This thesis is a theoretical analysis of sample two- and one-dimensional systems. The two-dimensional examples are the quantum Hall liquid and anomalous paired states. The most widely accepted effective theory of the quantum Hall liquid is based on the so-called Chern Simons Lagrangian, but it is not entirely satisfactory. We obtain the first derivation of an alternative effective theory from microscopic principles. Our formulation allows for a first principles derivation of physical quantities such as the effective mass and compressibility and contains the first analytical observation of the magnetoroton. The formalism developed along the way is also applied to paired states in anomalous supeconductors, a topic of much recent interest. The one-dimenisonal system is the carbon nanotube. Gas uptake in nanotube bundles is currently attracting a wealth of research with both applied and fundamental implications. We propose adsorption of gases on the surface of a single tube, finding strong correlations and symmetries that have not been observed yet. The properties of these states are directly relevant to other one-dimensional structures such as spin ladders and stripes and raise interesting and open questions.
This thesis is a theoretical analysis of sample two- and one-dimensional systems. The two-dimensional examples are the quantum Hall liquid and anomalous paired states. The most widely accepted effective theory of the quantum Hall liquid is based on the so-called Chern Simons Lagrangian, but it is not entirely satisfactory. We obtain the first derivation of an alternative effective theory from microscopic principles. Our formulation allows for a first principles derivation of physical quantities such as the effective mass and compressibility and contains the first analytical observation of the magnetoroton. The formalism developed along the way is also applied to paired states in anomalous supeconductors, a topic of much recent interest. The one-dimenisonal system is the carbon nanotube. Gas uptake in nanotube bundles is currently attracting a wealth of research with both applied and fundamental implications. We propose adsorption of gases on the surface of a single tube, finding strong correlations and symmetries that have not been observed yet. The properties of these states are directly relevant to other one-dimensional structures such as spin ladders and stripes and raise interesting and open questions.
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.
Many of the most vexing phenomena observed in condensed matter physics involve strongly correlated systems near quantum critical points in two spatial dimensions. Examples range from the sharing of critical exponents (superuniversality) among quantum Hall plateau transitions to the emergence of a charge-vortex symmetry near the field-tuned superconductor-insulator transition in thin films to the appearance of “anomalous” metallic states which evade localization. These problems elude theoretical understanding because they lack small parameters; standard perturbative approaches constitute poor, often uncontrolled approximations. This thesis concerns itself with the development of techniques reaching beyond the perturbative paradigm, focusing especially on duality, the idea that two seemingly different theories are actually one and the same. While a duality is most useful when it relates a strongly interacting theory to a weakly interacting one, even dualities between strongly coupled theories can reveal emergent symmetries and exotic gapped phases that may be obscure in one theory but not its dual. Recently, starting from a relativistic analogue of flux attachment, an entire web of quantum field theory dualities was proposed connecting a wide variety of quantum critical states in two spatial dimensions. This web of dualities has led to a surge of progress on a wide variety of condensed matter problems, several examples of which are presented in this thesis. One of the major challenges that comes with the proposal of new dualities is to “derive” them. In Chapter 2, we construct simple, explicit derivations of many members of the web of dualities in (particle-hole symmetric) models of relativistic current loops. In relativistic theories, flux attachment necessarily involves transmutation of both statistics and spin, an operation which can be made precise in the context of loop models. We further show that while non-relativistic theories are invariant under attachment of even numbers of flux quanta due to the periodicity of statistics, this symmetry is completely lost in relativistic theories due to the presence of spin, clarifying the interpretation of earlier loop models in which this symmetry appeared to be present. Motivated in part by this lack of statistical periodicity in relativistic flux attachment, in Chapter 3 we turn our attention to the metallic, “composite Fermi liquid” states occuring in quantum Hall systems at filling fraction ν = 1/2n. Famously, the state at ν = 1/2 is known to display particle-hole symmetry, which has recently led to the proposal of a manifestly particle-hole symmetric theory of relativistic, or Dirac, composite fermions that features prominently in the web of dualities. Surprisingly, however, an analogous “reflection symmetry” has also been observed in transport experiments about ν = 1/4. To explain this symmetry, we propose a series of relativistic composite fermion theories for the compressible states at ν = 1/2n, in which the reflection symmetry is incorporated as a mean field time-reversal symmetry of the composite fermions. These theories consist of electrically neutral Dirac fermions attached to 2n flux quanta via an emergent Chern-Simons gauge field. While not possessing an explicit particle-hole symmetry, these theories reproduce the known Jain sequence states proximate to ν = 1/2n, and we show that such states can be related by the observed reflection symmetry, at least at mean field level. In Chapter 4, we describe how duality can be used to access exotic gapped phases, in particular non- Abelian quantum Hall states. Using proposed non-Abelian bosonization dualities in two spatial dimensions, which morally relate U(N)k and SU(k)−N Chern-Simons-matter theories, we present pairing scenarios for which non-Abelian quantum Hall states can be obtained starting from theories of Abelian composite particles. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings ν = k/(kM +2) by starting from k layers of bosons at ν = 1/2 with M Abelian fluxes attached and then condensing k-clusters of the dual non-Abelian bosons. We further extend this constructions to obtain generalizations of the Read-Rezayi states with emergent global symmetries. In Chapter 5, we describe a context in which non-perturbative ideas from duality and perturbative results in the language of the renormalization group (RG) can inform one another: the interplay of quenched disorder and strong interaction effects near quantum critical points. In particular, we focus on the problem of quenched disorder at the superfluid-insulator transition of the O(N) model in the large-N limit. While a random mass is strongly relevant at the free fixed point, its effect is screened by the strong interactions of the Wilson-Fisher fixed point in this model, enabling a perturbative RG study of the interplay of disorder and interactions about this fixed point. In contrast to the spiralling flows obtained in earlier double-ε expansions, we show that the theory flows directly to a quantum critical point characterized by finite disorder and interactions, with critical exponents in remarkable agreement with numerical studies of the superfluid-Mott glass transition. With these results in hand, we apply duality to discuss the possible implications of this result for the dual Abelian Higgs and Chern-Simons-Dirac fermion theories when N = 1.
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.
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 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.
A phenomenon can hardly be found that accompanied physical paradigms and theoretical concepts in a more reflecting way than magnetism. From the beginnings of metaphysics and the first classical approaches to magnetic poles and streamlines of the field, it has inspired modern physics on its way to the classical field description of electrodynamics, and further to the quantum mechanical description of internal degrees of freedom of elementary particles. Meanwhile, magnetic manifestations have posed and still do pose complex and often controversially debated questions. This regards so various and utterly distinct topics as quantum spin systems and the grand unification theory. This may be foremost caused by the fact that all of these effects are based on correlated structures, which are induced by the interplay of dynamics and elementary interactions. It is strongly correlated systems that certainly represent one of the most fascinating and universal fields of research. In particular, low dimensional systems are in the focus of interest, as they reveal strongly pronounced correlations of counterintuitive nature. As regards this framework, the quantum Hall effect must be seen as one of the most intriguing and complex problems of modern solid state physics. Even after two decades and the same number of Nobel prizes, it still keeps researchers of nearly all fields of physics occupied. In spite of seminal progress, its inherent correlated order still lacks understanding on a microscopic level. Despite this, it is obvious that the phenomenon is thoroughly fundamental of nature. To resolve some puzzles of this nature is a key topic of this thesis. (excerpt from abstract)