Schrödinger invariance and strongly anisotropic critical systems

The extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated. For the special case of an anisotropy or dynamical exponent =z=2, the group of local scale transformation considered is the Schrödinger group, which can be obtained as the nonrelativistic limit of the conformal group. The requirement of Schrödinger invariance determines the two-point function in the bulk and reduces the three-point function to a scaling form of a single variable. Scaling forms are also derived for the two-point function close to a free surface which can be either spacelike or timelike. These results are reproduced in several exactly solvable statistical systems, namely the kinetic Ising model with Glauber dynamics, lattice diffusion, Lifshitz points in the spherical model, and critical dynamics of the spherical model with a nonconserved order parameter. For generic values of , evidence from higher-order Lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two-point function.     

On non-local representations of the ageing algebra

The ageing algebra is a local dynamical symmetry of many ageing systems, far from equilibrium, and with a dynamical exponent z=2$z=2$. Here, new representations for an integer dynamical exponent z=n$z=n$ are constructed, which act non-locally on the physical scaling operators. The new mathematical mechanism which makes the infinitesimal generators of the ageing algebra dynamical symmetries, is explicitly discussed for an n-dependent family of linear equations of motion for the order-parameter. Finite transformations are derived through the exponentiation of the infinitesimal generators and it is proposed to interpret them in terms of the transformation of distributions of spatio-temporal coordinates. The two-point functions which transform co-variantly under the new representations are computed, which quite distinct forms for n even and n odd. Depending on the sign of the dimensionful mass parameter, the two-point scaling functions either decay monotonously or in an oscillatory way towards zero.     

Hidden local gauge invariance in integrable magnetic chains

It is shown that local gauge transformations preserve the integrability of one-dimensional quantum Heisenberg chains. Abelian U(1) gauge transformations associated to z-rotations appear in the XXZ model which is worked out in detail. The exact energy spectrum derived by the Bethe ansatz turns out to be gauge-invariant whereas the eigenvectors are explicitly gauge-dependent. Isotropic XXX chains exhibit SU(2) ⊗ Z₂ gauge invariance properties and anisotropic XYZ chains possess discrete Z₂ ⊗ Z₂ gauge invariance.     

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.     

Anisotropic scaling and generalized conformal invariance at Lifshitz points

The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are alpha = 0.18(2), beta = 0.238(5), and gamma = 1.36(3). Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.     

Scale invariance in complex seismic system and its uses in gaining precursory information before large earthquakes: Importance of methodology

Research on precursory information about the scale invariance of seismicity before large earthquakes has been an interesting topic for geophysicists in recent years. However, in some cases, it is difficult to capture this precursory information. Our study results show that seeking a pertinent selection of methodology is really needed for the purpose of gaining precursory information in the application of the analytical methods related to scale invariance. We investigated the scale invariance of the interevent time series of the seismic sequences for the Minle and Songpan regions in China by applying the methods of R/S$R/S$ Hurst analysis, local scaling property, correlation dimension, and generalized dimension spectrum. It is indicated that there are clearly precursory changes of local scaling property before the Minle M6.1 and Songpan M7.2 earthquakes, as well as precursory changes of the Hurst exponent and generalized dimension spectrum before the Songpan M7.2 earthquakes, while there are no clearly precursory changes of the Hurst exponent and generalized dimension spectrum before the Minle M6.1 earthquake, as well as the precursory change of the correlation dimension before the above large earthquakes, which signifies that there is a difference in the capability for capturing precursory information among the above four methods. This result suggests that the selection of an appropriate methodology is quite necessary to obtain the precursory information on the scale invariance in the use of the above four methods. If we do not select an appropriate methodology, we might not obtain any precursory information.     

On logarithmic extensions of local scale-invariance

Ageing phenomena far from equilibrium naturally present dynamical scaling and in many situations this may be generalised to local scale-invariance. Generically, the absence of time-translation-invariance implies that each scaling operator is characterised by two independent scaling dimensions. Building on analogies with logarithmic conformal invariance and logarithmic Schrödinger-invariance, this work proposes a logarithmic extension of local scale-invariance, without time-translation-invariance. Carrying this out requires in general to replace both scaling dimensions of each scaling operator by Jordan cells. Co-variant two-point functions are derived for the most simple case of a two-dimensional logarithmic extension. Their form is compared to simulational data for autoresponse functions in several universality classes of non-equilibrium ageing phenomena.     

