Large Fractals in Condensed Matter Physics

Abstract

The concept of fractal[1] entered condensed matter physics in the 1970’s through a problem in electrical conduction in disordered solids (with random potential) that eluded solution for quite sometime. Since then it has set a new trend in condensed matter research and scenarios ranging from percolation to quasi crystals, to Penrose tiles to subtle objects like electron wavefunctions and electron-state distributions in disordered and quasi-disordered systems, have received new insight by the use of the concept of fractal. We will briefly touch upon some of the prominent examples and develop the case of self-similar wavefunctions in some detail to investigate the famous problem of metal-insulator transition driven entirely by disordered potential. The examples of application of fractals fall in two categories: classical and quantum mechanical. The examples of classical systems are infinite clusters formed at the percolation threshold, diffusion limited aggregates, colloidal emulsions and many from the realm of chaos. The prominent candidates in the domain of quantum systems are electron wavefunctions and energy state distributions (density of states) in random potentials, quasi crystals and Penrose tiles. The latter two systems offer almost periodic potentials that can be commensurate or incommensurate with the lattice periodicity. In either situation one learns new physics[2].