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UBC Theses and Dissertations

The influence of interference on the Kondo effect Malecki, Justin 2010

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The Influence of Interference on the Kondo Effect by Justin Malecki B.Sc.(Hons), University of Waterloo, 2004  a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in  the faculty of science (Physics)  The University of British Columbia (Vancouver) November 2010 c Justin Malecki, 2010  Abstract The Kondo effect, wherein a local magnetic moment is screened via interactions with a continuum of quantum excitations, occurs in quantum dots with an odd number of electrons. By placing a quantum dot in an Aharanov-Bohm interferometer, one is able to probe the effects of electron interference on the manifestation of the Kondo effect. In this thesis, we present a theoretical study of the Kondo effect in a model system of a quantum dot embedded in an Aharanov-Bohm interferometer connected to two conducting leads. By transforming to the scattering basis of the direct inter-lead tunneling, we are able to describe precisely how the Kondo screening of the dot spin occurs. We calculate the Kondo temperature and zero-temperature conductance and find that both are influenced by the Aharanov-Bohm interferometer as well as the electron density in the leads. We also calculate the form of an additional potential scattering term that arises at low energies due to the breaking of particle-hole symmetry. In addition to these analytic results, a numerical renormalization group analysis of the system is presented. We fully describe the influence of the Aharanov-Bohm interferometer on the renormalization group flow of the quantum dot model and obtain strong support for the derived form of the Kondo temperature. A method for extracting the phase shifts of the strongcoupling fixed point from the numerical data is described. These phase shifts are compared with those derived analytically, providing further support for our conclusions.  ii  Preface Much of the content of all of the chapters and appendices (with the exception of Appendices C and D) of this thesis is contained in the following manuscript: J. Malecki and I. Affleck, “The Influence of Interference on the Kondo Effect” Physical Review B 82, 165426 (2010). Significant additions have been made to chapter 1 to provide a more comprehensive introduction to the subject, to § 3.1 to provide more details on the derivation of the numerical renormalization group, as well as less significant additions to the remaining chapters to provide a more detailed description of the methods and results. These additions do not appear in the above publication and were written solely by the author. The first author of the manuscript referenced above (JM) performed the majority of the research that is described in the manuscript as well as performing the majority of the work in producing the manuscript itself. The conception, scope, and method of the research was developed collectively by both JM and the second author (IA). All of the analytical and numerical calculations described in the manuscript were conducted by JM. The manuscript was written entirely by JM but heavily influenced by significant consultation with IA. In particular, parts of the second last paragraph of section 1.3 were derived from content written by IA in an earlier draft.  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1 Historical Development of the Kondo Effect . . . . . . . . . . 1.2 The Kondo Effect in Quantum Dots . . . . . . . . . . . . . . .  2 4  1.3 Quantum Dots in an Aharanov-Bohm Interferometer . . . . . 9 1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Model & Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1 Theoretical Model & Transformations . . . . . . . . . . . . . . 16 2.2 Additional Potential Scattering . . . . . . . . . . . . . . . . . 23 2.2.1 Asymmetric dot εd 6= −U/2 . . . . . . . . . . . . . . . 23 2.2.2  Symmetric dot εd = −U/2 . . . . . . . . . . . . . . . . 24 iv  2.3 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.1 2.3.2  Kondo Temperature . . . . . . . . . . . . . . . . . . . 28 S-Matrix and Conductance . . . . . . . . . . . . . . . . 31  3 Numerical Renormalization Group Analysis . . . . . . . . . 41 3.1 Derivation of the Numerical Renormalization Group . . . . . . 41 3.2 Renormalization Group Flow & Kondo Temperature . . . . . 47 3.2.1 Fixed points of the single-channel Anderson impurity 3.2.2 3.2.3  model . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Fixed points of the Aharanov-Bohm quantum dot model 48 Kondo Temperature from the NRG . . . . . . . . . . . 58  3.3 Phase Shifts and VR . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.1 Phase shifts from the NRG . . . . . . . . . . . . . . . . 63 3.3.2  NRG evidence for VR . . . . . . . . . . . . . . . . . . . 69  4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A Details of the transformation to the scattering basis . . . . 88 B Potential scattering phase shift . . . . . . . . . . . . . . . . . 91 B.1 Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 B.2 Continuum Model . . . . . . . . . . . . . . . . . . . . . . . . . 92 C Implementation of the Numerical Renormalization Group . 94 D Clebsch-Gordon Coefficients . . . . . . . . . . . . . . . . . . . 102 D.1 S ⊗ 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 D.2 S ⊗ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104  v  List of Tables Table 3.1 Summary of fixed points for the single-channel Anderson impurity model. . . . . . . . . . . . . . . . . . . . . . . . . 47 Table 3.2 The lowest energies and associated total charge Q and total spin S quantum numbers of the free orbital (FO) NRG fixed point of the Aharanov-Bohm interferometer (ABI) model for odd N. The numerical values for the single-particle excitation energies were obtained by diagonalizing the Hamiltonian of eq. (3.20) using a value of Ṽp = 3.0 and Λ = 2.5. All energies within a particular box are equal by eq. (3.23).  