Development of a (4-ε)-dimensional theory for the density of states of a disordered system near the Anderson transition
Kapitza Institute of Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 117334, Russian Federation
The calculation of the density of states for the Schrödinger equation with a Gaussian random potential is equivalent to the problem of a second-order transition with a ’wrong’ sign for the coefficient of the quartic term in the Ginzburg-Landau Hamiltonian. The special role of the dimension d = 4 for such a Hamiltonian can be seen from different viewpoints but is fundamentally determined by the renormalizability of the theory. The construction of an ε 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-ε)-dimensional theory, but this requires successive consideration of four types of theories: a nonrenormalizable theories for d > 4, nonrenormalizable and renormalizable theories in the logarithmic situation (d = 4), and a super-renormalizable theories for d < 4. An approximation is found for each type of theory giving asymptotically exact results. In the (4-ε)-dimensional theory, the terms of leading order in 1/ε are only retained for N~1 (N is the order of the perturbation theory) while all degrees of 1/ε are essential for large N in view of the fast growth of their coefficients. The latter are calculated in the leading order in N from the Callan-Symanzik equation with the results of Lipatov method used as boundary conditions. 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 ε expansion for the problem of conductivity near the Anderson transition.