Finite size scaling for O(N) φ4-theory at the upper critical dimension

A finite size scaling theory for the partition function zeroes and thermodynamic functions of O(N) φ⁴-theory in four dimensions is derived from renormalization group methods. The leading scaling behaviour is mean-field like with multiplicative logarithmic corrections which are linked to the triviality of the theory. These logarithmic corrections are independent of N for odd thermodynamic quantities and associated zeroes and are N dependent for the even ones. Thus a numerical study of finite size scaling in the Ising model serves as a non-perturbative test of triviality of φ⁴₄-theories for all N.     

Universality in quantum chaos and the one-parameter scaling theory

The one-parameter scaling theory is adapted to the context of quantum chaos. We define a generalized dimensionless conductance, g, semiclassically and then study Anderson localization corrections by renormalization group techniques. This analysis permits a characterization of the universality classes associated to a metal (g-->infinity), an insulator (g-->0), and the metal-insulator transition (g-->g(c)) in quantum chaos provided that the classical phase space is not mixed. According to our results the universality class related to the metallic limit includes all the systems in which the Bohigas-Giannoni-Schmit conjecture holds but automatically excludes those in which dynamical localization effects are important. The universality class related to the metal-insulator transition is characterized by classical superdiffusion or a fractal spectrum in low dimensions (d < or = 2). Several examples are discussed in detail.     

Field induced order in magnetic systems: Marginal case

The bond operator representation and the one-loop renormalization group treatment are used to study the spin-1 Heisenberg antiferromagnetic with single-ion anisotropy and transversal magnetic fields in three-dimensional cubic lattices. We start from a disordered spin-liquid phase to an ordered phase, at a critical field H_c1 above which the system enters an XY-antiferromagnetic phase. This transition is interpreted as belonging to a universality class with a dynamical critical exponent z=1. In this marginal case logarithmic corrections are found to the physical quantities. These theoretical predictions are compared with the scaling of the magnetization as a function of field and temperature for the organic compound NiCl₂-4SC(NH₂)₂.     

Local waiting times in critical systems

The quiet times at a fixed point in space are investigated in a system close to or at a non-equilibrium phase transition. The statistics for such first-return times follow from the universality class of the dynamics and the ensemble: for a power-law waiting time distribution the exponent depends on the dimension and the underlying model. We study the two-dimensional Manna sandpile, with both the continously driven self-organized version and the tuned one. The latter has an absorbing state or depinning phase transition at a critical value of the control parameter. The connection to a driven interface in a random medium gives the exponent of the waiting time distribution. In the open ensemble, differences ensue due to the spatial inhomogeneity and the properties of the driving signal. For both ensembles, the waiting time distributions are found to exhibit logarithmic corrections to scaling.Received: 13 September 2004, Published online: 23 December 2004PACS: 05.70.Ln Nonequilibrium and irreversible thermodynamics - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 52.25.Fi Transport properties     

Anderson transition in two-dimensional systems with spin-orbit coupling

We report a numerical investigation of the Anderson transition in two-dimensional systems with spin-orbit coupling. An accurate estimate of the critical exponent nu for the divergence of the localization length in this universality class has to our knowledge not been reported in the literature. Here we analyze the SU(2) model. We find that for this model corrections to scaling due to irrelevant scaling variables may be neglected permitting an accurate estimate of the exponent nu=2.73+/-0.02.     

Antisymmetric and other subleading corrections to scaling in the local potential approximation

For systems in the universality class of the three-dimensional Ising model we compute the critical exponents in the local potential approximation (LPA), that is, in the framework of the Wegner–Houghton equation. We are mostly interested in antisymmetric corrections to scaling, which are relatively poorly studied. We find the exponent for the leading antisymmetric correction to scaling ω_A≈1.691 in the LPA. This high value implies that such corrections cannot explain asymmetries observed in some Monte Carlo simulations.     

Time distribution and loss of scaling in granular flow

Two cellular automata models with directed mass flow and internal time scales are studied by numerical simulations. Relaxation rules are a combination of probabilistic critical height (probability of toppling p) and deterministic critical slope processes with internal correlation time t_c equal to the avalanche lifetime, in model A, and ,in model B. In both cases nonuniversal scaling properties of avalanche distributions are found for , where is related to directed percolation threshold in d=3. Distributions of avalanche durations for are studied in detail, exhibiting multifractal scaling behavior in model A, and finite size scaling behavior in model B, and scaling exponents are determined as a function of p. At a phase transition to noncritical steady state occurs. Due to difference in the relaxation mechanisms, avalanche statistics at approaches the parity conserving universality class in model A, and the mean-field universality class in model B. We also estimate roughness exponent at the transition. Received: 29 May 1998 / Revised: 8 September 1998 / Accepted: 10 September 1998     

Universal amplitude ratios in the 2D four-state Potts model

We present a Monte Carlo study of various universal amplitude ratios of the two-dimensional q=4 Potts model. We simulated the model close to criticality in both phases taking care to keep the systematic errors, due to finite size effects and logarithmic corrections in the scaling functions, under control. Our results are compatible with those recently obtained using the form-factor approach and with the existing low temperature series for the model.     