52  Table 3.3 The lowest energies and associated total charge Q and total spin S quantum numbers of the local moment (LM) NRG fixed point of the ABI model for N odd. The single-particle energy levels are the same as in Table 3.2 using Ṽp = 3.0 and Λ = 2.5. All energies within a particular box are equal by eq. (3.23). . . . . . . . . . . . . . . . . . . . . . . . . . 53 Table 3.4 The lowest energies and associated total charge Q and total spin S quantum numbers of the strong coupling (SC) NRG fixed point of the ABI model for odd N. The NRG parameters used are Ṽp = 3.0 and ϕ = 1.047. The same parameters were used to determine the energy levels of HN,SC where a value of Ṽp′ = 2.885 was found to reproduce the NRG data.  . . . . . . . . . . . . . . . . . . . . . . . . . . 55  vi  Table 3.5 The parameters for the best fit of eqs. (3.45) and (3.46) to the U/D = 0.001 data in Figure 3.10. . . . . . . . . . . . . 73 Table C.1 Definition of the states |iiN +1 with their associated values of (Q, S, S z ). . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Table C.2 Definition of the states |Q, S; ω, iiN +1 for i = 1 . . . 8 from which the coefficients aij (S) and index functions nij , Qi , Si and Sijz of eq. (C.3) can be inferred. . . . . . . . . . . . . . 98 Table C.3 Definition of the states |Q, S; ω, iiN +1 for i = 9 . . . 16. . . . . 99  vii  List of Figures Figure 1.1  A schematic representation of a semiconductor quantum dot in real space (top) and its associated energy profile (bottom) horizontally aligned with the top diagram. The quantum dot is labeled by QD and the remaining labels are  Figure 1.2  described in the text. The curve surrounding the energy level εd closest to the lead chemical potentials indicates √ the broadening of the level by ΓL + ΓR . . . . . . . . . . The four energy states of a single-level quantum dot with local Coulomb energy U > 0 and on-site energy εd = −U/2. In this parameter regime, a local moment with two degenerate spin states is energetically favored. . . . . .  5  8  Figure 1.3  Schematic diagrams of the two categories of theoretical models discussed in the text. Dotted lines indicate quantum tunnel junctions. . . . . . . . . . . . . . . . . . . . . . 12  Figure 2.1  The lattice model described by the Hamiltonian of eq. (2.1). 17  Figure 2.2  The second order Feynman diagrams that contribute to VR . The solid line labelled by k, µ represents a single particle excitation of Ψscr kµ while the dashed line represents the quantum dot spin degree of freedom. . . . . . . . . . . . . 25  viii  Figure 2.3  The flux dependence of the Kondo temperature for a value  Figure 2.4  of τ ′ = 0.4, νJ = 0.3, and γ = 1. Here we see an increase in the flux dependence as the electron density is lowered. . 29 The τ ′ dependence of the Kondo temperature for a value of kF = π/(6a), νJ = 0.3, and γ = 1. This exhibits the variety of behaviour that can be seen for different values of  Figure 2.5  the flux and that the Kondo temperature always vanishes as τ ′ → ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . 30 The kF dependence of the Kondo temperature for a value of τ ′ = 0.4, νJ = 0.3, γ = 1, and various values of the magnetic flux. . . . . . . . . . . . . . . . . . . . . . . . . 31  Figure 2.6  Figure 2.7  The conductance as a function of dot level εd for various values of the direct interlead transmission probability given by Tb = sin2 2δ. Here, we assume the particle-hole symmetric value of half-filling, kF = π/(2a) and γ = 1. . . 36 The conductance plotted as a function of magnetic flux for νJ = 0.287, kF = π/2a, εd = −U/2 and γ = 1. Each of the different coloured lines indicates a different value of the inter-lead coupling τ ′ . The solid lines are the prediction with VR = 0 (equivalently δR = 0) with the dotted lines showing the finite VR correction. . . . . . . . . . . . . . . 38  Figure 2.8  The conductance plotted as a function of inter-lead coupling τ ′ when kF = π/(2a) and εd = −U/2. Here, νJ = 0.287 and ϕ = 0. . . . . . . . . . . . . . . . . . . . . . . . 39  Figure 2.9  The conductance plotted as a function of kF for τ ′ = 0.4, γ = 1 and for multiple values of the magnetic flux. We do not consider the small correction due to VR so that the conductance is given by eq. (2.75). . . . . . . . . . . . . . 40  ix  Figure 3.1  The lowest energy levels with quantum numbers Q = 1, S = 0 as produced by the NRG as a function of odd N. The values of the predicted fixed point energies from Tables 3.2 and 3.3 are indicated by arrows on the left side and energies from Table 3.4 are indicated on the right. The parameters used to generate this plot are Γ/D = 0.0003142, U/D = 0.001, νVp = 1.05, and ϕ = 1.047. Here we see the unstable FO fixed point is approached for 5 < N < 10, the unstable LM fixed point for 19 < N < 33, and the stable  Figure 3.2  SC fixed point for N > 60. . . . . . . . . . . . . . . . . . . 57 The lowest non-zero energy level with quantum numbers Q = 1, S = 0 as produced by the NRG as a function of odd N. The different lines correspond to different values of the flux ϕ. The parameters used to generate this plot are the same as in Figure 3.1. The value of NK , related to the Kondo temperature via eq. (3.33), is indicated by the  Figure 3.3  arrow and is the same for all values of ϕ. . . . . . . . . . 58 The lowest NRG energy levels of the final stable fixed point (circles) are plotted as a function of flux ϕ and compared with those given by the fixed point Hamiltonian of eq. (3.28) (solid lines). The parameters used to generate this plot are the same as in Figure 3.1 except here we use a value of νVp = 0.525. The single value of Ṽp′ was tuned in order to fit the NRG fixed point energy levels for ϕ = 0. This single parameter is able to reproduce the predicted flux dependence. . . . . . . . . . . . . . . . . . . . . . . . 