Abstract The extended quasiparticle picture is adapted to non-Fermi systems by suggesting a Padé approximation which interpolates between the known small scattering-rate expansion and the deviation from the Fermi energy. The first two energy-weighted sum rules are shown to be fulfilled independent of the interpolating function for any selfenergy. For various models of one-dimensional Fermions scattering with impurities the quality of the Padé approximation for the spectral function is demonstrated and the reduced density matrix or momentum distribution is reproduced not possessing a jump at the Fermi energy. Though the two-fold expansion is necessary to realize the spectral function and reduced density, the extended quasiparticle approximation itself is sufficient for the description of transport properties due to cancellation of divergent terms under integration. The T-matrix approximation leads to the delay time as the time two particles spend in a correlated state. This contributes to the reduced density matrix and to an additional part in the conductivity which is presented at zero and finite temperatures. Besides a localization at certain impurity concentrations, the conductivity shows a maximum at small temperatures interpreted as onset of superconducting behaviour triggered by impurities. The Tan contact reveals the same universal behaviour as known from electron–electron scattering. Graphic abstract
Abstract The extended quasiparticle picture is adapted to non-Fermi systems by suggesting a Padé approximation which interpolates between the known small scattering-rate expansion and the deviation from the Fermi energy. The first two energy-weighted sum rules are shown to be fulfilled independent of the interpolating function for any selfenergy. For various models of one-dimensional Fermions scattering with impurities the quality of the Padé approximation for the spectral function is demonstrated and the reduced density matrix or momentum distribution is reproduced not possessing a jump at the Fermi energy. Though the two-fold expansion is necessary to realize the spectral function and reduced density, the extended quasiparticle approximation itself is sufficient for the description of transport properties due to cancellation of divergent terms under integration. The T-matrix approximation leads to the delay time as the time two particles spend in a correlated state. This contributes to the reduced density matrix and to an additional part in the conductivity which is presented at zero and finite temperatures. Besides a localization at certain impurity concentrations, the conductivity shows a maximum at small temperatures interpreted as onset of superconducting behaviour triggered by impurities. The Tan contact reveals the same universal behaviour as known from electron–electron scattering. Graphic abstract
Expanding the two-particle Green's functions determines the self-energy and the polarization as well as the response function on the same footing. The correlation energy is calculated with the help of the extended quasiparticle picture, which accounts for off-shell effects. The corresponding response function leads to the same correlation energy as the self-energy in agreement with perturbation theory, provided one works in the extended quasiparticle picture. A one-dimensional quantum wire of fermions is considered and ground-state properties are calculated in the high-density regime within the extended quasiparticle picture and Born approximation. While the on-shell selfenergies are strictly zero due to Pauli-blocking of elastic scattering, the off-shell behavior shows a rich structure of a gap in the damping of excitation, which is closed when the momentum approaches the Fermi one. The consistent spectral function is presented, completing the first two energy-weighted sum rules. The excitation spectrum shows a splitting due to holons and antiholons as non-Fermi liquid behavior. A renormalization procedure is proposed by subtracting an energy constant to render the Fock exchange energy finite. The effective mass derived from meanfield approximation shows a dip analogous to the onset of Peierls instability. The reduced density matrix or momentum distribution is calculated with the help of a Pad\'e regularization repairing deficiencies of the perturbation theory. A seemingly finite step at the Fermi energy indicating Fermi-liquid behavior is repaired in this way.
The theoretical investigation of spectral functions and pseudogap in systems with strongly correlated electrons is discussed, with the emphasis on the single-band t-J model as relevant for superconducting cuprates. The evidence for the pseudogap features from numerical studies of the model is presented. One of the promising methods to study spectral functions is the method of equations of motion. The latter can deal systematically with the local constraints and projected fermion operators inherent for strongly correlated electrons. In the evaluation of the self energy the decoupling of spin and single-particle fluctuations is performed. In an undoped antiferromagnet the method reproduces the selfconsistent Born approximation (SCBA). For finite doping the approximation evolves into a paramagnon contribution which retains large incoherent contribution in the hole part. On the other hand, the contribution of longer-range spin fluctuations is essential for the emergence of the pseudogap. The latter shows up at low doping in the effective truncation of the large Fermi surface, reduced electron density of states and at the same time reduced quasiparticle density of states at the Fermi level.
