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Journal of Physics Condensed Matter

Towards a simulation of disordered systems with interactions

A. MacKinnon

Published Jul 8, 2008• 2 citations
Publisher

Abstract

We consider the problem of Anderson localization in disordered systems with interactions. We present a numerical approach in quasi-one-dimensional (quasi-1D) systems which combines aspects of the transfer matrix and Green's function recursion methods with the density matrix renormalization group. The method is applied to spinless fermions in 1D and a generalization to finite cross-sections is outlined.

DOI
10.1088/0953-8984/20/30/304212

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We consider the problem of Anderson localization in disordered systems with interactions. We present a numerical approach in quasi-one-dimensional (quasi-1D) systems which combines aspects of the transfer matrix and Green's function recursion methods with the density matrix renormalization group. The method is applied to spinless fermions in 1D and a generalization to finite cross-sections is outlined.
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.
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.
Using the density matrix renormalization group algorithm, we investigate the lattice model for spinless fermions in one dimension in the presence of a strong interaction and disorder. The phase sensitivity of the ground state energy is determined with high accuracy for systems up to a size of 60 lattice constants. This quantity is found to be log normally distributed. The fluctuations grow algebraically with system size with a universal exponent of $\ensuremath{\approx}2/3$ in the localized region of the phase diagram. Surprisingly we find, for an attractive interaction, a delocalized phase of finite extension. The boundary of this delocalized phase is determined.
Effects of randomness have supplied fundamental problems in condensed matter physics and localization due to interference of quantum mechanical electrons are well studied as the Anderson localization. Although we have well established understanding of the localization of non-interacting electrons, information of the correlated electrons with randomness is still missing. It was mainly due to lack of reliable numerical techniques for the correlated electrons. For the one dimensional correlated systems without randomness, lots of numerical results are collected by the Density Matrix Renormalization Group (DMRG) method and consistent understanding with analytical predictions has been achieved. In this paper, we plan to apply DMRG for the random electron systems by calculating direct responses of the system with electric field. At first, random systems without interaction are carefully investigated. Then we try to treat both of interaction and randomness in one dimensional systems.