Universal crossing probabilities and incipient spanning clusters in directed percolation

Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. In a dynamical interpretation, the crossing probability is the probability that, on a system with size L, an epidemic spreading without immunization remains active at time t. Since the system is strongly anisotropic, the shape dependence in space-time enters through the effective aspect ratio r _eff = ct/L ^z, where c is a non-universal constant and z the anisotropy exponent. A particular attention is paid to the influence of the initial state on the universal behaviour of the crossing probability. Using anisotropic finite-size scaling and generalizing a simple argument given by Aizenman for isotropic percolation, we also obtain the behaviour of the probability to find n incipient spanning clusters on a finite system at time t. The numerical results are in good agreement with the conjecture. Received 10 February 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: turban@lpm.u-nancy.fr RID="b" ID="b"UMR CNRS 7556     

Critical dynamics of the three-dimensional Ising model: A Monte Carlo study

Extensive Monte Carlo simulations have been performed to analyze the dynamical behavior of the three-dimensional Ising model with local dynamics. We have studied the equilibrium correlation functions and the power spectral densities of odd and even observables. The exponential relaxation times have been calculated in the asymptotic one-exponential time region. We find that the critical exponentz=2.09 ±0.02 characterizes the algebraic divergence with lattice size for all observables. The influence of scaling corrections has been analyzed. We have determined integrated relaxation times as well. Their dynamical exponentz _int agrees withz for correlations of the magnetization and its absolute value, but it is different for energy correlations. We have applied a scaling method to analyze the behavior of the correlation functions. This method verifies excellent scaling behavior and yields a dynamical exponentz _scal which perfectly agrees withz.     

The elliptical dimension of space-time atmospheric stratification of passive admixtures using lidar data

State-of-the-art airborne lidar data of passive scalars have shown that the spatial stratification of the atmosphere is scaling: the vertical extent (Δz) of structures is typically ≈Δx^H_z where Δx is the horizontal extent and H_z is a stratification exponent. Assuming horizontal isotropy, the volumes of the structures therefore vary as ΔxΔxΔx^H_z=Δx^D_s where the “elliptical dimension” D_s characterizes the rate at which the volumes of typical non-intermittent structures vary with scale. Work on vertical cross-sections has shown that 2+H_z=2.55±0.02 (close to the theoretical prediction 23/9).In this paper we extend these (x, z) analyses to (z, t). In the absence of overall advection, the lifetime Δt of a structure of size Δx varies as Δx^H_t with H_t=2/3 so that the overall space-time dimension is D_st=29/9=3.22…. However, horizontal and vertical advection lead to new exponents: we argue that the temporal stratification exponent H_t≈1 or ≈0.7 depending on the relative importance of horizontal versus vertical advection velocities. We empirically test these space-time predictions using vertical-time (z, t) cross-sections using passive scalar surrogates (aerosol backscatter ratios from lidar) at ∼3 m resolution in the vertical, 0.5-30 s in time and spanning 3-4 orders of magnitude in scale as well as new analyses of vertical (x, z) cross-sections (spanning over 3 orders of magnitude in both x, z directions). In order to test the theory for density fluctuations at arbitrary displacements in (Δz, Δt) and (Δx, Δz) spaces, we developed and applied a new Anisotropic Scaling Analysis Technique (ASAT) based on nonlinear coordinate transformations. Applying this and other analyses to data spanning more than 3 orders of magnitude of space-time scales we determined the anisotropic scaling of space-time finding the empirical value D_st=3.13±0.16. The analyses also show that both cirrus clouds and aerosols had very similar space-time scaling properties. We point out that this model is compatible with (nonlinear) “turbulence” waves, hence potentially explaining the observed atmospheric structures.     

Dynamic Monte Carlo renormalization group

A new and simple method of applying the idea of real space renormalization group theory to the analysis of Monte Carlo configurations is proposed and applied to the Glauber kinetic Ising model in two and three dimensions, and to the Kawasaki model in two dimensions. Our method, if correct, utilizes how the system approaches its equilibrium; in contrast to most other Monte Carlo investigations there is no need to wait until equilibrium is established. The renormalization analysis takes only a small fraction of the computer time needed to produce the Monte Carlo configurations, and the results are obtained as the system relaxes atT =T _c , the critical temperature. The values obtained for the dynamical critical exponent,z, are 2.12 (d=2) and 2.11 (d=3) for the Glauber model, the 3.90 for the two-dimensional Kawasaki model. These results are in good agreement with those obtained by other methods but with smaller error bars in three dimensions.     

Relaxation theory of spin-3/2 Ising system near phase transition temperatures

<正>Dynamics of a spin-3/2 Ising system Hamiltonian with bilinear and biquadratic nearest-neighbour exchange interactions is studied by a simple method in which the statistical equilibrium theory is combined with the Onsager's theory of irreversible thermodynamics.First,the equilibrium behaviour of the model in the molecular-field approximation is given briefly in order to obtain the phase transition temperatures,i.e.the first- and second-order and the tricritical points.Then,the Onsager theory is applied to the model and the kinetic or rate equations are obtained.By solving these equations three relaxation times are calculated and their behaviours are examined for temperatures near the phase transition points.Moreover,the z dynamic critical exponent is calculated and compared with the z values obtained for different systems experimentally and theoretically,and they are found to be in good agrement.     

