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arXiv:cond-mat/0209213

Disorder and Interactions on a 1D Chain

Jonathan M. Carter, A. MacKinnon

Published Sep 9, 2002• 0 citations

Abstract

It has become increasingly clear that a full understanding of the physics of electrons in disordered systems requires an approach in which both disorder and interactions are taken into account. Work on small numbers of electrons has suggested that interactions can substantially increase the localisation length in 1 dimensional systems and possibly lead to delocalisation in 2D. It has not been clear, however, to what extent these results are applicable to systems with a finite density of electrons. Among numerical approaches the transfer matrix method combined with finite size scaling has been particularly successful for non--interacting systems, whereas the density matrix renormalisation group successfully describes interacting systems. We have developed a new approach which combines elements of both these methods and have successfully applied it to spinless electrons with nearest neighbour interactions.

DOI: 10.48550/arxiv.cond-mat/0209213

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It has become increasingly clear that a full understanding of the physics of electrons in disordered systems requires an approach in which both disorder and interactions are taken into account. Work on small numbers of electrons has suggested that interactions can substantially increase the localisation length in 1 dimensional systems and possibly lead to delocalisation in 2D. It has not been clear, however, to what extent these results are applicable to systems with a finite density of electrons. Among numerical approaches the transfer matrix method combined with finite size scaling has been particularly successful for non--interacting systems, whereas the density matrix renormalisation group successfully describes interacting systems. We have developed a new approach which combines elements of both these methods and have successfully applied it to spinless electrons with nearest neighbour interactions.

Anderson localization is a quantum phenomenon in which disorder localizes electronic wave functions. In this work, we propose a new approach to study Anderson localization based on the density matrix formalism. Drawing an analogy to the standard transfer matrix method, we extract the localization length from the modular density matrix in quasi-one-dimensional systems. This approach successfully captures the metal-insulator transition in the three-dimensional Anderson model and in the two-dimensional Anderson model with spin-orbit coupling. It can be also readily extended to multi-orbital systems. We further generalize the formalism to interacting systems, showing that the one-dimensional spinless attractive model exhibits the expected metallic phase, consistent with previous studies. More importantly, we demonstrate the existence of a two-dimensional metallic tendency in the presence of Hubbard interactions and disorder. This method offers a new perspective on Anderson localization and its interplay with interactions.

Abstract We have seen that disordered systems give rise to a host of interesting effects related to the localization of states. Thus, from the outset we have explicitly to take account of the localization in the theoretical framework, which makes computer experiments on finite systems an important first-principles method. This situation immediately poses several essential questions, such as: (i) how can the properties of macroscopic systems be deduced from the results for finite systems; and (ii) how can the concept of Anderson localization be extended to disordered systems with electron-electron interactions in a numerical model in which disorder and electron correlation are treated on an equal footing? In this chapter we address these questions. In this section we introduce the concept of renormalization for disordered systems. Specifically, we examine real-space renormalization schemes.