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Disorder in multi-channel Luttinger liquids

Maximilian Jones

Published Jul 1, 2018• 0 citations
OpenAlex

Abstract

This thesis examines the effects of disorder upon a bundle of coupled one dimensional (ID) systems. Each 1D system is described as a Luttinger liquid, and the coupling between channels is weak enough such that this description remains valid. The coupling can be of either a density-density or current-current type. We consider continuous disorder in each channel, and derive renormalisation group (RG) equations governing the strength of the disorder. We analyse the effects of disorder in two specific examples: a lattice of identical channels, and two distinct channels. In both cases, close to the simultaneous metal-insulator transition, we arrive at coupled Berezinskii-Kosterlitz-Thouless (BKT) equations. Away from the simultaneous transition we analyse the RG \nequations numerically. Inter-channel interactions are found to shift the metal-insulator boundary, and destroy the mixed insulator-conductor phase close to the simultaneous transition.

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This thesis examines the effects of disorder upon a bundle of coupled one dimensional (ID) systems. Each 1D system is described as a Luttinger liquid, and the coupling between channels is weak enough such that this description remains valid. The coupling can be of either a density-density or current-current type. We consider continuous disorder in each channel, and derive renormalisation group (RG) equations governing the strength of the disorder. We analyse the effects of disorder in two specific examples: a lattice of identical channels, and two distinct channels. In both cases, close to the simultaneous metal-insulator transition, we arrive at coupled Berezinskii-Kosterlitz-Thouless (BKT) equations. Away from the simultaneous transition we analyse the RG \nequations numerically. Inter-channel interactions are found to shift the metal-insulator boundary, and destroy the mixed insulator-conductor phase close to the simultaneous transition.
We study the phase transition between conducting and insulating states taking place in disordered multichannel Luttinger liquids with interchannel interactions. We derive renormalization-group equations which are perturbative in disorder but nonperturbative in interaction. In the vicinity of the simultaneous phase transition in all channels, these equations become a set of coupled Berezinskii-Kosterlitz-Thouless equations, which we analyze within two models: an array of identical wires and a two-channel model with distinct channels. We show that a competition between disorder and interaction results in a variety of phases, expected to be observable at intermediate temperatures where the interaction and disorder are relevant but weak hybridization and the charge-density-wave interaction may be ignored.
We study effects of quenched disorder on coupled two-dimensional arrays of Luttinger liquids (LL's) as a model for stripes in high-Tc{T}_{c}Tc​ compounds. In the framework of a renormalization-group analysis, we find that weak inter-LL charge-density-wave couplings are always irrelevant as opposed to the pure system. By varying either disorder strength, intra- or inter-LL interactions, the system can undergo a delocalization transition between an insulator and a strongly anisotropic metallic state with LL-like transport. This state is characterized by short-ranged charge-density-wave order, the superconducting order is quasi-long-ranged along the stripes and short ranged in the transversal direction.
The present thesis adresses three different topics that are all in direct connection to the statistical physics of disordered systems. The first chapter, which is the main part of this work, is devoted to driven interfaces in random media. The main focus is on the case of an ac driving. Our investigations start with the associated mean field theory. At the beginning we consider the case of a constant driving force and discuss the critical behaviour close to the depinning transition. Subsequently, we examine the velocity hysteresis curve for small frequencies by use of analytical and numerical methods. It turns out that all observables under consideration exhibit scaling behaviour on approaching the adiabatic case. Next, we analyse perturbation theory for ac-driven interfaces. Tt turns out, that perturbation theory can only work for sufficiently high interface dimensionalities D>4. The failure of perturbation theory in low dimensions is discussed. The second part of this thesis is concerned with the influence of electroelastic coupling on the behaviour of domain walls in ferroelectrics. For the case of electrostrictive coupling in the high temperature phase we find, that merely the interface tension coefficients are changed in an anisotropic fashion. On the contrary, a piezo effect in the paraphase entails a long range term. After deriving the interface hamiltonian in the harmonic approximation, we determine the roughness of domain walls due to random field disorder using an Imry-Ma type argument. On large length scales, domain walls turn out to be flat. In the last chapter, we consider one-dimensional electronic systems, so-called Luttinger liquids. The subject of the first part is the competition of two phases of one-dimensional disordered fermionic systems, namely the Mott and the Anderson insulator. By relating both, the compressibility as well as a finite optical conductivity to a vanishing mass of charged excitations, the existence of an intermediate Mott glass phase can be excluded. Combining informations about the renormalisation group flow and the energy of topological excitations, as well as using simple scaling arguments, we construct the phase diagram. In the second part we study the replica trick to describe the quantum creep and the linear conductivity at finite temperatures for Luttinger liquids with relevant disorder. We provide approximate solutions to the Euler-Lagrange equations for the replica action. The replica limit turns out to be problematic. At the example of a toy model we illustrate a method for the replica limit and demonstrate the emergent difficulties.
We investigate the effect of both strong and weak potential scattering caused by local impurities and extended (line) defects in the array of Luttinger liquid wires. We find that in both cases a finite range inter-wire interaction stabilises metallic state. Based on calculation of the scaling dimensions of one-particle scattering operators, we construct the phase diagram for low-temperature transport along the array. We find that unlike local impurity case where only conducting and insulating states are realised (metal-insulator transition driven by interactions), the extended line defects may bring the system into a mixed state where conducting or insulating behavior can be observed depending on bare strength of the scatterer (metal-insulator transition driven by disorder).
We investigate the stability of conducting and insulating phases in multichannel Luttinger liquids with respect to embedding a single impurity. We devise a general approach for finding critical exponents of the conductance in the limits of both weak and strong scattering. In contrast to the one-channel Luttinger liquid, the system state in certain parametric regions depends on the scattering strength which results in the emergence of a bistability. Focusing on the two-channel liquid, the method developed here enables us to provide a generic analysis of phase boundaries governed by the most relevant (i.e., not necessarily single-particle) scattering mechanism. The present approach is applicable to channels of different nature as in fermion-boson mixtures, or to identical ones as on the opposite edges of a topological insulator. We show that interaction per se cannot provide protection in the particular case of topological insulators realized in narrow Hall bars.
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.