Scale and conformal invariance in quantum field theory

We study the relation between invariance under rigid and local changes of length scale. In two dimensions, we complete an argument of Zamolodchikov showing that the rigid invariance implies the local under broad conditions. In three or more dimensions we are unable to find either a general proof or a counterexample, but we find some new conformally invariant systems.     

Topology invariance in percolation thresholds

An universal invariant for site and bond percolation thresholds ( and respectively) is proposed. The invariant writes where and are positive constants, and d the space dimension. It is independent of the coordination number, thus exhibiting a topology invariance at any d. The formula is checked against a large class of percolation problems, including percolation in non-Bravais lattices and in aperiodic lattices as well as rigid percolation. The invariant is satisfied within a relative error of for all the twenty lattices of our sample at d=2, d=3, plus all hypercubes up to d=6. Received: 7 July 1997 / Accepted: 5 November 1997     

Theory of self-avoiding walks on percolation fractals

A phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self-avoiding walks (SAW) is used to derive a new expression for the Flory exponent describing the average radius of gyration of SAWs on fractals. We focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of SAWs. The mean-field version of our new formula corresponds to the Aharony and Harris expression, who used the standard Flory approach for its derivation.On leave from Santipur College, Nadia 741404, India.     

Theta-point exponent for polymer chains on percolation fractals

We derive a new expression for the Flory exponent describing the average radius of gyration of polymer chains at the theta point. For this we make use of the appropriate distribution function for the radius of gyration. We start from Euclidean lattices and extend the results to percolation fractals, by taking into account the basic geometry and the topology of such structures. We show that such basic features have a very prominent effect on the Flory exponent of the chain polymer on fractals at the theta point.     

Scale invariance and inflation

A scale invariant model for early universe inflationary cosmology is developed. In order to realize dilatation invariance and spontaneous symmetry breaking we introduce two scalar fields, a dilaton and an inflaton. The scale invariant theory encompasses the Brans-Dicke and induced-gravity models as limiting cases. The model is solved numerically for a wide class of initial conditions. We find that the inflationary epoch is generically characterized by a two phase evolution of the universe: A single or double exponential era and a power-law expansion. Onset of gravity triggers double exponential evolution of the scale factor. We further examine inflation in the Brans-Dicke theory and find that scale invariance is restored in the course of spontaneous symmetry breaking.     

Numerical investigations of discrete scale invariance in fractals and multifractal measures

Fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. Here, we focus on lattice multifractals which exhibit complex exponents associated with observable log-periodicity. We perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments. A novel numerical approach is proposed to extract the log-frequencies. In the non-lattice case, there is no visible log-periodicity, i.e., no preferred scaling ratio since the set of complex exponents spreads irregularly within the complex plane. A non-lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non-lattice one. An algorithm for the construction of the lattice sequence is proposed explicitly.     

Conformally invariant fractals and potential theory

The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a Q-state Potts cluster, is solved in two dimensions. The dimension &fcirc;(straight theta) of the boundary set with local wedge angle straight theta is &fcirc;(straight theta) = pi / straight theta-25-c / 12 (pi-straight theta)(2) / straight theta(2pi-straight theta), with c the central charge of the model. As a corollary, the dimensions D(EP) of the external perimeter and D(H) of the hull of a Potts cluster obey the duality equation (D(EP)-1) (D(H)-1) = 1 / 4. A related covariant MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.     

Scale invariance and superfluid turbulence

We construct a Schroedinger field theory invariant under local spatial scaling. It is shown to provide an effective theory of superfluid turbulence by deriving, analytically, the observed Kolmogorov 5/3 law and to lead to a Biot–Savart interaction between the observed filament excitations of the system as well.     

Scale invariance rule in electromagnetic scattering

The volume integral equation formalism is used to prove the scale invariance rule for an arbitrarily sized scatterer with an arbitrary shape, morphology, and orientation. The only assumptions are that the scatterer is made of optically isotropic linear materials and is embedded in a homogeneous, linear, isotropic, and nonabsorbing infinite medium.     

Scale invariance and spontaneous symmetry breaking

Spontaneous breaking of gauge symmetries is studied in theories with nonlinearly realized scale invariance. The classically sliding vacuum expectation values are fixed through quantum corrections. The anomaly of the dilatation current determines the vacuum energy density as well as the dilaton mass. The coupling of gravity to matter is modified in such a way that the cosmological constant vanishes.     

Euler characteristic in percolation theory

The notion of the Euler characteristic is introduced in percolation theory, which, in fact, was implicitly used from the very beginning of studying percolation problems. An exact formula is given for the case of the ball problem, along with some of its generalizations.     

Conformal invariance in perturbation theory

Asymptotic conformal invariance is proved to be true at all orders in perturbation theory. The correct Ward Identities for broken conformal invariance are derived: they are the extension of the Callan Symanzik equation from scale to conformal transformations.     

Scalar quantum field theory on fractals

We investigate scale invariant measures over multiple variables for scalar field theories by imitating Wiener’s construction of the measure on the space of functions of one variable. We assign random fields values on the vertices of simple geometric shapes (triangles, squares, tetrahedra) which are subdivided into a finite number of similar shapes. We find several Gaussian measures with anomalous scaling associated with these field variables. A non-Gaussian fixed point arises from the Ising model on a fractal. In the continuum limit, we construct correlation functions that vary as a power of the distance. It is either a positive power (analogous to the Wiener process) or a negative power depending on the subdivision scheme used; however it is an irrational number for all the examples. This suggests that in the continuum limits it corresponds to quantum field theories (random fields) on spaces of fractional dimension.     

Scale invariance and lack of self-averaging in fragmentation

We derive exact statistical properties of a recursive fragmentation process. We show that introducing a fragmentation probability 0相似文献   

Scale invariance in dynamic fragmentation of quartz

In the work, we studied statistical regularities in fragmentation of cylindrical quartz specimens under dynamic loading. The original equipment used in the study ensured integrity of the specimens after impact for the determination of fragment size distribution (spatial scaling) and allowed recording fractoluminescence pulses at newly formed fracture surfaces for the determination of pulse spacing distribution (temporal scaling). The results of experimental data processing show that the size distributions for both the spatial parameter (fragment size) and the temporal parameter (fractoluminescence pulse spacing) are described by a power function. This enables us to refer dynamic fragmentation of quartz to phenomena exhibiting self-organized criticality.