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Register a new userDiscrete scale invariance and complex dimensions
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Authors
Didier Sornette
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We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the seventies, complex exponents have been studied in the eighties in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete scale invariance and its associated complex exponents may appear ``spontaneously in euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete scale invariance. The main motivation to study discrete scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes.
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This paper is a study of the behavior of experimentally observed stress-strain force during the fracture of a quantum wire. The magnitude of the force oscillates as a function of time and can be phenomenologically regarded as a sign of discrete-scale invariance. In the theory of discrete-scale invariance, termination of the wire is regarded as a phase transition. We estimate the critical point and exponents.
We consider the discrete surface growth process with relaxation to the minimum [F. Family, J. Phys. A {bf 19} L441, (1986).] as a possible synchronization mechanism on scale-free networks, characterized by a degree distribution $P(k) sim k^{-lambda}$, where $k$ is the degree of a node and $lambda$ his broadness, and compare it with the usually applied Edward-Wilkinson process [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London Ser. A {bf 381},17 (1982) ]. In spite of both processes belong to the same universality class for Euclidean lattices, in this work we demonstrate that for scale-free networks with exponents $lambda<3$ this is not true. Moreover, we show that for these ubiquitous cases the Edward-Wilkinson process enhances spontaneously the synchronization when the system size is increased, which is a non-physical result. Contrarily, the discrete surface growth process do not present this flaw and is applicable for every $lambda$.
We study the energy spectrum of a two-dimensional electron in the presence of both a perpendicular magnetic field and a potential. In the limit where the potential is small compared to the Landau level spacing, we show that the broadening of Landau levels is simply expressed in terms of the structure factor of the potential. For potentials that are either periodic or random, we recover known results. Interestingly, for potentials with a dense Fourier spectrum made of Bragg peaks (as found, e.g., in quasicrystals), we find an algebraic broadening with the magnetic field characterized by the hyperuniformity exponent of the potential. Furthermore, if the potential is self-similar such that its structure factor has a discrete scale invariance, the broadening displays log-periodic oscillations together with an algebraic envelope.
We derive exact statistical properties of a class of recursive fragmentation processes. We show that introducing a fragmentation probability 0<p<1 leads to a purely algebraic size distribution in one dimension, P(x) ~ x^{-2p}. In d dimensions, the volume distribution diverges algebraically in the small fragment limit, P(V)sim V^{-gamma} with gamma=2p^{1/d}. Hence, the entire range of exponents allowed by mass conservation is realized. We demonstrate that this fragmentation process is non-self-averaging. Specifically, the moments Y_alpha=sum_i x_i^{alpha} exhibit significant fluctuations even in the thermodynamic limit.
We show that a one-dimensional chain of trapped ions can be engineered to produce a quantum mechanical system with discrete scale invariance and fractal-like time dependence. By discrete scale invariance we mean a system that replicates itself under a rescaling of distance for some scale factor, and a time fractal is a signal that is invariant under the rescaling of time. These features are reminiscent of the Efimov effect, which has been predicted and observed in bound states of three-body systems. We demonstrate that discrete scale invariance in the trapped ion system can be controlled with two independently tunable parameters. We also discuss the extension to n-body states where the discrete scaling symmetry has an exotic heterogeneous structure. The results we present can be realized using currently available technologies developed for trapped ion quantum systems.
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