Transition Probabilities and Dynamic Structure Function in the ASEP Conditioned on Strong Flux

We consider the asymmetric simple exclusion processes (ASEP) on a ring constrained to produce an atypically large flux, or an extreme activity. Using quantum free fermion techniques we find the time-dependent conditional transition probabilities and the exact dynamical structure function under such conditioned dynamics. In the thermodynamic limit we obtain the explicit scaling form. This gives a direct proof that the dynamical exponent in the extreme current regime is z=1 rather than the KPZ exponent z=3/2 which characterizes the ASEP in the regime of typical currents. Some of our results extend to the activity in the partially asymmetric simple exclusion process, including the symmetric case.     

Dynamics of selfavoiding tethered membranes. I. Model A dynamics (Rouse model)

The dynamical scaling properties of selfavoiding polymerized membranes with internal dimension D are studied using model A dynamics. It is shown that the theory is renormalizable to all orders in perturbation theory and that the dynamical scaling exponent z is given by . This result applies especially to membranes (D=2) but also to polymers (D=1). Received: 5 September 1997 / Accepted: 17 November 1997     

Reducible scale invariance at short distances

It is proven that, using reducible scale invariance at short distances, conformal symmetry implies canonical (Bjorken) scaling, provided diagonal dimensions of dilatation multiplets occuring in the operator product expansion of two electromagnetic currents have the canonical value l_n = 2 + n. If the electromagnetic current itself belongs to such multiplets then the hadron production cross section in e⁺e^? annihilation falls off faster than $\text{1}\text{s}$ at asymptotic energy.     

Dynamics of selfavoiding tethered membranes. II. Inclusion of hydrodynamic interaction (Zimm model)

The dynamical scaling properties of selfavoiding polymerized membranes with internal dimension D embedded into d dimensions are studied including hydrodynamical interactions. It is shown that the theory is renormalizable to all orders in perturbation theory and that the dynamical scaling exponent z is given by z=d. The crossover to the region, where the membrane is crumpled swollen but the hydrodynamic interaction irrelevant is discussed. The results apply as well to polymers (D=1) as to membranes (D=2). Received: 5 September 1997 / Accepted: 17 November 1997     

Conformal invariance and conserved quantities of dynamical system of relative motion

This paper discusses in detail the conformal invariance by infinitesimal transformations of a dynamical system of relative motion. The necessary and sufficient conditions of conformal invariance and Lie symmetry are given simultaneously by the action of infinitesimal transformations. Then it obtains the conserved quantities of conformal invariance by the infinitesimal transformations. Finally an example is given to illustrate the application of the results.     

Atomic force microscopy study of growth kinetics: Scaling in TiN-TiB2 nanocomposite films on Si(1 0 0)

We used the reactive unbalanced close-field dc-magnetron sputtering growth of TiN-TiB₂ on Si(1 0 0) at room temperature to determine if scaling theory provides insight into the kinetic mechanisms of two-phase nanocomposite thin films. Scaling analyses along with height-difference correlation functions of measured atomic force microscopy (AFM) images have shown that the TiN-TiB₂ nanocomposite films with thickness ranging from 70 to 950 nm exhibit a kinetic surface roughening with the roughness increasing with thickness exponentially. The roughness exponent α and growth exponent β are determined to be ∼0.93 and ∼0.25, respectively. The value of dynamic exponent z, calculated by measurement of the lateral correlation length ξ, is ∼3.70, agreeing well with the ratio of α to β. These results indicate that the surface growth behavior of sputter-deposited TiN-TiB₂ thin films follows the classical Family-Vicseck scaling and can be reasonably described by the noisy Mullins diffusion model, at which surface diffusion serves as the smoothing effect and shot noise as the roughening mechanism.     

Critical exponents of the Ising model on low-dimensional fractal media

The critical behavior of the Ising model on fractal substrates with noninteger Hausdorff dimension d_H<2 and infinite ramification order is studied by means of the short-time critical dynamic scaling approach. Our determinations of the critical temperatures and critical exponents β, γ, and ν are compared to the predictions of the Wilson-Fisher expansion, the Wallace-Zia expansion, the transfer matrix method, and more recent Monte Carlo simulations using finite-size scaling analysis. We also determined the effective dimension (d_ef), which plays the role of the Euclidean dimension in the formulation of the dynamic scaling and in the hyperscaling relationship d_ef=2β/ν+γ/ν. Furthermore, we obtained the dynamic exponent z of the nonequilibrium correlation length and the exponent θ that governs the initial increase of the magnetization. Our results are consistent with the convergence of the lower-critical dimension towards d=1 for fractal substrates and suggest that the Hausdorff dimension may be different from the effective dimension.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.