59  x  Figure 3.4  Select energy levels with quantum numbers Q = 1, S = 0 as produced by the NRG as a function of odd N. The different lines correspond to different values of the inter-lead coupling Vp and the arrows of the same line type indicate the value of NK at which the Kondo crossover takes place for each case. The parameters used to generate this plot  Figure 3.5  are the same as in Figure 3.1. . . . . . . . . . . . . . . . . 61 A plot of TK as extracted from the NRG (symbols) as a function of the input value of νVp . The different symbols describe data with different input values of νJ giving rise to different effective Kondo couplings. The solid line indi-  Figure 3.6  cates the prediction described in eq. (3.32). . . . . . . . . . 62 Energy level diagram of the single-particle NRG energy levels of the two channels. The shift of each relative to the Fermi energy (here indicated by the dotted line) defines the phase shift in each channel. . . . . . . . . . . . . . . . 65  Figure 3.7  The phase shift of the two channels as a function of Vp from the NRG data with an effective Kondo coupling νJ = 0.255 and zero flux. In this case, the two channels are independent with the screening channel phase shift starting at π/2 for νVp = 0 and the non-screening channel phase shift starting at 0 for νVp = 0. The symbols are the phase shifts obtained from the full NRG many-body energy levels. The ascending solid lines are the phase shifts obtained from the single-particle energy levels of a single non-interacting Wilson chain with potential scattering Vp and the descending solid lines are π/2 minus the ascending lines. The solid lines do not take into account the small correction due to the additional potential scattering VR that occurs in the screening channel. . . . . . . . . . . . . . . . . . . . . . . . 67  xi  Figure 3.8  The phase shifts of the two channels as a function of Vp . The symbols denote phase shifts derived from the NRG data while the lines are the analytic predictions. The solid black line is the curve expected if there is no additional potential scattering (i.e. VR = 0). We have set ϕ = 0 to generate this plot. . . . . . . . . . . . . . . . . . . . . . . . 70  Figure 3.9  The additional potential scattering VR as derived from the NRG phase shifts (symbols) and from the analytic tightbinding model (lines) for various values of the effective  Kondo coupling J. We have set ϕ = 0 to generate this plot. 71 Figure 3.10 The additional potential scattering as determined from the NRG phase shifts as a function of effective Kondo coupling J. Each different symbol uses a fixed value of the dot Coulomb repulsion U while varying Γ such that the range of J, given in eq. (3.31), is roughly the same for each iteration. The solid line presents the best fit third degree polynomial to the U/D = 0.001 points. In this data, νVp = 0.3 and ϕ = 0. . . . . . . . . . . . . . . . . . 72 Figure 3.11 The phase shifts of the strong-coupling fixed point as determined from the NRG (symbols) and compared with that predicted in eq. (3.36) (lines). Here, the effective Kondo coupling is νJ = 0.191 and νVp = 0.25. . . . . . . 73 Figure 3.12 The value of the additional potential scattering VR as derived from the NRG for various values of Λ, all using a value of νJ = 0.254. . . . . . . . . . . . . . . . . . . . . . 74  xii  Figure 4.1  A schematic representation of the geometry for a proposed experiment closely related to the model studied. The shaded areas are the metallic gates on top of a two dimensional electron gas with the quantum dot indicated by QD. The horizontal pathway acts as a quantum point contact, allowing only a single channel of electrons. The gate Vg tunes the energy levels of the dot whereas Vp creates a potential barrier in the middle of the direct horizontal channel through which the electrons can tunnel. . . . . . . 79  xiii  Glossary RG  renormalization group  NRG numerical renormalization group 2DEG two dimensional electron gas ABI  Aharanov-Bohm interferometer  FO  free orbital  LM  local moment  SC  strong coupling  xiv  Acknowledgements The research and results described in this thesis are the culmination of many years of hard work. During this time, I have depended on numerous others for assistance and guidance, often without expressing the gratitude they deserve. This oversight is an error that I will formally remedy right here. I will always be grateful to my parents, Marysia and John Malecki, who never failed to offer encouragement and support despite my failure to explain to them exactly what it was I was doing at school. I am equally grateful to my sister Kristen and brothers Jeffrey and Brian for the refuge of our unique brand of Maleckian camaraderie on which I perpetually rely. For their criticism of my research and guidance throughout my graduate career, I would like to thank my supervisory committee of Ian Affleck, Mona Berciu, Joshua Folk, and Jörg Rottler. In particular, I thank Ian for the model of scientific rigour that he embodies and so tirelessly defends. Though I came to him, like so many young graduate students, wishing to weave grand, novel tales of the nature of the universe, Ian taught me that scientific truth rarely comes by projecting ones imagination into the unknown but through careful and measured analysis of the known. The ideas of those with whom I have shared the thrill of mutual discovery populate these pages. For their past and continued inspiration and guidance, I thank Eran Sela, Lee Smolin, Stephon Alexander, Robert Mann, and Mike Hudson. Thanks also to Howard Burton on whose keen insight I relied (but not always followed) at key moments in my educational career.  