Abstract:
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.
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