Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and −1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x=0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L ^1.78(2). Anomalous scaling is not found for the characteristic energy, which essentially scales as L ². When x=0.25, a crossover to L ² scaling of the entropy is seen near L=12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L. PACS numbers: 75.10.Nr, 75.40.Mg, 75.50.Lk     

Limit Theorems for Monolayer Ballistic Deposition in the Continuum

We consider a deposition model in which balls rain down at random towards a 2-dimensional surface, roll downwards over existing adsorbed balls, are adsorbed if they reach the surface, and discarded if not. We prove a spatial law of large numbers and central limit theorem for the ultimate number of balls adsorbed onto a large toroidal surface, and also for the number of balls adsorbed on the restriction to a large region of an infinite surface.     

Geometrical aspects of the Z-invariant Ising model

We discuss a geometrical interpretation of the Z-invariant Ising model in terms of isoradial embeddings of planar lattices. The Z-invariant Ising model can be defined on an arbitrary planar lattice if and only if certain paths on the lattice edges do not intersect each other more than once or self-intersect. This topological constraint is equivalent to the existence of isoradial embeddings of the lattice. Such embeddings are characterized by angles which can be related to the model coupling constants in the spirit of Baxter's geometrical solution. The Ising model on isoradial embeddings studied recently by several authors in the context of discrete holomorphy corresponds to the critical point of this particular Z-invariant Ising model.     

Finite-Size Scaling of the Domain Wall Entropy Distributions for the 2D ± J Ising Spin Glass

The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and −1 bonds are studied for L × L square lattices with L ≤ 48, and p = 0.5, where p is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. When L is even, almost all domain walls have energy E _dw = 0 or 4. When L is odd, most domain walls have E _dw = 2. The probability distribution of the entropy, S _dw , is found to depend strongly on E _dw . When E _dw = 0, the probability distribution of |S _dw | is approximately exponential. The variance of this distribution is proportional to L, in agreement with the results of Saul and Kardar. For E _dw = k > 0 the distribution of S _dw is not symmetric about zero. In these cases the variance still appears to be linear in L, but the average of S _dw grows faster than L. This suggests a one-parameter scaling form for the L-dependence of the distributions of S _dw for k> 0. PACS: 75.10.Nr, 75.40.Mg, 75.60.Ch, 05.50.+q     

Phase transition in the Blume-Capel model with second neighbour interaction

A two dimensional antiferromagnetic spin-1 Ising model with negative next- nearest neighbour interaction (J ₂ <0) and under an external magnetic field is investigated by two methods: The mean-field theory and Finite-Size-Scaling based on transfer matrix (TMFSS) calculations. The ground state diagrams exhibit several new phases including frustrated ones. At finite temperature we obtain by these two methods quite rich phase diagrams, with several multicritical points. While Mean field approximation yields phase diagrams which are sometimes even qualitatively incorrect, accurate results are obtained from transfer matrix finite size scaling calculations. For a certain range of interaction parameters, the model is shown to violate the ordinary universality hypothesis. Received: 3 November 1997 / Revised: 31 March 1998 / Accepted: 7 April 1998     

Consistency of Microcanonical and Canonical Finite-Size Scaling

Typically, in order to obtain finite-size scaling laws for quantities in the microcanonical ensemble, an assumption is taken as a starting point. In this paper, consistency of such a Microcanonical Finite-Size Scaling Assumption with its commonly accepted canonical counterpart is shown, which puts Microcanonical Finite-Size Scaling on a firmer footing.     

