Finite-size scaling of correlation functions in finite systems

We propose the finite-size scaling of correlation functions in finite systems near their critical points.At a distance r in a ddimensional finite system of size L,the correlation function can be written as the product of|r|~(-(d-2+η))and a finite-size scaling function of the variables r/L and tL~(1/ν),where t=(T-T_c)/T_c,ηis the critical exponent of correlation function,andνis the critical exponent of correlation length.The correlation function only has a sigificant directional dependence when|r|is compariable to L.We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations.We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponentη.     

Scaling for level statistics from self-consistent theory of localization

Accepting validity of self-consistent theory of localization by Vollhardt and Wölfle, we derive the relations of finite-size scaling for different parameters characterizing the level statistics. The obtained results are compared with the extensive numerical material for space dimensions d = 2, 3, 4. On the level of raw data, the results of numerical experiments are compatible with the self-consistent theory, while the opposite statements of the original papers are related with ambiguity of interpretation and existence of small parameters of the Ginzburg number type.     

Critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder

A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.     

Conductance of finite systems and scaling in localization theory

The conductance of finite systems plays a central role in the scaling theory of localization (Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)). Usually it is defined by the Landauer-type formulas, which remain open the following questions: (a) exclusion of the contact resistance in the many-channel case; (b) correspondence of the Landauer conductance with internal properties of the system; (c) relation with the diffusion coefficient D(??, q) of an infinite system. The answers to these questions are obtained below in the framework of two approaches: (1) self-consistent theory of localization by Vollhardt and Wölfle, and (2) quantum mechanical analysis based on the shell model. Both approaches lead to the same definition for the conductance of a finite system, closely related to the Thouless definition. In the framework of the self-consistent theory, the relations of finite-size scaling are derived and the Gell-Mann-Low functions ??(g) for space dimensions d = 1, 2, 3 are calculated. In contrast to the previous attempt by Vollhardt and Wölfle (1982), the metallic and localized phase are considered from the same standpoint, and the conductance of a finite system has no singularity at the critical point. In the 2D case, the expansion of ??(g) in 1/g coincides with results of the ??-model approach on the two-loop level and depends on the renormalization scheme in higher loops; the use of dimensional regularization for transition to dimension d = 2 + ?? looks incompatible with the physical essence of the problem. The results are compared with numerical and physical experiments. A situation in higher dimensions and the conditions for observation of the localization law ??(??) ?? ?i?? for conductivity are discussed.     

Conductance distribution near the Anderson transition

Using a modification of the Shapiro approach, we introduce the two-parameter family of conductance distributions W(g), defined by simple differential equations, which are in the one-to-one correspondence with conductance distributions for quasi-one-dimensional systems of size L ^d–1 × L _z , characterizing by parameters L/ξ and L _z /L (ξ is the correlation length, d is the dimension of space). This family contains the Gaussian and log-normal distributions, typical for the metallic and localized phases. For a certain choice of parameters, we reproduce the results for the cumulants of conductance in the space dimension d = 2 + ? obtained in the framework of the σ-model approach. The universal property of distributions is existence of two asymptotic regimes, log-normal for small g and exponential for large g. In the metallic phase they refer to remote tails, in the critical region they determine practically all distribution, in the localized phase the former asymptotics forces out the latter. A singularity at g = 1, discovered in numerical experiments, is admissible in the framework of their calculational scheme, but related with a deficient definition of conductance. Apart of this singularity, the critical distribution for d = 3 is well described by the present theory. One-parameter scaling for the whole distribution takes place under condition, that two independent parameters characterizing this distribution are functions of the ratio L/ξ.     

Static critical behavior of 3D frustrated Heisenberg model on stacked triangular lattice with variable interlayer exchange coupling

Several Monte Carlo algorithms are used to examine the critical behavior of the 3D frustrated Heisenberg model on stacked triangular lattice with variable interlayer exchange coupling for values of the interlayer-to-intralayer exchange ratio R = |′/J| in the interval between 0.01 and 1.0. A finite-size scaling technique is used to calculate the static magnetic and chiral critical exponents α (specific heat), γ and γ_k (susceptibility), β and β_k(magnetization), ν and ν_k(correlation length), and the Fisher exponent η. It is shown that 3D frustrated Heisenberg models on stacked triangular lattice with R > 0.05 constitute a new universality class of critical behavior. At lower R, a crossover from 3D to 2D critical behavior is observed.     