xv  It would have been much more difficult to survive while conducting my research were it not for the aid provided by the Natural Sciences and Engineering Research Council of Canada, the Government of British Columbia, the Physics and Astronomy Department of the University of British Columbia, and the Canadian Institute for Advanced Research. I am grateful to these organizations and the taxpayers, students, and investors that support them for their generosity without any assurance of a return on their investment. Of my fellow students and researchers who I have come to know as colleagues and friends, I could write at length how each provided their own unique brand of fellowship and support. Alas, a list of their names must suffice with apologies to those inadvertently forgotten: Conan Weeks, Glen Goodvin, Lara Thompson, Mya Warren, Dominic Marchand, Adrian del Maestro, Ion Garate, Mohammadsadegh Mashayekhi, Hamed Karimi, Rodrigo Pereira, Ming-Shyang Chang, Jesko Sirker, Nicolas Laflorencie, Amy Liu, Mike McDermott, Anand Thirumulai, Ramandeep Gill, Martha Milkeraitis, Tom Davis, Kristin Woodley, Alejandro Gaita and Silke Weinfurtner. For helping provide inspiration, I would like to thank Sean Cronin, The Drive Street Band, the PHAS and the Furious, Glenn Gould, Don Ellis, Brad Mehldau, Dave Holland, The Acorn, Tortoise, John Zorn, Tom Waits, François Houle, Joëlle Léandre, Bruno Hubert, Jónsi, Nico Muhly, Bramwell Tovey and the Vancouver Symphony Orchestra, David Pay, Chris, P.D.’s Hot Shop, Landyachtz, the unofficial guru I met in Tatlow park, and Newton. Finally, anything I write about my partner Sarah Dawn Etherington is bound to fail, in the most absurd fashion, to encompass the debt of gratitude I owe her. We traveled together to an unknown land so that we may explore our respective realms of the mind. Though I often felt disoriented in my own exploration, Sarah Dawn’s clear vision of what is truly important was a beacon so that I did not become lost. As we continue to travel together into the unknown future, I can only look back and, with the greatest humility, say thank you.  xvi  Chapter 1 Introduction Within the academic field of condensed matter physics, the Kondo effect [1, 2] has become an important paradigm in the study of strongly-correlated manybody quantum phenomena. The Kondo effect is invoked to explain seemingly disparate observations such as anomalous resistivity in metals at low temperature [3, 4], electronic transport through quantum dots [5, 6, 7, 8], and electrons scattering off magnetic impurities on the surface of metals [9, 10] to name just a few. Furthermore, models giving rise to the Kondo effect involving one or a few impurities can provide a simple yet non-trivial framework that illuminates some of the key aspects of less tractable models such as the Hubbard model [11, 12] or Kondo lattice models [2] of heavy Fermion materials [13]. This thesis describes research that provides new insights into the rich phenomenology of the Kondo effect. We begin with an introduction and brief historical review of the Kondo effect then proceed to describe in more detail the manifestation of the Kondo effect in quantum dots and, more specifically, quantum dots in Aharanov-Bohm interferometers. This is followed by a more detailed overview of the remainder of the thesis.  1  1.1  Historical Development of the Kondo Effect  The study of magnetic impurities in metals began with the observation that the electrical resistance achieves a minimum value at finite temperature T in many metallic samples, the first such observation being made in samples of gold [3]. The reason for this anomalous resistance remained a mystery until Jun Kondo’s seminal paper [4] which was motivated by observed correlations between the character of the resistance minimum and the concentration of 3d transition metal impurities in the metallic sample [14]. Earlier, Anderson modeled such impurities by a local resonant energy level εd and localized Coulomb repulsion U interacting with the conduction electrons (which later came to be known as the “Anderson model”) and showed that, for particular ranges of the model parameters, such impurities give rise to local magnetic moments [15]. As such, Kondo calculated the contribution to the resistance by the presence of dilute magnetic impurities interacting antiferromagnetically with the conduction electrons with strength J [4]. In particular, Kondo showed that the term third order in J behaves as ln T which, when taken together with the phonon contribution to the resistivity which goes as T 5 , gives rise to a minimum value of the resistance at finite T . While providing a satisfactory explanation of the resistance minimum, the presence of the ln T term, which diverges as T → 0, presented a new problem that came to be known as the “Kondo problem”. The solution of the Kondo problem began with Anderson’s introduction  of an approximate scaling analysis [16] showing that the magnitude of the antiferromagnetic interaction J increases as high energy modes are eliminated to produce an effective model applicable at low temperatures. This scaling analysis is based on perturbation theory in J and so must necessarily become invalid at a temperature scale TK , known as the Kondo temperature, where 2  J becomes so large that higher order terms are larger than those calculated. Nevertheless, such a scaling analysis led to the hypothesis that J continues to increase smoothly for temperatures T < TK , eventually diverging as T → 0 where the magnetic moment of the impurity is fully screened by forming a singlet with a conduction electron. That is, even if J is initially small at high temperatures, there exists a crossover at scale T ≈ TK to a low-energy  regime where J becomes very strong and the electrons screen the impurity. This hypothesis was confirmed by the introduction of a more sophisticated scaling analysis known as the renormalization group (RG). In particular, it was Wilson’s application of his numerical renormalization group (NRG) to the Kondo problem that provided the necessary non-perturbational implementation of the RG to prove the complete screening of the magnetic impurity at T = 0 [17]. Furthermore, thermodynamic properties of the system such as the magnetic susceptibility χ and the specific heat c were calculated using the NRG and found to be significantly influenced at low temperatures by the presence of a magnetic impurity. Most famously, it was shown that the ratio χ/c, later known as the Wilson ratio, is enhanced by a factor of 2 at T = 0 compared to non-interacting electrons. The NRG was later applied to the Anderson model, showing that such magnetic screening also occurs in this more general scenario [18, 19]. This screening of magnetic impurities soon came to be known as the “Kondo effect”.  