Fractons and Luttinger liquids

We consider the concept of fractons as particles or quasiparticles which obey a specific fractal statistics in connection with a one-dimensional Luttinger liquid theory. We obtain a dual statistics parameter ν̃ = ν+1 which is identified with the controlling parameter e−2φ of the Luttinger model. In this way, a bosonic system characterized by a fractal index if [h] = if [2] = 1 is considered as a conformal field theory with central charge c[ν = 0] = 1 = if [2] with a compactified radius R = 1 √ ν̃ = 1. Thus, we have a mapping of a bosonic theory to a fermionic one and vice-versa, i.e., the duality symmetry h̃ = 3− h of the universal class h of fractons defined in the interval 1 < h < 2 is satisfied. PACS numbers: 05.30.-d; 67.40.Db; 71.27.+a; 11.10.Kk