Effects of quenched disorder in the two-dimensional Potts model: a Monte Carlo study

Motivated by recent experiments on phase behavior of systems confined in porous media, we have studied the effect of randomness on the nature of the phase transition in the two-dimensional Potts model. To model the effects of the porous matrix we introduce a random distribution of couplings P(J(ij))=pdelta(J(ij)-J1)+(1-p)delta(J(ij)-J2) in the q state Potts Hamiltonian. An extensive Monte Carlo study is made on this system for q=5. We studied two different cases of disorder (a) J(1)/J(2)-->infinity and p=0.8 and (b) J(1)/J(2)=10 and p=0.5. We observed, in both cases, that the weak first order transition that appears in the pure case, changes to a second-order transition. A finite size scaling analysis shows that the correlation length exponent nu is close to 1 and the best fit to the dependence of the specific heat on system size is logarithmic. This suggests that both cases belong to the universality class of the Ising model. In contrast, the magnetic exponents point to a different universality class.     

Polymers and g|φ|4 theory in four dimensions

We investigate the approach to the critical point and the scaling limit of a variety of models on a four-dimensional lattice, including g|φ|₄⁴ theory and the self-avoiding random walk. Our results, both theoretical and numerical, provide strong evidence for the triviality of the scaling limit and for logarithmic corrections to mean field scaling laws, as predicted by the perturbative renormalization group. We relate logarithmic corrections to scaling to the triviality of the scaling limit. Our numerical analysis is based on a novel, high-precision Monte Carlo technique.     

Wang-Landau study of the 3D Ising model with bond disorder

We implement a two-stage approach of the Wang-Landau algorithm to investigate the critical properties of the 3D Ising model with quenched bond randomness. In particular, we consider the case where disorder couples to the nearest-neighbor ferromagnetic interaction, in terms of a bimodal distribution of strong versus weak bonds. Our simulations are carried out for large ensembles of disorder realizations and lattices with linear sizes L in the range L=8-64L=8{-}64. We apply well-established finite-size scaling techniques and concepts from the scaling theory of disordered systems to describe the nature of the phase transition of the disordered model, departing gradually from the fixed point of the pure system. Our analysis (based on the determination of the critical exponents) shows that the 3D random-bond Ising model belongs to the same universality class with the site- and bond-dilution models, providing a single universality class for the 3D Ising model with these three types of quenched uncorrelated disorder.     

T-Duality Simplifies Bulk-Boundary Correspondence

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice \({{{\mathbb{Z}}}^{4}}\), for the weakly coupled n-component \({|\varphi|^{4}}\) spin model for all \({n \ge 1}\), and for the continuous-time weakly self-avoiding walk. For the \({|\varphi|^{4}}\) model, we prove that the critical two-point function has |x|^?2 (Gaussian) decay asymptotically, for \({n \ge 1}\). We also determine the asymptotic decay of the critical correlations of the squares of components of \({\varphi}\), including the logarithmic corrections to Gaussian scaling, for \({n \ge 1}\). The above extends previously known results for n = 1 to all \({n \ge 1}\), and also observes new phenomena for n > 1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the “watermelon” network consisting of p weakly mutually- and self-avoiding walks, for all \({p \ge 1}\), including the logarithmic corrections. This extends a previously known result for p = 1, for which there is no logarithmic correction, to a much more general setting. In addition, for both models, we study the approach to the critical point and prove the existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the n = 0 case of the \({|\varphi|^{4}}\) model, and provides a unified treatment of both models, and of all the above results.     

Melting and wetting transitions in the three-state chiral clock model

The melting transition of the two-dimensional, three-state, asymmetric or chiral clock model is examined. Evidence from scaling arguments and analysis of perturbation series is presented, indicating that the chiral symmetry-breaking operator is relevant at the symmetric (or pure Potts) critical point with a crossover exponent of ø ≈ 0.2. The remainder of the commensurate-disordered phase boundary therefore appears to be in a new universality class, distinct from the pure three-state Potts transition. An interfacial wetting transition that plays an important role in the crossover between the two types of critical behavior is discussed. The location and exponents of this wetting transition are obtained both in a low-temperature limit using generating function techniques and in a systematic low-temperature expansion of the transfer matrix.     