These compelling numerical results found an elegant analytic description in Nozières Fermi liquid theory [20, 21] applicable for temperatures T ≪ TK . Applying Landau’s general Fermi liquid theory specifically to magnetic impurity models, Noziéres assumes complete screening of the impurity at T = 0 and that excitations at temperatures T ≪ TK are given by quasiparticles that are in one-to-one correspondence with the Fermionic excitations of the highenergy model. The strong binding of a conduction electron to the impurity in a singlet gives rise to spin-independent scattering of the remaining conduction electrons. As this scattering is localized to the site of the impurity,  3  the effect of the screened impurity is to induce a phase shift in the conduction electrons scattering off the impurity, the value of which becomes π/2 at T = 0 [20, 22]. For non-zero T ≪ TK , J is still very large and so disallows real processes that flip the impurity spin however virtual spin-flip processes can induce effective interactions amongst the conduction electrons in the vicinity of the impurity. The precise strength of these effective interactions can be obtained by comparison with Wilson’s NRG and treated perturbatively in the calculation of physical quantities in the regime T ≪ TK . In this way, a full analytic description of the low-energy dynamics of the system is obtained. Although the Kondo problem can be thought to be solved with the description given above, theoretical understanding of the Kondo effect continued to develop. Of note, the Bethe ansatz technique was applied to such impurity models, allowing (at least in principle) the exact analytic calculation of thermodynamic properties over all temperature ranges [23, 24, 25]. Later, a mapping of the low-energy model onto a conformal field theory allowed the application of new exact calculational techniques to such quantum impurity models [26, 27]. This approach proved especially fruitful in that it can readily be applied to more complicated models, involving multiple conduction channels [28, 29] or multiple impurities [30] where non-Fermi liquid behaviour can occur.  1.2  The Kondo Effect in Quantum Dots  Interest in this field of study resurged with the observation of the Kondo effect in semiconductor quantum dot devices [5, 6]. An example of such a device is represented in Figure 1.1. To construct a semiconductor quantum dot, one starts with a heterostructure composed of layers of semiconductors of different chemical compositions (typically GaAs and AlGaAs) such that a two dimensional electron gas (2DEG) is formed at the interface of two such layers. Metallic electrodes are then laid on top of the heterostructure and 4  Vg Lead L  ΓL  QD  ΓR  Lead R  ∆ε µL  µR εd  E  x Figure 1.1: A schematic representation of a semiconductor quantum dot in real space (top) and its associated energy profile (bottom) horizontally aligned with the top diagram. The quantum dot is labeled by QD and the remaining labels are described in the text. The curve surrounding the energy level εd closest √ to the lead chemical potentials indicates the broadening of the level by ΓL + ΓR . a voltage bias applied which depletes the concentration of electrons in the 2DEG in the vicinity of the electrodes. In this way, one can define regions in which the electrons are allowed to travel as well as the rate of tunneling between such regions based on the geometry of the electrodes and the voltage applied. A quantum dot is constructed by laying the gates in such a way as to produce a small region of confined electrons, typically of the order of 100nm. The potential that confines electrons to the dot is typically manipulated by a 5  single voltage applied to one electrode on the edge of the dot which we shall call the gate voltage Vg . This effectively changes the energy of the small droplet of electrons relative to the surrounding 2DEG. Two additional energies, ΓL and ΓR , describing the coupling of electronic states on the droplet to the 2DEG on the left and right sides of the dot respectively are controlled by a voltage applied to two additional electrodes that mediate tunneling onto and off of the dot. Source and drain contacts are connected to the heterostructure, one on each side of the quantum dot, so that a voltage bias Vds = (µL − µR )/e can be applied and the current I through the quantum dot can be measured. Here, µL and µR are the chemical potentials in the left and right leads respectively.  As with any quantum particle, confining electrons to a small region gives rise to a quantization of the energy states that the electrons can occupy. The precise spectrum will depend on details of the geometry of the dot but the level spacing will have a characteristic scale that we call ∆ε. Tunneling √ between the dot and 2DEG broadens each energy level by ΓL + ΓR . Hence, quantization of both the energy spectrum and particle number N of the dot can be maintained approximately so long as ΓL , ΓR ≪ ∆ε, that is, so long as the quantum dot is sufficiently isolated from the surrounding 2DEG. An electron can tunnel onto the dot from the surrounding 2DEG, increasing the particle number to N + 1 and increasing the energy by U if N is odd or U + ∆ε if N is even. The factor U is due to the Coulomb repulsion with the other electrons on the dot whereas the ∆ε factor in the latter case is due to the fact that, when N is even, the highest occupied state on the dot will be doubly occupied so that the tunneling electron must occupy the next highest state. As