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arXiv:cond-mat/0410586 · Physical Review B

Disorder and interactions in one-dimensional systems

Jonathan M. Carter, A. MacKinnon

Published Jul 19, 2005• 22 citations
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Abstract

We present a numerical approach to the study of disorder and interactions in quasi-one-dimensional (1D) systems which combines aspects of the transfer matrix method and the density matrix renormalization group which have been successfully applied to disorder and interacting problems, respectively. The method is applied to spinless fermions in 1D, and the existence of a conducting state is demonstrated in the presence of attractive interactions.

DOI
10.1103/physrevb.72.024208

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We present a numerical approach to the study of disorder and interactions in quasi-one-dimensional (1D) systems which combines aspects of the transfer matrix method and the density matrix renormalization group which have been successfully applied to disorder and interacting problems, respectively. The method is applied to spinless fermions in 1D, and the existence of a conducting state is demonstrated in the presence of attractive 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.
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.
In a previous paper [Bischoff and Jeckelmann, Phys. Rev. B 96, 195111 (2017)] we introduced a density-matrix renormalization group method for calculating the linear conductance of one-dimensional correlated quantum systems and demonstrated it on homogeneous spinless fermion chains with impurities. Here we present extensions of this method to inhomogeneous systems, models with phonons, and the spin conductance of electronic models. The method is applied to a spinless fermion wire-lead model, the homogeneous spinless Holstein model, and the Hubbard model. Its capabilities are demonstrated by comparison with the predictions of Luttinger liquid theory combined with Bethe ansatz solutions and other numerical methods. We find a complex behavior for quantum wires coupled to interacting leads when the sign of the interaction (repulsive/attractive) differs in the wire and leads. The renormalization of the conductance given by the Luttinger parameter in purely fermionic systems is shown to remain valid in the Luttinger liquid phase of the Holstein model with phononic degrees of freedom.
The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White.This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated.It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer.The approach is particularly suitable for one-, or quasi-one-dimensional problems.The method involves truncating the Hilbert space in a systematic and optimised manner.Results for the ground state energy are thus variational bounds.The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems.We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described.More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties.This approach has been applied to the anisotropic spin-1/2 Heisenberg chain.Finally, we discuss recent work which has borne promising applications in two dimensions -the Ising model and the two-dimensional Hubbard model.
Simulation methods for real materials often require the solution of extremely large inhomogeneous systems of noninteracting fermions. Here, the authors show how the density matrix renormalization group (DMRG), the most successful method for 1D interacting systems, can be adapted to noninteracting fermions, giving a significant improvement both over conventional DMRG and over the full solution of the noninteracting problem. The authors test the method on a range of one- and two-dimensional examples for both static and dynamic properties, demonstrating its applicability to systems up to a million sites.