Ph.D Thesis
Ph.D Student Gitelman Dor Physical Properties of Self-Similar Systems- applications to Fractals and Quasiperiodic Tilings Department of Physics Professor Eric Akkermans

Abstract

We study several systems and phenomena characterized by a discrete scale invariance. We first study phase transitions on self-similar fractals. We find that the discrete scaling symmetry which leads to an exponential growth of the Laplace operator eigenvalues and is responsible to for new and distinct critical behavior. Due to this behavior, scaling relations such as Rushbrooke does not hold. We argue that our results can be understood in a broader context which relates this behavior to substitutions. We then explain the relevance of the fractal self-similar structure using the Harris criterion. We show that breaks of scaling relations, which can be thought of as a competition between a pure critical behavior and the geometric disorder caused by the fractal structure, leads to a new type of phase transitions on fractals.

We then study the renormalization group (RG) equations of several phenomena which are all characterized by a set of complex exponents. We show that they can be mapped onto a generalized occurrence matrix relates the RG flow onto a generalized substitutions. We show that the complex RG flow of Quantum Einstein Gravity found in previous works can be interpreted using Efimov physics. We then show how to obtain spectral properties of self-similar fractals using substitutions.

Finally, we study topological properties of finite length two letters substitutions. We show that for such substitutions, one can define a real space torus for distinct lengths. We then propose exchange rules which produce an ensemble of topologically equivalent 2d .lattices. We show that for the case of Fibonacci substitutions those exchange rules can be interpreted as a phase factors which relates to a specific unitary group. Going to the Fourier space we show that the torus structure is preserved and that the corresponding winding numbers are derived from an algebraic structure which is generic to any substitution. Those winding numbers are known to be related to spectral properties of the quasiperiodic structure, through the gap labeling theorem