a first approximation to estimate U, the region of electrons on the dot can be described by a capacitance C so that U is the charging energy e2 /(2C) [31]. By decreasing the value of the gate voltage Vg , one can shift the quantized electronic spectrum of the dot down relative to the chemical potentials  6  in the leads1 . Once the spectrum has been shifted down sufficiently far so as to overcome the Coulomb repulsion U (or U +∆ε if N is even), it becomes energetically neutral for tunneling to occur and the number of electrons on the dot is able to fluctuate from N to N + 1 and back. If one then applies a small voltage bias Vds at temperatures larger than the Kondo temperature, one will measure a non-zero current only for values of Vg close to where this number fluctuation can occur [32, 33]. This phenomena is called Coulomb blockade and allows the quantum dot to act as a single-electron transistor [31]. Let us now assume that the system is at temperatures T ≪ ∆ε so that one only needs consider the energy level εd closest to the chemical potentials of the leads since thermal excitation to the other energy levels will be suppressed.  Furthermore, let us assume that Vg is tuned such that there is an odd number of electrons on the quantum dot. In this case, the ground state of the dot will have a single electron of either spin up or spin down occupying the energy level εd . Subsequent small variations of Vg can tune the value of εd to the desired parameter range. Combined with the Coulomb repulsion U, this is precisely the Anderson model discussed above which was shown to give rise to local magnetic states when εd ≈ −U/2 as demonstrated in Figure 1.2. As discussed above, interactions between a local moment and a continuum of electrons such as those in the leads gives rise to the Kondo effect. More specifically, although real fluctuations of the number of electrons on the dot are highly suppressed in this parameter regime due to U, virtual processes can occur quantum mechanically whereby an electron tunnels onto then off of the dot in a time less than ~/U. Such processes are capable of flipping the spin of the quantum dot and, taken together, give rise to an effective antiferromagnetic exchange interaction J ∝ (ΓL +ΓR )/(νU) between the spin of the dot and the spin of the electrons in the leads [34] where ν is the density of states of the leads evaluated at the Fermi energy. In this way, 1  Herein, we shall use “lead” to refer to the continuum of states present collectively in the contact and 2DEG on either the right or the left side of the quantum dot.  7  E  0  − U2  0  2εd + U  εd  εd  Figure 1.2: The four energy states of a single-level quantum dot with local Coulomb energy U > 0 and on-site energy εd = −U/2. In this parameter regime, a local moment with two degenerate spin states is energetically favored. just as for a magnetic atom embedded in a bulk metal, the effective spin of the dot is screened by the electrons in the leads for temperatures T < TK . The numerous virtual transitions that lead to this screening give rise to a many-body resonance [35, 36, 37], often called the Kondo resonance, that is peaked at the Fermi energy and has a width TK (assuming µL ≈ µR ). This does not occur for even N as there will be no net spin of the quantum dot in the ground state. However, the physical manifestation of the Kondo effect is markedly different in quantum dots than in bulk metals. Whereas one observes an increase in the resistivity of bulk samples as T is lowered below TK , one observes an increase in the conductance through a quantum dot, eventually approaching the maximum conductance of 2e2 /h as T → 0 (when ΓL = ΓR ). These two apparently opposite phenomena can be understood by considering the channels for conduction in each case. For bulk samples, the development of 8  the Kondo resonance scatters electrons from one momentum state to another near the Fermi energy leading to a rise in the resistivity. For the quantum dot, all conduction channels must pass through the dot and the Kondo resonance helps to mix the states at the Fermi energy between the two leads, enhancing the conductance [38]. This behaviour in quantum dots, mediated by the Kondo effect, was first predicted theoretically in 1988 [7, 8], later extended to non-equilibrium situations [39, 40], and subsequently observed in 1998 [5, 6]. For a quantitative review of the theory of low-temperature transport through quantum dots focussing on the Kondo effect, see ref. [41].  1.3  Quantum Dots in an Aharanov-Bohm Interferometer  A significant step in the understanding of the Kondo effect in quantum dots came with the fabrication of a quantum dot embedded in an Aharanov-Bohm interferometer (ABI) [42]. In such ABI devices, there are two paths or “arms” connecting the left and right leads and a quantum dot is fabricated in only one of these arms. For ABIs that are smaller than the coherence length of the 2DEG, the multiply-connected geometry allows electrons to interfere by taking two separate paths, similar to the classic double-slit experiment. By immersing the device in a magnetic field B perpendicular to the ABI, a magnetic flux Φ is generated in the region between the two arms. This flux gives rise to an Aharanov-Bohm phase [43] ϕ = 2πΦ/Φ0 in the wave function of an electron that has encircled the device where Φ0 = h/e is the magnetic flux quantum. In this way, the relative phase of electrons traversing the two arms of the ABI can be continuously varied by varying the magnetic field giving rise to interference induced periodic oscillations of the measured conductance through the device. The primary focus of most of the experimental investigations of a quantum dot in an ABI has been on directly measuring the transmission phase 9  that an electron accumulates on traversing the quantum dot. These studies can be placed in one of two categories: those studying so-called “closed” ABIs [42, 44, 45, 46, 47, 48] wherein the two leads are