Improved Scaling of the Magnetic Heat Capacity in La<Subscript>0.85</Subscript>Ag<Subscript>0.15</Subscript>MnO<Subscript>3</Subscript> Manganite

The heat capacity in a La_0.8 Ag_0.15 MnO₃ manganite has been measured near the Curie temperature T _C in applied magnetic fields up to 26 kOe to study the scaling critical behavior and to obtain the universality class. The conventional scaling fails in application to the manganites with a hysteresis and the strong sensitivity of T _C to a magnetic field. However, the application of the improved scaling procedure designed by us allows yielding the good scaling the magnetic heat =0.23 capacity in La_0.85Ag_0.15MnO₃, which may belong to a new universality class for systems with the strong spin-orbital coupling of t _2g -electrons, namely, double -Heisenberg with the critical exponent of the heat capacity α = ?0.23 and the critical exponent of the correlation radius v=0.7433. This new universality class is consistent with the crystal, magnetic and orbital symmetries for the La_0.85Ag_0.15MnO₃. Scaling failure in the vicinity of T _C in the range of t/H ^1/2ν ≈ [?0.033;0.024] is understood by finite-size and other disordering effects when T →T _C. It is remarkable that finite-size effect is consistent with grain size, L ≈ 50 μm, in the La_0.85Ag_0.15MnO₃. The correlation radius, Lt ^ν ≈ 30.28 Å, estimated from the finite-size effect is of the same order of magnitude with the sizes of the ferromagnetic fluctuations and drops in manganites.     

Scaling and complex avalanche dynamics in the Abelian sandpile model

We study the two-dimensional Abelian Sandpile Model on a squarelattice of linear size L. We introduce the notion of avalanche’sfine structure and compare the behavior of avalanches and waves oftoppling. We show that according to the degree of complexity inthe fine structure of avalanches, which is a direct consequence ofthe intricate superposition of the boundaries of successive waves,avalanches fall into two different categories. We propose scalingansätz for these avalanche types and verify them numerically.We find that while the first type of avalanches (α) has a simplescaling behavior, the second complex type (β) is characterized by anavalanche-size dependent scaling exponent. In particular, we define an exponent γto characterize the conditional probability distribution functions for these typesof avalanches and show that γ _α = 0.42, while 0.7 ≤ γ _β ≤ 1.0depending on the avalanche size. This distinction provides aframework within which one can understand the lack of aconsistent scaling behavior in this model, and directly addresses thelong-standing puzzle of finite-size scaling in the Abelian sandpile model.     

Das Auger-Spektrum derL III- Schale von Wismut

A thin Bi layer is irradiated by X-rays so thatL-Auger electrons are emitted. A magnetic lens spectrometer is used to measure the electron spectrum. Energy, transition, and relativ intensity are given for 14 lines. Under the most favourable conditions the number ofL _III ionisations is about ten times that ofL _II ionisations. In this case only a small intensity ofL _II-Auger electrons is superposed on theL _III-Auger spectrum. The ratiod of intensities of line groupL _III M N to line groupL _III M M is found by extrapolation to bed=0·46±0·02. This combined with earlier results gives anL _III-Auger yielda ₃= 0·64±0·04. TheL _III fluorescenc yield isω ₃=0·36±0·04, correspondingly. A further application of the experimental method is described.     

Extraordinary photons with unusual angular momentum

A series of novel state-vector functions (SVFs), which is the general solution of the Schrödinger equation for a photon, are constructed. Each set of these functions consists of a triplet of eigen-SVFs: The triplet can be broken down into a pair of nonzero l-order functions and a single zero-order function. The photons, described with a triplet of eigen-SVFs, possess all the quantum characteristics of a photon: In addition to common attributes like energy E = ? _ω , and momentum p _z = ? _κ , they also exhibit different angular momenta (AM) L _z+ = l?, L _z? = l?, and L _z0 = 0, where l?1. In other words, in addition to usual eigenvalues L _z±= ±?, there are unusual nonzero l-order eigenvalues L _z± = ±l? and a zero-order eigenvalue L _z0 = 0 for AM of a photon. By a series of SVFs, the pattern from nonzero l-order and zero-order Laguerre-Gaussian modes of a laser beam is explained well from a quantum mechanical point of view.     

Generalized Drude scattering rate from the memory function formalism: an independent verification of the Sharapov-Carbotte result

An explicit perturbative computation of the Mori’s memory function was performed by Götzeand Wölfle (GW) to calculate generalized Drude scattering (GDS) rate for the case ofelectron-impurity and electron-phonon scattering in metals by assuming constant electronicdensity of states at the Fermi energy. In the present investigation, we go beyond thisassumption and extend the GW formalism to the case in which there is a gap around theFermi surface in electron density of states. The resulting GDS is compared with a recentone by Sharapov and Carbotte (SC) obtained through a different route. We find goodagreement between the two at finite frequencies. However, we find discrepancies in the dcscattering rate. These are due to a crucial assumption made in SC namely ω ? | Σ(?+ ω) ? Σ^?(?)|. No such high frequency assumption is made in the memory functionbased technique.     

