Physical consequences of complex dimensions of fractals

Abstract

It has been realized that fractals may be characterized by complex dimensions, arising from complex poles of the corresponding zeta function, and we show here that these lead to oscillatory behavior in various physical quantities. We identify the physical origin of these complex poles as the exponentially large degeneracy of the iterated eigenvalues of the Laplacian, and discuss applications in quantum mesoscopic systems such as oscillations in the fluctuation Σ 2 (E) of the number of levels, as a correction to results obtained in Random Matrix Theory. We present explicit expressions for these oscillations for families of diamond fractals, also studied as hierarchical lattices.PACS numbers: 05.45.Df, 05.60. Gg, Fractals, such as the well-known Sierpinski gasket, have been thoroughly studied in physics and in mathematics. In addition to their own intriguing properties, they provide a useful testing ground to investigate properties of disordered classical or quantum systems [1], addressing such fundamental physical issues as Anderson localization, the renormalization group, and phase transitions [2]. In addition to condensed matter and statistical physics, fractals have been considered in other contexts such as gravitational systems [3,4], and in quantum field theory [5]. Despite the large amount of work dedicated to the study of the spectra of deterministic fractals, explicit expressions for spectral functions such as heat kernels or spectral zeta functions, from which many physical quantities can be derived, have remained elusive. It is well known that the heat kernel Z(t) and zeta function ζ(s) play central roles in various fields of physics: from mesoscopic physics [6], to black holes [7], to quantum field theory on curved spaces such as de Sitter and anti De Sitter spaces [8], to the physics of the Casimir effect [9]. This is largely due to their relation to the notion of the partition function in statistical physics [10], and to the ubiquity of Schwinger's proper-time formalism [11].An important step was to identify the leading contribution to Weyl's small time expansion of Z(t), showing that it is determined by the fractal's spectral dimension d s [12], rather than by its fractal (Hausdorff) dimension d h , as initially conjectured. The fact that fractals are characterized by a set of more than one dimension, as opposed to standard Euclidean spaces, illustrates the richness and peculiarity of self-similar structures. Spectral properties of deterministic fractals have recently been considered anew in mathematics, and the notion of complex valued fractal dimensions has been introduced [12, 13], leading to new results for the zeta function [14][15][16]. In this Let- * On leave from Department of Physics, Technion Israel Institute of Technology, 32000 Haifa, Israel.ter we use and extend these results to study the resulting (log-periodic) oscillations in the heat kernel and related physical quantities. We illustrate these ideas with a special class of fractals known as diamond fractals. These diamond fractals per...

Geometry of fractional spaces

“…In fact, it is known that the heat kernel trace for a Laplacian on fractals displays log-oscillations in the scale [153,154]. Oscillatory behaviour has been found analytically and numerically for various fractals [155,156,157,158], and (4.28) illustrates a rather universal phenomenon. This is one of the most crucial points of the physical scenario that will emerge in [42], where it shall be given adequate space.…”

Asymptotic Safety, Fractals, and Cosmology

“…For similar work on fractal features in different approaches we must refer to the literature [69][70][71][72][73][74][75][76][77][78][79][80][81][82]. (3) As for possible physics implications of the RG flow predicted by QEG, ideas from particle physics, in particular the "RG improvement", have been employed in order to study the leading quantum gravity effects in black holes [83,84], cosmological space-times [51,52,[85][86][87][88][89][90][91][92][93] or possible observable signatures from Asymptotic Safety at the LHC [94][95][96][97].…”

Log-periodic Critical Amplitudes: A Perturbative Approach

“…In some cases, when the scale invariance is discrete [16,36,14,32,29,18,2,1], the amplitude of the powerlaw acquires a periodicity, often called log-periodic oscillations: see [35,21] for reviews on the topic. These oscillatory amplitudes are usually more difficult to calculate [13,12,7,26] than the critical exponents.…”