the only way in which an electron can enter or leave the ABI, and those studying “open” ABIs [49, 50, 51, 52, 53, 54] where at least one additional lead is placed in each of the two arms. In closed geometries, the Onsager relation [55, 56, 57] for the conductance [58] G reads G(B) = G(−B). Since the magnetic field B enters the expression for the conductance through ϕ only via terms cos(ϕ + β) [59] where β is the relative phase difference between the upper and lower arm, the Onsager relation restricts one to measure only β = 0, π. This “phase rigidity” occurs due to multiple traversals of the ABI so that the intrinsic transmission phase of the quantum dot is obscured.2 By grounding the additional leads in an open geometry, backscattered electrons are collected and effectively removed from the device so that open ABIs function like a double-slit experiment which allows for a continuum of relative phases to be measured. The results from these experimental studies were often surprising and generated a lot of speculation. After much theoretical analysis and debate [62, 63, 64, 65, 66, 67] (see [68] for a review of earlier papers), the following consensus has emerged regarding the transmission phase shift of a quantum dot based on analysis of the Anderson model3 generalized to ABI geometries. In the Coulomb blockade regime, T ≫ TK , the transmission phase increases by π across each Coulomb blockade peak with a sudden phase lapse of −π in each conductance valley [62, 68, 70]. For T ≪ TK , the phase climbs 2  However, deviations from this phase rigidity in closed ABIs have been reported [48, 60]. The authors of these papers propose that the presence of multiple conduction channels in the ABI allow for a continuum of phases β without violating the Onsager relation because each conduction path will have a different area and therefore a different flux ϕ. Theoretical models based on this idea are able to explain the data so that phase rigidity may hold strictly only for one-dimensional ABIs. Other methods [61] have also been proposed for how to measure the transmission phase shift in closed geometries. 3 Though some attempt to introduce more realistic models for the quantum dot based on a specific experimental configuration [69].  10  by only π/2 across the Coulomb blockade peaks surrounding a region of odd electrons within which the phase remains relatively constant [63], realizing the aforementioned π/2 phase shift indicative of the Kondo effect. Deviations from the predictions of [63] observed in [51, 52] are attributed to effects that go beyond the single-level Anderson model, requiring the inclusion of multiple energy levels on the quantum dot [68]. However, in experimental systems with only one electron on the quantum dot [54], the energy level spacing is large enough to realize the prediction of [63] based on the singlelevel Anderson model. Although experimental efforts on ABIs have almost solely focussed on measuring the transmission phase shift of a quantum dot (with the exception of [45, 46, 48] which tune and measure the Fano line shape [71], and [50, 47] which discuss decoherence), theoretical studies have investigated a much larger variety of phenomena arising from quantum dots in ABIs. All of these theoretical papers model the quantum dot as an Anderson impurity and model the ABI as either a small “Fano-Kondo” device where electrons tunnel directly between leads in one arm and directly onto and off of the quantum dot in the other [72, 73, 74, 75, 76, 77, 78, 79] or a larger device where at least one of the arms have a finite length [63, 80, 81, 82, 83, 84, 85, 86]. These two scenarios are depicted in Figure 1.3. For the smaller Fano-Kondo devices, the conductance [72, 73] and thermopower [75] was found to exhibit an asymmetric Fano-like dependence [71] on the energy level of the quantum dot. For both large and small ABIs, when the quantum dot is tuned to the Kondo regime that favors a local moment, a flux dependent Kondo temperature has been proposed using different methods [79, 83, 84, 85, 86] and the high and low-temperature conductance has been described [63, 73, 76, 78, 79, 80, 81, 83, 84, 86]. The majority of these calculations relate the conductance to the non-equilibirum Keldysh Green’s < A functions GR LR (retarded), GLR (advanced), and GLR (lesser) for electrons in the left and right leads. These Green’s functions are then related to exactly  11  Lead L  Φ  Lead R  Fano-Kondo Geometry  Lead L  Φ  Lead R  Extended Ring Geometry Figure 1.3: Schematic diagrams of the two categories of theoretical models discussed in the text. Dotted lines indicate quantum tunnel junctions. known non-interacting Green’s functions for operators in the ring and leads as well as the interacting retarded dot Green’s function GR dd which must be calculated either analytically (often in some approximation) or numerically (see [87, 88, 89] for details on this general procedure). Other studies consider more theoretical considerations, such as showing the equivalence of the “scattering” versus “tunneling” picture [77] or addressing questions regarding the coherence of transmission through a quantum dot [74, 76, 82]. While some NRG work was reported, this only studied the electron occupancy of the quantum dot [73] or the density of states on the quantum dot [63], both of which can be approximately related to the conductance. It should be noted that most of the theoretical papers discussed here assume a particle-hole symmetric dispersion relation and Fermi energy in the leads.  