July 20, 2007

Electrons on the edge are fractal

Understanding a material’s transition from a metallic to an insulating state hinges on the fractal nature of electrons

Figure 1: The beauty of fractals. The fractal nature of the electron wave function is shown to strongly influence the transition of two-dimensional materials from a metallic to an insulating state.

A team of researchers from RIKEN’s Discovery Research Institute, Wako, and the US universities of Chicago and Santa Barbara has elucidated the role played by electrons in the transition of a two-dimensional disordered material from an insulating to a metallic state.

Most materials are either in a conducting state (that is, they are metallic) or an insulating state. However, some materials can suddenly switch from being a conductor of electric current to an insulator and vice versa, for example through the application of pressure, illumination with light or other means. This process is known to physicists as metal–insulator transition and originates in the way electrons are spread throughout a material. If the electrons are tightly bound at fixed locations and cannot move, the material is in an insulating state. In the metallic state, the electrons are spread widely throughout the material and can move freely from one end to the other.

At the transition from a metallic to an insulating state, characteristics of the electrons change rapidly and the electron distribution becomes fractal. This means that the electrons assume patterns that are self-replicating—the same structural motif repeats itself from a small localized scale to a large global scale (Fig. 1).

Now, the Japanese and US research team has studied metal–insulator transitions occurring in a plane. They demonstrated that the so-called conformal field theories (CFTs)—a powerful mathematical tool—are ideally suited for investigating these planar systems. “CFTs should give us an ultimate theoretical description of metal–insulator transitions in two dimensions,” explains RIKEN’s Akira Furusaki, a member of the team. The team’s findings, obtained from applying CFT principles, have now been published in the journal Physical Review Letters1.

In particular, the researchers evaluated the fractal nature of the electrons at the center of the two-dimensional plane, and at its borders and corners. They found the properties of the electrons are quite different at the borders and corners compared with those at the center. More importantly, the researchers demonstrate that these different properties at the borders and corners are fundamentally linked to each other.

“These findings are only a first step towards pinning down which CFT models describe the metal–insulator transitions,” explains Furusaki. More studies will be necessary to explore the capabilities of the CFTs. This will not only increase our understanding of metal–insulator transitions, but also of other related disordered systems such as turbulence in air or water.


  1. Obuse, H., Subramaniam, A. R., Furusaki, A., Gruzberg, I. A. & Ludwig, A. W. W. Multifractality and conformal invariance at 2D metal-insulator transition in the spin-orbit symmetry class. Physical Review Letters 98, 156802 (2007). |  | (Link)