Fractal Index, Central Charge and Fractons

Abstract

We introduce the notion of fractal index associated with the universal class h of particles or quasiparticles, termed fractons, which obey specific fractal statistics. A connection between fractons and conformal field theory(CFT)quasiparticles is established taking into account the central charge c[ν] and the particle-hole duality ν ←→ 1 ν , for integer-value ν of the statistical parameter.In this way, we derive the Fermi velocity in terms of the central charge as v ∼ c[ν] ν+1 . The Hausdorff dimension h which labelled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

“…Thus, the universality classes of the quantum Hall transitions are classified in terms of h and this result is obtained from our analysis considering the modular group for these transitions [1,6]. We have also, in our context, made a connection between Physics and Number Theory relating the fractal geometry and dilogarithm function through of the concept of fractal index introduced by us [3]. We have also the possibility of fractal sets with irrational values for the Hausdorff dimension.…”

“…We can check that all experimental data for the occurrence of FQHE satisfies a symmetry principle discovered by us, that is, the duality symmetry between universal classes h of particles or quasiparticles with any value of spin [1][2][3][4][5]. For example, we have the dual filling factors (f,f ) =…”

“…Now, as was introduced in [3], we have also established a connection between fractal geometry and Rogers dilogarithm function.…”

“…In Ref. [3] we have also defined a topological Theory is manifest because, on the one hand, we have the FQHE characterized in terms of a geometrical parameter related to the fractal curves and, on the other hand, the dilogarithm function appears in various branches of mathematics besides number theory, such as [8]:…”

Fractal von Neumann entropy

“…Besides this we have also considered some connection with conformal field theories [3] . Another direction of research is the connection between Number Theory and Physics as explicited by our formulation [1,3]. There exists other possibilities of application, for example, we are considering the study of entanglement states for fracton quantum computing [5].…”