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Bosonization of Nonequilibrium Quantum Wire Networks

Ngo Dinh, Stéphane

Published Jan 1, 2013• 0 citations
Publisher

Abstract

We develop a general approach to nonequilibrium nanostructures formed by one-dimensional channels coupled by tunnel junctions and/or by impurity scattering. As important applications of the formalism we consider tunneling into a voltage-biased quantum wire containing weak backscattering defects, electronic Fabry-Pérot and Mach-Zehnder interferometers realized in the integer quantum Hall systems, and the relaxation dynamics of a nonequilibrium Luttinger liquid after an interaction quench.

DOI
10.5445/ir/1000036847

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We develop a general approach to nonequilibrium nanostructures formed by one-dimensional channels coupled by tunnel junctions and/or by impurity scattering. As important applications of the formalism we consider tunneling into a voltage-biased quantum wire containing weak backscattering defects, electronic Fabry-Pérot and Mach-Zehnder interferometers realized in the integer quantum Hall systems, and the relaxation dynamics of a nonequilibrium Luttinger liquid after an interaction quench.
We develop a general approach to nonequilibrium nanostructures formed by one-dimensional channels coupled by tunnel junctions and/or by impurity scattering.The formalism is based on nonequilibrium version of functional bosonization.A central role in this approach is played by the Keldysh action that has a form reminiscent of the theory of full counting statistics.To proceed with evaluation of physical observables, we assume the weak-tunneling regime and develop a real-time instanton method.A detailed exposition of the formalism is supplemented by two important applications: (i) tunneling into a biased Luttinger liquid with an impurity, and (ii) quantum-Hall Fabry-Pérot interferometry.
We evaluate tunneling rates into/from a voltage (U)(U)(U) biased quantum wire containing a weak backscattering defect. Interacting electrons in such a wire form a true nonequilibrium state of the Luttinger liquid (LL). This state is created due to inelastic electron backscattering leading to the emission of nonequilibrium plasmons with typical frequency $\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\le}U$. The tunneling rates are split in two edges. The tunneling exponent at the Fermi edge is positive and equals that of the equilibrium LL, while the exponent at the side edge ${E}_{F}\ensuremath{-}U$ is negative if Coulomb interaction is not too strong.
The quantum physics of one dimensional systems exhibits many remarkable and exotic physical phenomena. These are the result of the strong correlations which exist in such systems due to the reduced dimensionality and in many cases can be described by Luttinger Liquid theory. Likewise quantum impurity systems, where a bath of particles couples to the same impurity, exhibit well known non perturbative phenomena like the Kondo effect. In this thesis we study the physics of quantum impurities in interacting environments - the intersection of these two areas. Such Luttinger-impurity systems can be realized experimentally in a range of different settings from quantum wires and carbon nano tubes to edges of fractional quantum hall systems and cold atom gases. We show that many of the models used to describe these experiments are integrable and can be studied by means of the Bethe Ansatz. This powerful and exact method provides us with the exact eigenstates and energy levels of the model Hamiltonian. The models considered include two which describe a Lutttinger liquid coupled to a back scattering impurity, the Kane-Fisher and weak tunnelling model as well as Luttinger liquids coupled a quantum dot in various geometries: at the boundary, sidecoupled and embedded. To incorporate both the backscattering nature of the impurities and the interacting bulk particles a new formulation of the Bethe Ansatz is developed and implemented. The eigenstates and spectra of the models are found and used to investigate their ground state and thermodynamic properties.
We develop a theoretical framework to determine distribution functions in nonequilibrium systems coupled to equilibrium reservoirs, by using the nonequilibrium Green's function technique. As a paradigmatic example, we consider the nonequilibrium distribution function in a nanowire under a bias voltage. We model the system as a tight-binding chain connected to reservoirs with different electrochemical potentials at both ends. For electron scattering processes in the wire, we consider both elastic scattering from impurities and inelastic scattering from phonons within the self-consistent Born approximation. We demonstrate that the nonequilibrium distribution functions, as well as the electrostatic potential profiles, in various scattering regimes are well described within our framework. This scheme will contribute to advancing our understanding of quantum many-body phenomena driven by nonequilibrium distribution functions that have different functional forms from the equilibrium ones.
Bosonization technique for one-dimensional fermions out of equilibrium is developed in the framework of the Keldysh action formalism. We first demonstrate how this approach is implemented for free fermions and for the problem of nonequilibrium Fermi edge singularity. We then employ the technique to study an interacting quantum wire attached to two electrodes with arbitrary energy distributions. The nonequilibrium electron Green's functions, which can be measured via tunneling spectroscopy technique and carry the information about energy distribution, zero-bias anomaly, and dephasing, are expressed in terms of functional determinants of single-particle ``counting'' operators. The corresponding time-dependent scattering phase is found to be intrinsically related to ``fractionalization'' of electron-hole excitations in the tunneling process and at boundaries with leads. Results are generalized to the case of spinful particles as well to Green's functions at different spatial points (relevant to the problem of dephasing in Luttinger liquid interferometers). For double-step distributions, the dephasing rates are oscillatory functions of the interaction strength.