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 ${ \tilde{ \nu}}=\nu+1$ which is identified with the controlling parameter $e^{ -2\phi}$ of the Luttinger model. In this way, a bosonic system characterized by a fractal index $i_{ f}[h]=i_{ f}[2]=1$ is considered as a conformal field theory with central charge $c[\nu=0]=1=i_{ f}[2]$ with a compactified radius $R=\frac{ 1}{ \sqrt{ \tilde{ \nu}}}=1$. Thus, we have a mapping of a bosonic theory to a fermionic one and vice-versa, i.e., the duality symmetry ${ { \tilde{ h}} =3-h}$ of the universal class $h$ of fractons defined in the interval 1< h <2 is satisfied.
Categories: cond-mat.mes-hall hep-th quant-ph