In this thesis I present a theoretical study of three different\nstrongly correlated systems: Luttinger liquids, spin-1 chains and\nthe two-dimensional spin-1/2 Heisenberg model.\n\nFor every sub-field of our research I summarize the background and\nprovide further details to the work presented in the appended\npapers. Chapter 1 corresponds to paper I, Chapter 2 to paper II,\nChapter 3 and 4 to paper III and IV.\n\nIn paper I, we calculate with bosonization techniques the local\ndensity of states of a finite segment of Luttinger liquid. Besides of\nbeing of experimental relevance for Scanning Tunneling Microscopy\nexperiments on carbon nanotubes and cleaved overgrowth wires, the\nwave functions of a "Luttinger liquid in a box" very nicely\nvisualize the fundamental bosonic nature of the low energy\nexcitations.\n\nIn paper II, we analyze with field theory methods how the presence\nof a single-ion anisotropy affects the behavior of a spin-1 chain.\nWe determine in the framework of the Non Linear Sigma Model how the gaps\nof the anisotropic chain depend on an applied staggered magnetic\nfield. Our theoretical analysis is meant to model compounds like\n$\\rm R_2BaNiO_5$ that show coexistence of Haldane phase and\nlong-range order.\n\nIn paper III, we compute with quantum Monte Carlo simulations the\nlocal response to a uniform magnetic field around vacancies in the\ntwo-dimensional Heisenberg model. From the full understanding of the\nnumerical results, we can make quantitative predictions of relevance\nfor Nuclear Magnetic Resonance and susceptibility experiments. In\nparticular, we expect, even in the thermodynamic limit, a finite\nCurie contribution to the total susceptibility arising from the\nimpurities.\n\nFinally in paper IV, we consider interaction effects among\nimpurities. We compute the potential between static vacancies in\nseveral low-dimensional spin-1/2 antiferromagnets comparing\nthe numerics with exact results from conformal field theory and\nlinear spin wave theory. The potential is directly related to the\nlocal valence bond order and its magnitude is a loyal measure of the\namount of quantum correlations present into the system.
In this thesis I present a theoretical study of three different\nstrongly correlated systems: Luttinger liquids, spin-1 chains and\nthe two-dimensional spin-1/2 Heisenberg model.\n\nFor every sub-field of our research I summarize the background and\nprovide further details to the work presented in the appended\npapers. Chapter 1 corresponds to paper I, Chapter 2 to paper II,\nChapter 3 and 4 to paper III and IV.\n\nIn paper I, we calculate with bosonization techniques the local\ndensity of states of a finite segment of Luttinger liquid. Besides of\nbeing of experimental relevance for Scanning Tunneling Microscopy\nexperiments on carbon nanotubes and cleaved overgrowth wires, the\nwave functions of a "Luttinger liquid in a box" very nicely\nvisualize the fundamental bosonic nature of the low energy\nexcitations.\n\nIn paper II, we analyze with field theory methods how the presence\nof a single-ion anisotropy affects the behavior of a spin-1 chain.\nWe determine in the framework of the Non Linear Sigma Model how the gaps\nof the anisotropic chain depend on an applied staggered magnetic\nfield. Our theoretical analysis is meant to model compounds like\nrmR2BaNiO5 that show coexistence of Haldane phase and\nlong-range order.\n\nIn paper III, we compute with quantum Monte Carlo simulations the\nlocal response to a uniform magnetic field around vacancies in the\ntwo-dimensional Heisenberg model. From the full understanding of the\nnumerical results, we can make quantitative predictions of relevance\nfor Nuclear Magnetic Resonance and susceptibility experiments. In\nparticular, we expect, even in the thermodynamic limit, a finite\nCurie contribution to the total susceptibility arising from the\nimpurities.\n\nFinally in paper IV, we consider interaction effects among\nimpurities. We compute the potential between static vacancies in\nseveral low-dimensional spin-1/2 antiferromagnets comparing\nthe numerics with exact results from conformal field theory and\nlinear spin wave theory. The potential is directly related to the\nlocal valence bond order and its magnitude is a loyal measure of the\namount of quantum correlations present into the system.