Interface Motion in Disordered Ferromagnets

We consider numerically the depinning transition in the random-field Ising model. Our analysis reveals that the three and four dimensional model displays a simple scaling behavior whereas the five dimensional scaling behavior is affected by logarithmic corrections. This suggests that d = 5 is the upper critical dimension of the depinning transition in the random-field Ising model. Furthermore, we investigate the so-called creep regime (small driving fields and temperatures) where the interface velocity is given by an Arrhenius law.     

Criticality of Parasitic Disease Transmission in a Diffusive Population

Through using the methods of finite-size effect and short time dynamic scaling, we study the critical behavior of parasitic disease spreading process in a diffusive population mediated by a static vector environment. Through comprehensive analysis of parasitic disease spreading we find that this model presents a dynamical phase transition from disease-free state to endemic state with a finite population density. We determine the critical population density, above which the system reaches an epidemic spreading stationary state. We also perform a scaling analysis to determine the order parameter and critical relaxation exponents. The results show that the model does not belong to the usual directed percolation universality class and is compatible with the class of directed percolation with diffusive and conserved fields.     

New ordered phases in a class of generalized XY models

It is well known that the 2D XY model exhibits an unusual infinite order phase transition belonging to the Kosterlitz-Thouless (KT) universality class. Introduction of a nematic coupling into the XY Hamiltonian leads to an additional phase transition in the Ising universality class [D. H. Lee and G. Grinstein, Phys. Rev. Lett. 55, 541 (1985)]. Using a combination of extensive Monte Carlo simulations and finite size scaling, we show that the higher order harmonics lead to a qualitatively different phase diagram, with additional ordered phases originating from the competition between the ferromagnetic and pseudonematic couplings. The new phase transitions belong to the 2D Potts, Ising, or KT universality classes.     

Finite-temperature scaling of quantum coherence near criticality in a spin chain

We explore quantum coherence, inherited from Wigner-Yanase skew information, to analyzequantum criticality in the anisotropic XY chain model at finite temperature. Based on theexact solutions of the Hamiltonian, the quantum coherence contained in a nearest-neighborspin pairs reduced density matrix ρ is obtained. The first-order derivative of thequantum coherence is non-analytic around the critical point at sufficient low temperature.The finite-temperature scaling behavior and the universality are verified numerically. Inparticular, the quantum coherence can also detect the factorization transition in such amodel at sufficient low temperature. We also show that quantum coherence contained indistant spin pairs can characterize quantum criticality and factorization phenomena atfinite temperature. Our results imply that quantum coherence can serve as an efficientindicator of quantum criticality in such a model and shed considerable light on therelationships between quantum phase transitions and quantum information theory at finitetemperature.     

Superconductor-insulator transition of Josephson-junction arrays on a honeycomb lattice in a magnetic field

We study the superconductor to insulator transition at zero temperature in aJosephson-junction array model on a honeycomb lattice with f flux quantum perplaquette. The path integral representation of the model corresponds to a (2 + 1)-dimensional classical model, which isused to investigate the critical behavior by extensive Monte Carlo simulations on largesystem sizes. In contrast to the model on a square lattice, the transition is found to befirst order for f = 1 /3 and continuous for f = 1 / 2 but in a different universality class.The correlation-length critical exponent is estimated from finite-size scaling of vortexcorrelations. The estimated universal conductivity at the transition is approximately fourtimes its value for f =0. The results are compared with experimental observations on ultrathinsuperconducting films with a triangular lattice of nanoholes in a transverse magneticfield.     

Universal behavior of crossover scaling functions for continuous phase transitions

We consider two different systems exhibiting a continuous phase transition into an absorbing state. Both models belong to the same universality class; i.e., they are characterized by the same scaling functions and the same critical exponents. Varying the range of interactions, we examine the crossover from the mean-field-like to the non-mean-field scaling behavior. A phenomenological scaling form is applied in order to describe the full crossover region, which spans several decades. Our results strongly support the hypothesis that the crossover function is universal.     

How to discriminate easily between directed-percolation and Manna scaling

Here we compare critical properties of systems in the directed-percolation (DP) universality class with those of absorbing-state phase transitions occurring in the presence of a non-diffusive conserved field, i.e., transitions in the so-called Manna or C-DP class. Even if it is clearly established that these constitute two different universality classes, most of their universal features (exponents, moment ratios, scaling functions,...) are very similar, making it difficult to discriminate numerically between them. Nevertheless, as illustrated here, the two classes behave in a rather different way upon introducing a physical boundary or wall. Taking advantage of this, we propose a simple and fast method to discriminate between these two universality classes. This is particularly helpful in solving some existing discrepancies in self-organized critical systems as sandpiles.