Universality in the partially anisotropic three-dimensional Ising lattice

Using transfer-matrix extended phenomenological renormalization-group methods, we study the critical properties of the spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths \(\mathop J\limits^ \to = (J',J',J)\). The universality of both fundamental critical exponents y _t and y _h is confirmed. It is shown that the critical finite-size scaling amplitude ratios \(U = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chi\), and \(Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }\) are independent of the lattice anisotropy parameter Δ=J′/J. For the Y² invariant of the three-dimensional Ising universality class, we give the first quantitative estimate Y²≈2.013 (shape L×L×∞, periodic boundary conditions in both transverse directions).     

Observation of the Talbot effect for ultrasonic waves

The diffraction of ultrasonic radiation on an amplitude diffraction grating in the near-field area (Fresnel diffraction) has been studied. The effect of self-imaging of the grating (Talbot effect) has been detected for ultrasonic radiation at distances from the grating in the range from z = 0 to z = 2L _T, where L _T is the Talbot length. The fractional Talbot effect, i.e., the ultrasonic image of the grating with the period d/2, has been observed.     

Thermodynamic measurements in a fractional quantum hall effect and the composite-fermion approach

A quantitative comparison of the magnetocapacitance measurements on high-quality samples in the fractional quantum Hall effect with the composite-fermion approach has been performed. For this comparison, the relation between the electron chemical potential μ _e and quasi-particle spectrum has been derived. The effect of the temperature T has been calculated in the two-level approximation. The calculation quantitatively describes the decrease in the measured chemical-potential jump at filling factors of ν = 1/3, 2/5 with increasing T. In the compressible range 1/3 < ν < 2/5, the slope of the temperature dependence dμe/dν(T) is also in agreement with the calculation. The discrepancy of the composite-fermion approach with the experimental data is in the incorrectly predicted gaps and their dependence on the denominator of a fraction.     

Similarity and dimensional theory for galaxies: Explanation of long-known results of observations

A practical extension of the similarity and dimensional theory to the case of several similarity parameters is proposed. On this basis, for galaxies an explanation is given for the empirical correlations noticed in the last quarter of the 20th century: the Tully–Fisher relation, the concept of a fundamental plane, etc. For galaxies, apart from the virial, there is another similarity parameter whose choice is arbitrary. Here, it is introduced in the simplest form for an empirical determination:Π₁ = U ₀/U _d, U ₀ is the observed velocity, the scale U _d = (GL)^1/5, where L is the object luminosity, G is the gravitational constant.     

On neutrino oscillations and the time-energy uncertainty relation

We consider neutrino oscillations as a nonstationary phenomenon based on the Schrödinger evolution equation and mixed neutrino states with definite flavor. We demonstrate that for such states, invariance under translations in time does not take place. We show that the time-energy uncertainty relation plays a crucial role in neutrino oscillations. We compare neutrino oscillations with K ⁰ ? -K ⁰, B _d ⁰ ? B _d ⁰ , and other oscillations.     

Forced snaking: Localized structures in the real Ginzburg-Landau equation with spatially periodic parametric forcing

We study spatial localization in the real subcritical Ginzburg-Landau equation u _t = m ₀ u + Q(x)u + u _xx + d|u|² u ?|u|⁴ u with spatially periodic forcing Q(x). When d>0 and Q ≡ 0 this equation exhibits bistability between the trivial state u = 0 and a homogeneous nontrivial state u = u ₀ with stationary localized structures which accumulate at the Maxwell point m ₀ = ?3d ²/16. When spatial forcing is included its wavelength is imprinted on u ₀ creating conditions favorable to front pinning and hence spatial localization. We use numerical continuation to show that under appropriate conditions such forcing generates a sequence of localized states organized within a snakes-and-ladders structure centered on the Maxwell point, and refer to this phenomenon as forced snaking. We determine the stability properties of these states and show that longer lengthscale forcing leads to stationary trains consisting of a finite number of strongly localized, weakly interacting pulses exhibiting foliated snaking.     

Path integral approach for spaces of nonconstant curvature in three dimensions

In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D _3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D _3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D _3d-II. I solve the path integral on D _3d-I in the (u, v, w) system and on D _3d-II in the (u, v, w) system and in spherical coordinates.     

Duality-mediated critical amplitude ratios for the (2 + 1)-dimensional S = 1XY model

The phase transition for the (2 + 1)-dimensional spin-S = 1XY model was investigated numerically. Because of the boson-vortex duality, the spin stiffness ρ _s in the ordered phase and the vortex-condensate stiffness ρ _v in the disordered phase should have a close relationship. We employed the exact diagonalization method, which yields the excitation gap directly. As a result, we estimate the amplitude ratios ρ _s,v /Δ (Δ: Mott insulator gap) by means of the scaling analyses for the finite-size cluster with N ≤ 22 spins. The ratio ρ _s /ρ _v admits a quantitative measure of deviation from selfduality.     

Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

We consider random Schrödinger equations on \({\mathbb{R}^{d}}\) for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ _t the solution with initial data ψ₀. The space and time variables scale as \({x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}}\) with 0 < κ <  κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ _t converges weakly to the solution of a heat equation in the space variable x for arbitrary L ² initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [10] the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.