Strongly correlated systems show a variety of intriguing properties and have been extensively studied over the years. In this thesis two strongly correlated systems are discussed: the two-dimensional antiferromagnet and the Luttinger liquid. The Luttinger liquid has attracted much interest as a paradigm of non-Fermi liquid behavior. In the first part of the thesis, basic properties of Fermi liquids are briefly described while those of Luttinger liquids are considered in more detail. The differences in spectral properties of Luttinger liquids and Fermi liquids are discussed, as are the consequences for photoemission experiments. In this context Luttinger liquids with boundaries, treated in Papers I, II, and III, are introduced. The exact single-electron Green's function at finite temperature is calculated with bosonization methods, for both periodic and finite open-boundary systems. The corresponding local spectral density is analyzed. For a system with open boundaries, the well-known zero temperature bulk behavior always crosses over to a boundary dominated regime for small energies, while at finite temperature the low energy behavior becomes quadratic. For a periodic finite-size system we also calculate the Coulomb-blockade-like oscillations and study their dependence on interaction and temperature. The two-dimensional antiferromagnet has attracted much attention due to its connection to the undoped phase of the ceramic high-temperature superconductors. In the second part of the thesis including Papers IV and V, the two-dimensional Heisenberg antiferromagnet defined on the triangular and the honeycomb lattices is studied. We use the Schwinger-boson mean-field theory to investigate the stability of the ground state with respect to frustration. For the honeycomb lattice model we find an increased Néel stability as compared to the classical case. The dynamic structure factor is considered for various temperatures and the spin-wave velocity and the susceptibility are presented as functions of frustration for spin S=1/2. In the case of the intrinsically frustrated triangular lattice model we obtain, in the limit of infinite spin, results in complete agreement with the linear spin-wave theory. In contrast to the latter, however, two of the three modes acquire a mass for finite spin. The origin of this effect is discussed.
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 dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional.
This thesis presents theoretical studies of strongly correlated systems as well as topologically ordered systems in 1D. Non-Fermi liquid behavior characteristic of interacting 1D electron systems is investigated with an emphasis on experimentally relevant setups and observables. The existence of end Majorana fermions in a 1D p-wave superconductor subject to periodic, incommensurate and disordered potentials is studied. The Tomonaga-Luttinger liquid (TLL), a model of interacting electrons in one spatial dimension, is considered in the context of two systems of experimental interest. First, a study of the electronic properties of single-walled armchair carbon nanotubes in the presence of transverse electric and magnetic fields is presented. As a result of their effect on the band structure and electron wave functions, fields alter the nature of the (effective) Coulomb interaction in tubes. In particular, it is found that fields couple to nanotube bands (or valleys), a quantum degree of freedom inherited from the underlying graphene lattice. As revealed by a detailed TLL calculation, it is predicted that fields induce electrons to disperse into their spin, band, and charge components. Fields also provide a means of tuning the shell-filling behavior associated with short tubes. The phenomenon of charge fractionalization is investigated in a one-dimensional ring. TLL theory predicts that momentum-resolved electrons injected into the ring will fractionalize into clockwise- and counterclockwise-moving quasiparticles. As a complement to transport measurements in quantum wires connected to leads, non-invasive measures involving the magnetic field profiles around the ring are proposed. Topological aspects of 1D p-wave superconductors are explored. The intimate connection between non-trivial topology (fermions) and spontaneous symmetry breaking (spins) in one-dimension is investigated. Building on this connection, a spin ladder system endowed with vortex degrees of freedom is proposed in order to study the effects that inhomogeneous potentials have on the topological phase diagram. Periodic vortex patterns yield a rich parameter space for tuning into a topologically non-trivial phase. This analysis hinges on the development of a topological invariant based on the wave function of Majorana fermions which inhabit the ends of the system and are robust to disorder. The method is generalized to aperiodic and disordered potentials. The topological phase diagram of such systems is studied; numerical and analytic results are found to be in close agreement.
In this thesis, I use a quantum Monte Carlo (QMC) method based on the stochastic series expansion (SSE) to study transport, quantum phase diagrams, and dynamics of several one- and quasi-one-dimensional spin systems, including spin-1/2 and spin-1 chains, a frustrated four-leg spin tube, and N-leg spin-1/2 ladders. I establish a diffusive channel for the finite temperature spin transport of the spin-1/2 antiferromagnetic Heisenberg chain. Extensive finite size studies are detailed, allowing to extract the diffusion kernel in the thermodynamic limit. The diffusion is found to exhibit no anomalous exponent, moreover it is shown to survive for finite values of exchange anisotropy and magnetic fields, however with a strong suppression away from the isotropic point and for increasing magnetic fields. In the case of the spin-1 chain, I obtain the quantum phase diagram and spin dynamics in the combined presence of finite single-ion anisotropies and external magnetic fields. I also prove the existence of magnon modes and multi-particle continua and determine their evolutions as a function of anisotropies and magnetic fields. Moreover, I calculate the magnetic field dependence of the nuclear magnetic relaxation as a function of the single ion anisotropy. For the quasi-one-dimensional spin system of a frustrated four-leg tube, I investigate the regime of large rung coupling, where QMC is applicable. I evaluate the uniform spin susceptibility and the dynamic structure factor and show that in this regime the spin-tube exhibts a crossover from a gapped effective antiferromagnetic spin-2 chain behavior with massive magnons to a gapless Luttinger liquid phase with spinon excitations. Finally, I investigate the spin dynamics of N-leg spin-1/2 ladders with spacially isotropic interactions for N=1,..,5. I obtain the low-lying excitations of the spin ladders as a function of the number of legs and show that they exhibt an even-odd oscillation which is consistent with Haldane's conjecture and other known results for the thermodynamics of spin ladders.
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.