Scaling theory of Anderson's transition for interacting electrons

Tobias Brandes, Stefan Kettemann

Matthias Stosiek, Felix Weiner, Ferdinand Evers

Alexander M. Finkel’stein

The metal–insulator transition (MIT) observed in a two-dimensional dilute electron liquid raises the question about the applicability of the scaling theory of disordered electrons, the approach pioneered by Phil Anderson and his collaborators,8 for the description of this transition. In this context, we review here the scaling theory of disordered electrons with electron–electron interactions. We start with the disordered Fermi liquid, and show how to adjust the microscopic Fermi-liquid theory to the presence of disorder. Then we describe the non-linear sigma model (NLSM) with interactions. This model has a direct relation with the disordered Fermi liquid, but can be more generally applicable, since it is a minimal model for disordered interacting electrons. The discussion is mostly about the general structure of the theory emphasizing the connection of the scaling parameters entering the NLSM with conservation laws. Next, we show that the MIT, as described by the NLSM with interactions, is a quantum phase transition and identify the parameters needed for the description of the kinetics and thermodynamics of the interacting liquid in the critical region of the transition. Finally, we discuss the MIT observed in Si -MOSFETs. We consider it as an example of the Anderson transition in the presence of the electron interactions. We demonstrate that the two-parameter RG equations, which treat disorder in the one-loop approximation but incorporate the full dependence on the interaction amplitudes, describe accurately the experimental data in Si -MOSFETs including the observed non-monotonic behavior of the resistance and its strong drop at low temperatures. The fact that this drop can be reproduced theoretically, together with the argument that Anderson localization should occur at strong disorder, justified the existence of the MIT within the scaling theory.

N. Grewe

Kyung-Yong Park, Hyunjung Lee, Ki‐Seok Kim

Although quantum phase transitions involved with Anderson localization have been investigated for more than a half century, the role of spin polarization in these metal-insulator transitions has not been clearly addressed as a function of both the range of interactions and the range of energy scales. Based on the Anderson-Hartree-Fock study, we reveal that the spin polarization has nothing to do with Anderson metal-insulator transitions in three dimensions as long as effective interactions between electrons are long-ranged Coulomb type. On the other hand, we find that metal-insulator transitions appear with magnetism in the case of Hubbard-type local interactions. In particular, we show that the multifractal spectrum of spin-$\ensuremath{\uparrow}$ electrons differs from that of spin-$\ensuremath{\downarrow}$ electrons at the high-energy mobility edge, which indicates the existence of spin-dependent universality classes for metal-insulator transitions. One of the most fascinating and rather unexpected results is the appearance of half metals at intermediate energy scales above the high-energy mobility edge in Anderson-type insulators of the Fermi energy; that is, only spin-$\ensuremath{\uparrow}$ electrons are delocalized, while spin-$\ensuremath{\downarrow}$ electrons are Anderson type localized.

Gabriel Lemarié, DELANDE, Dominique