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.
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.
A quantum impurity attached to an interacting quantum wire gives rise to an array of new phenomena. Using the Bethe Ansatz we solve exactly models describing two geometries of a quantum dot coupled to an interacting quantum wire: a quantum dot that is (i) side coupled and (ii) embedded in a Luttinger liquid. We find the eigenstates and determine the spectrum through the Bethe Ansatz equations. Using this we derive exact expressions for the ground-state dot occupation. The thermodynamics are then studied using the thermodynamics Bethe Ansatz equations. It is shown that at low energies the dot becomes fully hybridized and acts as a backscattering impurity or tunnel junction depending on the geometry and furthermore that the two geometries are related by changing the sign of the interactions. Although remaining strongly coupled for all values of the interaction in the wire, there exists competition between the tunneling and backscattering leading to a suppression or enhancement of the dot occupation depending on the sign of the bulk interactions.
A Luttinger liquid coupled to a quantum impurity describes a large number of physical systems. The Hamiltonian consists of left- and right-moving fermions interacting among themselves via a density-density coupling and scattering off a localized transmitting and reflecting impurity. We solve exactly the Hamiltonian by means of an incoming-outgoing scattering Bethe basis which properly incorporates all scattering processes. A related model, the weak-tunneling model, wherein the impurity is replaced by a tunnel junction, is solved by the same method. The consistency of the construction is established through a generalized Yang-Baxter relation. Periodic boundary conditions are imposed and the resulting Bethe ansatz equations are derived by means of the off-diagonal Bethe ansatz approach. We derive the spectrum of the model for all coupling constant regimes and calculate the impurity free energy. We discuss the low energy behavior of the systems for both repulsive and attractive interactions.
In this Ph.D. thesis I present the background and providefurther details to the work presented in the appended papers. The background consists of brief overviews of Kondo physics, Luttinger liquid theory, and conformal field theory. My own research is divided into three parts: the first deals with the interplay between Luttinger and Kondo physics (described in Papers I and II), the secondaddresses scanning tunneling microscopy (STM) response of a Luttinger liquid with an edge or an impurity (described in Paper III), and the third explores the boundary Green's function of 1D electrons in the spin-incoherent regime (described in Paper IV). In Papers I and II a setup is proposed which allows for acontrolled study of Kondo and Luttinger liquid physics. Itconsists of a quantum box, biased by a gate voltage, and side-coupled to a quantum wire by a point contact. Close to the degeneracy points of the Coulomb blockaded box the setup can be described as a Luttinger liquid interacting with an effective Kondo impurity. Using boundary conformal field theory techniques we predict that for the case of spin-polarized electrons the differential capacitance of the box will exhibit distinctive Luttinger liquid scaling with temperature and gate voltage. In the limit of zero magnetic field the Luttinger liquid behavior gets masked by two-channel Kondo screening, leading to a logarithmic scaling of the differential capacitance with temperatureand gate voltage. These effects should be possible to studyexperimentally, using the recently developed single-electrontransistor (SET) electrometer technique.In Paper III the finite-temperature local spectral weight (LSW) of a Luttinger liquid with a hard wall boundary is calculated. Close to the boundary the LSW exhibits characteristic oscillations indicative of spin-charge separation. The line shape of the LSW is also found to have a Fano-like asymmetry, a feature originating from the combined effect of electron-electron interaction and scattering off the boundary. Our results can be used to predict how edges and impurities influencethe STM response of one-dimensional electron systemsat low temperatures and voltage bias. Applications to STM onsingle-walled carbon nanotubes are also discussed.Paper IV, finally, presents a study of the spin-incoherent regime of 1D strongly interacting electrons. For sufficiently low densities the potential energy dominates the kinetic energy, and one can easily reach thespin-incoherent regime where the spin exchange energy is much less than the temperature. We have generalized the description of the spin-incoherent regime to account for the presence of a boundary. By calculating the exact Green's function we find that the charge sector critical exponent is highly sensitive to the boundary, strongly modifying the tunneling of electrons close to it. Our approach also allows for a detailed description of the crossover between boundary and bulk regimes.
This thesis concerns correlation effects in quantum many-particle systems in one and two dimensions. Such systems show many exotic non-Fermi liquid phenomena, which can be treated analytically using non-perturbative field-theory methods.\nQuantum phase transitions between topologically ordered phases of matter, which do not break any symmetries, are studied. It is shown that although there is no local order parameter, a local measure from quantum information theory called reduced fidelity can detect such transitions.\nEntanglement in quantum impurity systems is also studied. The general expression for scaling corrections in entanglement entropy from boundary perturbations is derived within conformal field theory, showing that the asymptotic decay of Kondo screening clouds follow the same power-law as the impurity specific heat.\nFurthermore, the effects from spin-orbit interactions on Kondo physics in helical Luttinger liquids are studied. Such helical liquids occur on the edges of two-dimensional topological insulators. It is shown that Rashba and Dresselhaus interactions can potentially destroy Kondo singlet formation in such a system, and that the coupling to an electric field gives a mechanism to control transport properties.\nThe most recent work focuses on correlations in interacting one-dimensional Bose gases. The asymptotic expression for correlation functions in a generalized Gibbs ensemble, where all the local conservation laws appear, is obtained from Bethe Ansatz and conformal field theory.\nIn addition to the research papers with the above results, the thesis also contains an introductory text reviewing background material.
This thesis addresses a theoretical study of the problem of a single impurity embedded in a one-dimensional system of interacting electrons in presence of electron-phonon coupling. First we consider a system with a featureless point-like potential impurity, followed by the case of a resonant level hybridised with a Luttinger Liquid. The stress is made on a more fundamental problem of a featureless scatterer, for which two opposite limits in the impurity strength are considered: a weak scatterer and a weak link. We have found that, regardless of the transmission properties of phonons through the impurity, the scaling dimensions of the conductance in these limits obey the duality condition, \\( \\triangle_{WS}\\) \\( \\triangle_{WL}\\) = 1, known for the Luttinger Liquid in the absence of phonons. However, in the case when the strength of phonon scattering is correlated with electron scattering by the impurity, we find a nontrivial phase diagram with up to three fixed points and a possibility of a metal-insulator transition. We also consider the case of a weakly interacting electron-phonon system in the presence of a single impurity of an arbitrary scattering potential. In the problem of a resonant level attached to the Luttinger Liquid we show that the electron-phonon coupling significantly modifies the effective energy-dependent width of the resonant level in two different geometries, corresponding to the resonant and anti-resonant transmission in the Fermi gas.
In a first part, I study the electronic transport through a one-dimensional, finite-length quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (two-dimensional) leads, thereby bringing theory closer towards systems resembling setups realized in experiments. In particular, I compute the temperature dependence of the linear conductance of a system without bulk impurities. The appearance of new temperature scales introduced by the lengths of the overhanging parts of the leads and the wire implies a conductance function which is much more complex than the simple power-law behavior obtained in earlier approaches. My results can be used to optimize the experimental setups designed for a verification of Luttinger-liquid scaling. In a second part, I suggest a setup to study Luttinger-liquid behavior in quantum wires which allows to determine the Luttinger-liquid parameter from two independent measurements: transport through a quantum dot embedded in the wire and the charge on the dot. To this end, I identified novel Luttinger-liquid power-laws in the charging of the dot. In a more technical part, I extended an adaption of the functional renormalization group such that it can be used on correlated electrons in complex geometries like the wire-lead structures described above and arbitrary dot structures coupled to Luttinger-liquid leads.