The properties of fermions in one dimension are markedly different from those in higher dimensions. In two or three dimensions, the many-body behaviour of fermions is described by the Fermi liquid model, whose most prominent result is that, instead of studying the behaviour of individual particles, it is better to describe the emergent phenomena of the whole system, known as quasiparticles. These are collective modes of excitations born out of particle-particle interactions, which resemble the constituent single particles, albeit with renormalised mass. In one dimension, however, the formation of such a correlated state is unfavourable and the quasiparticles are completely absent. Instead, 1D fermions are described by a model known as the Tomonaga-Luttinger liquid. Central to this model is the process of bosonisation, a mathematical technique which maps the excitation of fermions to bosonic modes of spin-density and charge-density waves. As a prerequisite of the bosonisation technique, the fermionic single-particle dispersion is approximated as a linear function. Consequently, the Luttinger liquid model is limited in its scope of applicability. Specifically, it is valid only in the vicinity of the Fermi points, where the excitation energy is low. Recent advancement in the theoretical field has expanded our understanding of 1D interacting fermions beyond the limit of linearity. The mobile-impurity model, proposed by L. Glazman and his co-authors, is a generalised model of 1D fermions which extends the scope of applicability of the Tomanaga-Luttinger model, accounting for the curvature of single-particle dispersion relations. It extends the region of validity of the tradition TLL theory from the neighbourhood of the Fermi points to a band which follows the single-particle dispersion. This thesis presents the results of an experimental work which provides direct evidence of non-linear behaviour predicted by the new model. It revolves around the measurement of the spectral function of electrons in surface-gate-defined quantum wires on a double-quantum-well, GaAs/AlGaAs device, and the comparison and fitting of the result to the non-linear theory. The spectral function is a theoretical concept frequently used in discussions on many-body problems. Its physical interpretation makes it closely related to experimental observables, such as the tunnelling current between the double quantum wells in this work. This thesis consists of three parts. Part one concerns the theories of the various many-body models and the experimental technique: Chapter 1 briefly introduces the concept of Fermi liquid, before focusing on a detailed account of the Luttinger-liquid model. The mobile-impurity model will be introduced in the framework of the Luttinger liquid model afterwards, followed by a review of experimental works reported in the literature on the study of TLL physics. In Chapter 2, the principles of the experimental tunnelling spectroscopy technique are described. The second part of the thesis details the structure of the test device used in the experiment and its fabrication. In particular, we will discuss the intricacies of the three-dimensional EBL fabrication process developed specifically for this project. The setup and running of the measurement circuit will also be discussed in this part of the thesis, in the context of a prerequisite sample characterisation measurement. In the final part of the work, data from the main experiment will be presented and analysed, in search for the electron-electron interaction effects which are predicted by the theoretical discussions in the first chapter.
Abstract In this work we discuss extensions of the pioneering analysis by Dzyaloshinskii and Larkin [Sov. Phys. JETP 38, 202 (1974)] of correlation functions for one-dimensional Fermi systems, focusing on the effects of quasiparticle relaxation enabled by a nonlinear dispersion. Throughout the work we employ both, the weakly interacting Fermi gas picture and nonlinear Luttinger liquid theory to describe attenuation of excitations and explore the fermion-boson duality between both approaches. Special attention is devoted to the role of spin-exchange processes, effects of interaction screening, and integrability. Thermalization rates for electron- and hole-like quasiparticles, as well as the decay rate of collective plasmon excitations and the momentum space mobility of spin excitations are calculated for various temperature regimes. The phenomenon of spin–charge drag is considered and the corresponding momentum transfer rate is determined. We further discuss how momentum relaxation due to several competing mechanisms, viz. triple electron collisions, electron–phonon scattering, and long-range inhomogeneities affect transport properties, and highlight energy transfer facilitated by plasmons from the perspective of the inhomogeneous Luttinger liquid model. Finally, we derive the full matrix of thermoelectric coefficients at the quantum critical point of the first conductance plateau transition, and address magnetoconductance in ballistic semiconductor nanowires with strong Rashba spin–orbit coupling.
We study the temperature dependence of the electrical resistivity of interacting two-dimensional Fermi liquids. We perform a numerical simulation of the nonequilibrium state based on semiclassical Boltzmann transport theory. A two-dimensional system of quasiparticles on a square lattice serves as a model system. We use a single-orbital tight-binding model of the dispersion. For conceptual purposes, we choose a simple repulsive onsite interaction that leads to quasiparticle scattering and delta-potential scatterers as a model of nonmagnetic impurity scattering. Through our simulation, we demonstrate that deviations from the predictions of standard Fermi-liquid theory can arise due to the nontrivial scattering geometry of umklapp processes, in special cases even in the ultralow-temperature limit. We show through qualitative arguments how these unconventional scaling properties of the electrical resistivity, which are often interpreted as indication of a non-Fermi-liquid state, can arise due to special geometric conditions of the Fermi surface. The appearance of robust deviations from the predictions of Fermi-liquid theory within our simple model presents a viewpoint in order to interpret unconventional transport properties in electron-electron scattering dominated metallic systems.
Low energy properties of the metallic state of the two-dimensional t - J model are presented for second neighbor hopping with hole-doping ( ) and electron-doping ( ), with various superexchange energy J . We use a closed set of equations for the Greens functions obtained from the extremely correlated Fermi liquid theory . These equations reproduce the known low energies features of the large U Hubbard model in infinite dimensions. The density and temperature dependent quasiparticle weight, decay rate and the peak spectral heights over the Brillouin zone are calculated. We also calculate the resistivity, Hall conductivity, Hall number and cotangent Hall angle. The spectral features display high thermal sensitivity at modest T for density , implying a suppression of the effective Fermi-liquid temperature by two orders of magnitude relative to the bare bandwidth. The cotangent Hall angle exhibits a T 2 behavior at low T , followed by an interesting kink at higher T . The Hall number exhibits strong renormalization due to correlations. Flipping the sign of changes the curvature of the resistivity versus T curves between convex and concave. Our results provide a natural route for understanding the observed difference in the temperature dependent resistivity of strongly correlated electron-doped and hole-doped matter.