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arXiv:cond-mat/9608006

From single to many Impurities in one-dimensional systems

Thierry Giamarchi, Maurey, Hélène

Published Aug 2, 1996• 1 citation
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Abstract

We examine the effects of disorder in one-dimensional systems. We link the case of a few impurities, typical of a short quantum wire, to that of a finite density of scatterers more appropriate for a long wire or a macroscopic system. Finally we investigate the effects of long-range interactions on the transport in 1D systems. We predict in that case a conductivity behaving as $σ(T) \sim T^2$.}

DOI
10.48550/arxiv.cond-mat/9608006

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We examine the effects of disorder in one-dimensional systems. We link the case of a few impurities, typical of a short quantum wire, to that of a finite density of scatterers more appropriate for a long wire or a macroscopic system. Finally we investigate the effects of long-range interactions on the transport in 1D systems. We predict in that case a conductivity behaving as σ(T)∼T2σ(T) \sim T^2σ(T)∼T2.}
We study transport in interacting one-dimensional systems with a scattering object. In particular we are interested in inelastic scattering processes. Very recent theoretical [3] and experimental [54] research points toward a strong influence of incoherent scattering on the low-temperature transport features of one-dimensional systems with extended scatterers. Furthermore, if inelastic processes are strong a straightforward application of the renormalization group (RG) approach is not possible. In the field of transport in one-dimensional weakly interacting electron systems, this procedure is the state-of-the art method to extend results for weak interaction to the low temperature regime [18,27,43, 47, 52,57, 62, 67,74]. The underlying assumption, namely that the first-order correction in interaction incorporates all effects of the interaction and reproduces itself in higher orders, is widely accepted as a valid “educated guess” disregarding the fact that inelastic processes, if present, arise in subleading orders. Indeed, for the simple case of a single impurity in an infinite wire it is common believe that the approach is valid [6,74]. Having this in mind, systems with a great variety of additional degrees of freedom were treated in the same fashion: Junctions of wires [43] and more complex geometries [18], anti-resonant [47] and resonant [57, 62] scatterers, just to name a few. A careful examination of the interplay between the renormalization and possible inelastic processes or dephasing in these cases is missing. Our strategy is to calculate the second order in interaction for the single impurity situation, thereby establishing a framework with a real-space diagrammatic technique, and to apply this tool to more complex situations, i.e., a finite interacting region and an additional impurity. The diagrammatic technique enables us to identify specific diagrams where the particle number at a given energy is not conserved or the interaction can lead to dephasing in the system. In particular, by modelling the compound scatterer by an energy dependent S-matrix of the Breit-Wigner type, we are able to find a fingerprint of weak dephasing which carries some features of its counterpart in disordered Luttinger liquids [32].
Table of contents 1. Introduction 2. Non-Fermi-liquid features of Fermi liquids: 1D physics in higher dimensions 3. Dzyaloshinskii-Larkin solution of the Tomonaga-Luttinger model 4. Renormalization group for interacting fermions 5. Single impurity in a 1D system: scattering theory for interacting electrons 6. Bosonization solution 7. Transport in quantum wires 7.1 Conductivity and conductance 7.2 Dissipation in a contactless measurement 7.3 Conductance of a wire attached to reservoirs 7.4 Spin component of the conductance 7.5 Thermal conductance: Fabry-Perrot resonances of plasmons 8. Appendices
We use the nonequilibrium Green's function formalism and a self-consistent Hartree-Fock approximation to numerically study the effects of a single impurity and interactions between the electrons (with and without spin) on the conductance of a quantum wire. We study how the conductance varies with the wire length, the temperature, and the strengths of the impurity and interactions. The numerical results for the dependence of the conductance on the wire length and temperature are compared with the results obtained from a renormalization group analysis based on the Hartree-Fock approximation. For the spin-1∕21∕21∕2 model with a repulsive on-site interaction or the spinless model with an attractive nearest neighbor interaction, we find that the conductance increases with increasing wire length or decreasing temperature. This can be explained using the Born approximation in scattering theory. For a strong impurity, the conductance is significantly different for a repulsive and an attractive impurity; this is due to the existence of a bound state in the latter case. In general, the large density deviations close to the impurity have an appreciable effect on the conductance at short distances which is not captured by the renormalization group equations.
One-dimensional electron systems interacting with long-range Coulomb forces (quantum wires) show a Wigner crystal structure. We investigate in this paper the transport properties of such a Wigner crystal in the presence of impurities. Contrary to what happens when only short-range interactions are included, the system is dominated by 4kF{\mathit{k}}_{\mathit{F}}kF​ scattering on the impurities. There are two important length scales in such a problem: one is the pinning length above which the (quasi-)long-range order of the Wigner crystal is destroyed by disorder. The other length, ${\ensuremath{\xi}}_{\mathrm{cr}}$, is the length below which Coulomb interactions are not important and the system is behaving as a standard Luttinger liquid with short-range interactions. We obtain the frequency and temperature dependence of the conductivity. We show that such a system is very similar to a classical charge density wave pinned by impurities, but with important differences due to quantum fluctuations and long-range Coulomb interactions. Finally, we discuss our results in comparison with experimental systems.
In this thesis, we study the effects of both disorder and low carrier density in low-dimensional systems, employing a variety of analytical and numerical methods. The work is structured in two main parts, each addressing a distinct research direction. The first part focuses on the impact of disorder in one-dimensional and quasi-one dimensional systems. Using tools such as bosonization, renormalization group techniques, and exact diagonalization, we investigate two key aspects. First, we explore how specific spatial correlations in the disorder affect the properties of the metallic-insulating transition in a one-dimensional system, uncovering significant deviations from the behaviour typically associated with uncorrelated disorder. Second, we examine the effects of different types of disorder on a quasi-one-dimensional singlet superconductor, where we recover Anderson’s theorem under certain conditions and observe a competition between disorder and interactions. The second part of the thesis examines magneto-transport in two-dimensional, locally correlated materials. Using a microscopic quantum theory, we demonstrate pronounced deviations of the Hall constant and the conductivities of the system from their semiclassical predictions, in regimes where the Fermi energy is no longer the dominant energy scale.