Phase transitions in the mixed Ashkin-Teller model

By using a mean-field approximation (MFA) and Monte-Carlo (MC) simulations, we have studied the effect on the phase diagrams of mixed spins ( and S =1) in the Ashkin-Teller model (ATM) on a hypercubic lattice. By varying the strength describing the four spin interaction and the single ion potential, we have obtained by these two methods quite rich phase diagrams with several multicritical points. This model exhibits a new partially ordered phase which does not exist neither in the spin-1/2 ATM nor in the spin-1 ATM. While MFA yields phase diagrams which are sometimes qualitatively incorrect, accurate results are obtained from MC simulations. From the critical exponents which have been calculated using finite-size scaling ideas, we have shown that all phase transitions are Ising-like except for the paramagnetic-Baxter critical surface on which the critical exponents vary continuously, by varying only the strength of the coupling interaction independently of the value of the single ion potential. Received 5 July 1999 and Received in final form 4 July 2000     

On the Classical Limit of Quantum Mechanics

Contrary to the widespread belief, the problem of the emergence of classical mechanics from quantum mechanics is still open. In spite of many results of the standard approach, it is not yet clear how to explain within standard quantum mechanics the classical motion of macroscopic bodies. In this paper, we shall formulate the classical limit as a scaling limit in terms of an adimensional parameter ε. We shall take the first steps toward a comprehensive understanding of the classical limit, analyzing special cases of classical behavior in the framework of a precise formulation of quantum mechanics called Bohmian mechanics which contains in its own structure the possibility of describing real objects in an observer-independent way.     

Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

Consider a cellular automaton with state space {0,1} ² where the initial configuration _0 is chosen according to a Bernoulli product measure, 1s are stable, and 0s become 1s if they are surrounded by at least three neighboring 1s. In this paper we show that the configuration _n at time n converges exponentially fast to a final configuration , and that the limiting measure corresponding to is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents , , and , and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ² (i.e. for independent *-percolation on ), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents.This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement [Aizenman and Grimmett, J. Stat. Phys. 63: 817--835 (1991); Grimmett, Percolation, 2nd Ed. (Springer, Berlin, 1999)     

Anisotropy of the interface tension of the three-dimensional Ising model

We determine the interface tension for the 100, 110 and 111 interface of the simple cubic Ising model with nearest-neighbour interaction using novel simulation methods. To overcome the droplet/strip transition and the droplet nucleation barrier we use a newly developed combination of the multimagnetic algorithm with the parallel tempering method. We investigate a large range of inverse temperatures to study the anisotropy of the interface tension in detail.     

Octic Anharmonic Oscillators: Perturbed Coherent States and the Classical Limit

We use the Poincaré-Linstedt method to find a classical perturbation solution to the octic anharmonic oscillator. Next, we derive perturbed coherent states for this system, calculate the expectation value of the -operator in them and enforce a limiting process to retrieve the classical result from the corresponding quantum one. We have observed a frequency shift proportional to the sixth power of the amplitude for this system. Our results are in agreement with those obtained from Taylor-series method.     

A Nontrivial Scaling Limit for Multiscale Markov Chains

We consider Markov chains with fast and slow variables and show that in a suitable scaling limit, the dynamics becomes deterministic, yet is far away from the standard mean field approximation. This new limit is an instance of self-induced stochastic resonance which arises due to matching between a rare event timescale on the one hand and the natural timescale separation in the underlying problem on the other. Here it is illustrated on a model of a molecular motor, where it is shown to explain the regularity of the motor gait observed in some experiments.     

The Finite-Size Scaling Functions of the Four-Dimensional Ising Model

A finite-size scaling function of the Privman–Fisher form is proposed for the singular part of the free-energy density of the four-dimensional Ising model. It leads to the finite-size scaling relations available and to the prediction of new ones.     

Coupling between magnetic field and curvature in Heisenberg spins on surfaces with rotational symmetry

We study the nonlinear σ-model in an external magnetic field applied on curved surfaces with rotational symmetry. The Euler–Lagrange equations derived from the Hamiltonian yield the double sine-Gordon equation (DSG) provided the magnetic field is tuned with the curvature of the surface. A 2π skyrmion appears like a solution for this model and surface deformations are predicted at the sector where the spins point in the opposite direction to the magnetic field. We also study some specific examples by applying the model on three rotationally symmetric surfaces: the cylinder, the catenoid and the hyperboloid.     

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     

Limit probability distributions for an infinite-order phase transition model

The limiting probability distributions for the one-dimensional inhomogeneous spin system considered in a previous paper, which exhibits an infinite-order phase transition, are computed. It turns out that below the critical temperature or in the presence of an external magnetic field, the spins are completely polarized.