site percolation problem on square lattices whose sites are grouped in two types of energetically different patches is studied. Several lattices formed by collections of either randomly or orderly localized and no overlapped patches of different sizes are generated. The system is characterized by two parameters, namely, the size of each patch, l, and the energy difference between the two kind of sites, ΔE. Particles are adsorbed at equilibrium on the lattice. The critical coverage is determined by means of Monte Carlo simulations and finite-size scaling analysis. The percolative behavior of the system as a function of the parameters characterizing the heterogeneity of the energetic surface topography is presented and discussed.     

Study of the adsorption of interacting dimers on square lattices by using a cluster-exact approximation

A theoretical approach, based on exact calculation of the partition function on finite rectangular clusters, is introduced to study the adsorption of interacting homonuclear dimers on square lattices. An efficient algorithm allows us to calculate the detailed structure of the configuration space for m=(k×l) clusters with m varying from 8 to 48. The adsorption process has been monitored by following thermodynamic properties such as coverage versus chemical potential, internal energy and specific heat of the adlayer, etc. The analytical results are compared with those in [Surf. Sci. 411 (1998) 294], which were obtained by using Monte Carlo simulation and finite-size scaling techniques. The theoretical adsorption isotherms and phase diagrams (critical temperature versus coverage) for both attractive and repulsive lateral interactions are in good qualitative agreement with the computational data. This agreement between simulated and theoretical results supports the validity of the cluster-exact approximation proposed in this paper.     

Precise simulation of near-critical fluid coexistence

We present a novel method to derive liquid-gas coexisting densities, rho(+/-)(T), from grand canonical simulations (without knowledge of T(c) or criticality class). The minima of Q(L) identical with (2)(L)/(L) in an LxLxL box with m=rho-(L) are used to generate recursively an unbiased universal finite-size scaling function. Monte Carlo data for a hard-core square-well fluid and for the restricted primitive model electrolyte yield rho(+/-) to +/-1%-2% of rho(c) down to 1 part in 10(4)-10(3) of T(c) (and confirm well Ising character). Pressure mixing in the scaling fields is unequivocally revealed and indicates Yang-Yang ratios R(mu)=-0.04(4) and 0.2(6) for the two models, respectively.     

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     

Breakdown of universality in directed spiral percolation

Directed spiral percolation (DSP), percolation under both directional and rotational constraints, is studied on the triangular lattice in two dimensions (2D). The results are compared with that of the 2D square lattice. Clusters generated in this model are generally rarefied and have chiral dangling ends on both the square and triangular lattices. It is found that the clusters are more compact and less anisotropic on the triangular lattice than on the square lattice. The elongation of the clusters is in a different direction than the imposed directional constraint on both the lattices. The values of some of the critical exponents and fractal dimension are found considerably different on the two lattices. The DSP model then exhibits a breakdown of universality in 2D between the square and triangular lattices. The values of the critical exponents obtained for the triangular lattice are not only different from that of the square lattice but also different form other percolation models.Received: 12 March 2004, Published online: 23 July 2004PACS: 02.50.-r Probability theory, stochastic processes, and statistics - 64.60.-i General studies of phase transitions - 72.80.Tm Composite materials     

Superfluid density in the underdoped YBa2Cu3O7-x: evidence for d-density-wave order of the pseudogap

The investigation of the penetration depth lambda(ab)(T,p) in YBa2Cu3O7-x crystals allowed one to observe the following features of the superfluid density n(s)(T,p) proportional, variant lambda(-2)(ab)(T,p) as a function of temperature T0) depends on p slightly in the optimally and moderately doped regions (0.10相似文献   

Minimal path on the hierarchical diamond lattice

We consider the minimal paths on a hierarchical diamond lattice, where bonds are assigned a random weight. Depending on the initial distribution of weights, we find all possible asymptotic scaling properties. The different cases found are the small-disorder case, the analog of Lévy's distributions with a power-law decay at-, and finally a limit of large disorder which can be identified as a percolation problem. The asymptotic shape of the stable distributions of weights of the minimal path are obtained, as well as their scaling properties. As a side result, we obtain the asymptotic form of the distribution of effective percolation thresholds for finite-size hierarchical lattices.     

Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part II. Numerical methods

The scaling exponent and the scaling function for the 1D single-species coagulation model (A+AA) are shown to be universal, i.e., they are not influenced by the value of the coagulation rate. They are independent of the initial conditions as well. Two different numerical methods are used to compute the scaling properties of the concentration: Monte Carlo simulations and extrapolations of exact finite-lattice data. These methods are tested in a case where analytical results are available. To obtain reliable results from finite-size extrapolations, numerical data for lattices up to ten sites are sufficient.     

Finite-size scaling for mean-field percolation

By studying transfer matrix eigenvalues, correlation lengths for a mean field directed percolation model are obtained both near and far from the critical regime. Near criticality, finite-size scaling behavior is derived and an analytic technique is provided for obtaining the finite-size scaling function. Our methods involve the generating function, matched asymptotic expansions, and certain formulas developed for the study of eigenvalues of the transfer matrix for metastability.