We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: \ensuremath{\alpha}, which determines the strength of their singularities; and f, which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of \ensuremath{\alpha} values and the function f(\ensuremath{\alpha}). We apply this formalism to the ${2}^{\ensuremath{\infty}}$ cycle of period doubling, to the devil's staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.

Fractal measures and their singularities: The characterization of strange sets

We give a general introduction to quantum phase transitions in strongly-correlated electron systems. These transitions which occur at zero temperature when a non-thermal parameter $g$ like pressure, chemical composition or magnetic field is tuned to a critical value are characterized by a dynamic exponent $z$ related to the energy and length scales $\Delta$ and $\xi$. Simple arguments based on an expansion to first order in the effective interaction allow to define an upper-critical dimension $D_{C}=4$ (where $D=d+z$ and $d$ is the spatial dimension) below which mean-field description is no longer valid. We emphasize the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally we quote some recent inelastic neutron scattering experiments performed on heavy-fermions which lead to unusual scaling law in $\omega /T$ for the dynamical spin susceptibility revealing critical local modes beyond the itinerant magnetism scheme and mention new attempts to describe this local quantum critical point.

https://arxiv.org/pdf/cond-mat/0102119.pdf

Quantum Phase Transitions

We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularities; and f , which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the function f ( α ). We apply this formation to the 2 ∞ cycle of period doubling, to the devil's staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.

Since the first days of high-Tc superconductivity, the materials science and the physics of grain boundaries in superconducting compounds have developed into fascinating fields of research. Unique electronic properties, different from those of the grain boundaries in conventional metallic superconductors, have made grain boundaries formed by high-Tc cuprates important tools for basic science. They are moreover a key issue for electronic and large-scale applications of high-Tc superconductivity. The aim of this review is to give a summary of this broad and dynamic field. Starting with an introduction to grain boundaries and a discussion of the techniques established to prepare them individually and in a well-defined manner, the authors present their structure and transport properties. These provide the basis for a survey of the theoretical models developed to describe grain-boundary behavior. Following these discussions, the enormous impact of grain boundaries on fundamental studies is reviewed, as well as high-power and electronic device applications.

/pdf/grain-boundaries-in-high-t-c-superconductors-2widq11ljn.pdf

Grain boundaries in high- T c superconductors

We show that the pseudorelativistic physics of graphene near the Fermi level can be extended to three dimensional (3D) materials. Unlike in phase transitions from inversion symmetric topological to normal insulators, we show that particular space groups also allow 3D Dirac points as symmetry protected degeneracies. We provide criteria necessary to identify these groups and, as an example, present ab initio calculations of β-cristobalite BiO(2) which exhibits three Dirac points at the Fermi level. We find that β-cristobalite BiO(2) is metastable, so it can be physically realized as a 3D analog to graphene.

Dirac Semimetal in Three Dimensions

Universal Transition from Quasiperiodicity to Chaos in Dissipative Systems

We calculate the Lyapunov exponents for particles suspended in a random three-dimensional flow, concentrating on the limit where the viscous damping rate is small compared to the inverse correlation time. In this limit Lyapunov exponents are obtained as a power series in $ϵ$, a dimensionless measure of the particle inertia. Although the perturbation generates an asymptotic series, we obtain accurate results from a Pad\'e-Borel summation. Our results prove that particles suspended in an incompressible random mixing flow can show pronounced clustering when the Stokes number is large and we characterize two distinct clustering effects which occur in that limit.

/pdf/clustering-by-mixing-flows-1gcueo203p.pdf

Clustering by Mixing Flows

We consider particles suspended in a randomly stirred or turbulent fluid. When effects of the inertia of the particles are significant, an initially uniform scatter of particles can cluster together. We analyze this “unmixing” effect by calculating the Lyapunov exponents for dense particles suspended in such a random three-dimensional flow, concentrating on the limit where the viscous damping rate is small compared to the inverse correlation time of the random flow (that is, the regime of large Stokes number). In this limit Lyapunov exponents are obtained as a power series in a parameter which is a dimensionless measure of the inertia. We report results for the first seven orders. The perturbation series is divergent, but we obtain accurate results from a Pade-Borel summation. We deduce that particles can cluster onto a fractal set and show that its dimension is in satisfactory agreement with previously reported simulations of turbulent Navier-Stokes flows. We also investigate the rate of formation of caustics in the particle flow.

http://mcs.open.ac.uk/mw987/published%20papers/paper71.pdf

Unmixing in random flows

This paper will focus on the connection between renormalization of dynamical systems with two frequencies and the discrete 1-D Schroedinger equation with an incommensurate modulation. The connection will be explored by constructing a renormalization group formulation of both problems using a binary "Farey" address to label the incommensurability. This address is closely related to the continued fraction representation which has previously been used to investigate these problems. The emphasis of this paper is to introduce a new method rather than to present a completed calculation. Future investigations will determine whether this will eventually lead to deeper insights into these problems than can be obtained using the continued fractions.

https://iopscience.iop.org/article/10.1088/0031-8949/1985/T9/031/pdf

Renormalization of Quasiperiodic Mappings

We show how a generalization of continued fractions can be used to develop a renormalization-group formalism to study the behavior of maps of the two-torus Such maps may mimic the universal behavior of dynamical systems with three mutually incommensurate frequencies Numerical evidence indicates that ''chaos'' may occur in maps which are invertible While we do not see scaling at chaos onset, subcritical scaling is observed and explained by the number theory

Stellan Östlund

Papers

Universal Transition from Quasiperiodicity to Chaos in Dissipative Systems

Clustering by Mixing Flows

Unmixing in random flows

Renormalization of Quasiperiodic Mappings

Renormalization of mappings of the two-torus.