J. Phys. I France
Volume 4, Numéro 5, May 1994
Page(s) 655 - 673
DOI: 10.1051/jp1:1994168
J. Phys. I France 4 (1994) 655-673

Statistical properties of one-point Green functions in disordered systems and critical behavior near the Anderson transition

Alexander D. Mirlin1 and Yan V. Fyodorov2

1  Institut für Theorie der Kondensierten Materie, Universitât Karlsruhe, 76128 Karlsruhe, Germany
2  Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel

(Received 1 Octoher 1993, accepted 9 Fehruary 1994)

We investigate the statistics of local Green functions $G(E, x, x) = \langle x \vert(E - \hat{H})^{-1}\vert x\rangle$, in particular of the local density of states $\rho \propto Im G(E, x, x)$, with the Hamiltonian $\hat{H}$ describing the motion of a quantum particle in a d-dimensional disordered system. Corresponding distributions are related to a function which plays the role of an order parameter for the Anderson metal-insulator transition. When the system can be described by a nonlinear $\sigma$-model, the distribution is shown to possess a specific "inversion" symmetry. We present an analysis of the critical behavior near the mobility edge that follows from the abovementioned relations. We explain the origin of the non-power-like critical behavior obtained earlier for effectively infinite-dimensional models. For any finite dimension $d < \infty$ the critical behavior is demonstraied to be of the conventional power-law type wilh $d = \infty$ playing the rote of an upper critical dimension.

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