# 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 »

- Authors:

- 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}

}