Development of (4-epsilon)-dimensional theory for the density of states near the Anderson transition

Abstract

The density of states for the Schroedinger equation with a Gaussian random potential is determined by the functional integral corresponding to the phi^4 theory with a `wrong' sign of the interaction constant. The special role of the dimension d=4 for such a problem can be seen from different viewpoints but is fundamentally determined by the renormalizability of the theory. The construction of an epsilon-expansion in direct analogy with the phase-transition theory gives rise to the problem of a `spurious' pole. To solve this problem, a proper treatment of the factorial divergency of the perturbation series is necessary. Simplifications arising in high dimensions can be used for the development of a (4-epsilon)-dimensional theory, but this requires successive consideration of four types of theories: a nonrenormalizable theory for d>4, nonrenormalizable and renormalizable theories in the logarithmic situation (d=4), and a super-renormalizable theory for d<4. An approximation is found for each type of theory giving asymptotically exact results. The qualitative effect is the same in all four cases and consists in a shifting of the phase transition point in the complex plane. This results in the elimination of the `spurious' pole and in regularity of the density of states for all energies. A discussion is given of the calculation of high orders of perturbation theory and a perspective of the epsilon-expansion for the problem of conductivity near the Anderson transition.