Abstract

We base a scaling theory of localization on an expression for conductivity of a system of random elastic scatterers in terms of its scattering properties at a fixed energy. This expression, proposed by Landauer, is first derived and generalized to a system of indefinite size and number of scattering channels (a "wire"), and then an exact scaling theory for the one-dimensional chain is given. It is shown that the appropriate scaling variable is $f(\ensuremath{\rho})=\mathrm{ln}(1+\ensuremath{\rho})$ where $\ensuremath{\rho}$ is the dimensionless resistance, which has the property of "additive mean," and that scaling leads to a well-behaved probability distribution of this variable and to a very simple scaling law not previously given in the literature.

Keywords

ScalingDimensionless quantityScatteringChain (unit)PhysicsScaling lawDistribution (mathematics)Random variableMathematical physicsStatistical physicsEnergy (signal processing)Expression (computer science)Variable (mathematics)Simple (philosophy)Mathematical analysisMathematicsQuantum mechanicsGeometryStatisticsComputer science

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Publication Info

Year
1980
Type
article
Volume
22
Issue
8
Pages
3519-3526
Citations
1003
Access
Closed

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Cite This

P. W. Anderson, D. J. Thouless, Elihu Abrahams et al. (1980). New method for a scaling theory of localization. Physical review. B, Condensed matter , 22 (8) , 3519-3526. https://doi.org/10.1103/physrevb.22.3519

Identifiers

DOI
10.1103/physrevb.22.3519