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Title: Conductance of finite systems and scaling in localization theory

Abstract

The conductance of finite systems plays a central role in the scaling theory of localization (Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)). Usually it is defined by the Landauer-type formulas, which remain open the following questions: (a) exclusion of the contact resistance in the many-channel case; (b) correspondence of the Landauer conductance with internal properties of the system; (c) relation with the diffusion coefficient D({omega}, q) of an infinite system. The answers to these questions are obtained below in the framework of two approaches: (1) self-consistent theory of localization by Vollhardt and Woelfle, and (2) quantum mechanical analysis based on the shell model. Both approaches lead to the same definition for the conductance of a finite system, closely related to the Thouless definition. In the framework of the self-consistent theory, the relations of finite-size scaling are derived and the Gell-Mann-Low functions {beta}(g) for space dimensions d = 1, 2, 3 are calculated. In contrast to the previous attempt by Vollhardt and Woelfle (1982), the metallic and localized phase are considered from the same standpoint, and the conductance of a finite system has no singularity at the critical point. In the 2D case, the expansion of {beta}(g) in 1/gmore » coincides with results of the {sigma}-model approach on the two-loop level and depends on the renormalization scheme in higher loops; the use of dimensional regularization for transition to dimension d = 2 + {epsilon} looks incompatible with the physical essence of the problem. The results are compared with numerical and physical experiments. A situation in higher dimensions and the conditions for observation of the localization law {sigma}({omega}) {proportional_to} -i{omega} for conductivity are discussed.« less

Authors:
 [1]
  1. Kapitza Institute for Physical Problems (Russian Federation)
Publication Date:
OSTI Identifier:
22069261
Resource Type:
Journal Article
Journal Name:
Journal of Experimental and Theoretical Physics
Additional Journal Information:
Journal Volume: 115; Journal Issue: 5; Other Information: Copyright (c) 2012 Pleiades Publishing, Ltd.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1063-7761
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; DIFFUSION; FEYNMAN DIAGRAM; QUANTUM FIELD THEORY; QUANTUM MECHANICS; RENORMALIZATION; SIGMA MODEL; SINGULARITY

Citation Formats

Suslov, I. M., E-mail: suslov@kapitza.ras.ru. Conductance of finite systems and scaling in localization theory. United States: N. p., 2012. Web. doi:10.1134/S1063776112110143.
Suslov, I. M., E-mail: suslov@kapitza.ras.ru. Conductance of finite systems and scaling in localization theory. United States. doi:10.1134/S1063776112110143.
Suslov, I. M., E-mail: suslov@kapitza.ras.ru. Thu . "Conductance of finite systems and scaling in localization theory". United States. doi:10.1134/S1063776112110143.
@article{osti_22069261,
title = {Conductance of finite systems and scaling in localization theory},
author = {Suslov, I. M., E-mail: suslov@kapitza.ras.ru},
abstractNote = {The conductance of finite systems plays a central role in the scaling theory of localization (Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)). Usually it is defined by the Landauer-type formulas, which remain open the following questions: (a) exclusion of the contact resistance in the many-channel case; (b) correspondence of the Landauer conductance with internal properties of the system; (c) relation with the diffusion coefficient D({omega}, q) of an infinite system. The answers to these questions are obtained below in the framework of two approaches: (1) self-consistent theory of localization by Vollhardt and Woelfle, and (2) quantum mechanical analysis based on the shell model. Both approaches lead to the same definition for the conductance of a finite system, closely related to the Thouless definition. In the framework of the self-consistent theory, the relations of finite-size scaling are derived and the Gell-Mann-Low functions {beta}(g) for space dimensions d = 1, 2, 3 are calculated. In contrast to the previous attempt by Vollhardt and Woelfle (1982), the metallic and localized phase are considered from the same standpoint, and the conductance of a finite system has no singularity at the critical point. In the 2D case, the expansion of {beta}(g) in 1/g coincides with results of the {sigma}-model approach on the two-loop level and depends on the renormalization scheme in higher loops; the use of dimensional regularization for transition to dimension d = 2 + {epsilon} looks incompatible with the physical essence of the problem. The results are compared with numerical and physical experiments. A situation in higher dimensions and the conditions for observation of the localization law {sigma}({omega}) {proportional_to} -i{omega} for conductivity are discussed.},
doi = {10.1134/S1063776112110143},
journal = {Journal of Experimental and Theoretical Physics},
issn = {1063-7761},
number = 5,
volume = 115,
place = {United States},
year = {2012},
month = {11}
}