12  1.4  Overview  In this thesis, we reexamine the Fano-Kondo device using a combination of analytic and NRG methods. We only consider the Kondo regime where a local moment is favored on the quantum dot. We are able to reproduce many of the published results cited above as well as predicting, for the first time, non-trivial dependence of the Kondo temperature and zero-temperature conductance on the electron density in the leads. Such a dependence on electron density has not been investigated before given that a particle-hole symmetric Fermi energy has always been assumed in the leads. We calculate the generation of additional potential scattering terms that have often been neglected in previous studies but which do lead to small corrections to the zero-temperature conductance. Numerical confirmation of many of our results is provided for the first time using the NRG. Our analytic approach begins in chapter 2 with a tight-binding version of the Anderson model together with a direct tunneling term between the two leads and factors representing magnetic flux between the two conducting paths. Following [79, 90, 91, 92], we then perform an exact transformation to the “scattering basis” which diagonalizes the direct tunelling part of the Hamiltonian when the hybridization to the Anderson impurity is turned off. This gives a Hamiltonian containing no tunneling term directly between the leads, only the hybridization to the impurity, which assumes a more complicated dependence on flux, inter-lead tunneling, and particle momentum. The initial Hamiltonian contains two scattering channels, the even and odd combinations of the left and right lead operators, for example. After transforming to the scattering basis, only one linear combination of these appears in the Anderson hybridization; we refer to it as the “screening channel”. As we are primarily interested in the Kondo regime of the quantum dot, we perform a Schrieffer-Wolff transformation in the screening channel basis to obtain an effective Kondo model with an additional potential scattering term KR that is of order the bare Kondo coupling and which vanishes (to 13  this order) when the dot level is tuned to the symmetric value of εd = −U/2.  This latter term is discussed in § 2.2.1. Both the generated Kondo interaction and the potential scattering depend on the flux ϕ, the strength of the direct inter-lead coupling t′ , and the momentum of electrons in the leads. From the strength of this Kondo interaction we are able to obtain the dependence of the Kondo temperature on these model parameters as discussed in § 2.3.1. Next, in § 2.2.2, we integrate out high energy states to obtain a low energy effective Hamiltonian. In addition to renormalizing the Kondo interaction, this also generates a small potential scattering term, VR , of second order  in the bare Kondo coupling. Hence, VR contributes to the leading order term in the potential scattering when εd = −U/2 though there may be other  contributions as we discuss in the text. Otherwise, it is the KR term discussed above that provides the leading order contribution to the potential scattering. In § 2.3.2, we calculate the zero-temperature conductance in terms of the  effective S-matrix for low energy electrons via the Landauer formula. Below the Kondo temperature, a phase shift of π/2 occurs in the screening channel. To a good approximation, the low temperature S-matrix is simply determined by the unitary transformation to the scattering basis and this π/2 phase shift in the screening channel. A small correction to this S-matrix occurs due to the KR potential scattering term (or VR in the case that εd = −U/2). While this approach confirms the results of [73] in the special case of a half-filled band in the leads, we find that changing the electron density in the leads has a large effect. Although this method of calculating the conductance is not as generally applicable as the Keldysh Green’s function technique used in other theoretical analyses (as discussed in the previous section), our approach has the advantage of highlighting the separate contributions to the conductance by the Kondo effect (via the π/2 phase shift) and the interference of the ABI (via the unitary transformation to the screening channel) that is often obscured by a more complicated calculation. In this way, both in our choice of  14  model and in our analysis, we focus on the primary influences of interference on the Kondo effect without the distraction of too many details. We confirm some of these results by NRG calculations presented in chapter 3. We only consider the simplest case in the Kondo regime, ǫd = −U/2,  symmetric coupling of the left and right leads to the dot, and with a half-filled band. We begin by completely describing the renormalization group flow of our model, predicting the form of the various fixed points and crossover energy scales which are then confirmed in the NRG. Most notably, the Kondo temperature is extracted from the energy scale of the Wilson chain at which the crossover to the low energy strong-coupling fixed point occurs and agrees excellently with that predicted analytically. A method for extracting the phase shifts of the strong-coupling fixed point from the low-energy numerical excitation spectrum is then described. These phase shifts are then compared to those obtained analytically from the derived S-matrix. We obtain quite good agreement through this comparison, including the small corrections from VR . Throughout the manuscript, we assume units in which ~ = 1 unless otherwise noted.  15  Chapter 2 Model & Analysis 2.1  Theoretical Model & Transformations  We start with a tight-binding model depicted in Figure 2.1. The Hamiltonian for this model is H = H0 + H−+ + Htd + Hd " −2 # ! ∞ X X H0 = −t + c†j cj+1 + h.c. j=−∞    (2.1) (2.2)  j=1   H−+ = −t′ + h.c. h  i ϕ ϕ Htd = − td− ei 2 c†−1 + td+ e−i 2 c†1 d + h.c. c†−1 c1  Hd = εd d† d + Und↑ nd↓ .  (2.3) (2.4) (2.5)  Each annihilation operator for the leads, cj , and the Anderson impurity, d, is a spinor where the spin indices are The anti-commutation n often implied. o  † relationships of these operators are cjµ , cj ′ν = δjj ′ δµν and d†µ , dν = δµν . The number operator for dot electrons of spin µ is defined as ndµ ≡ d†µ dµ .  The various parameters are described in Figure 2.1. We will assume that all of the couplings are real. The magnetic flux has been introduced through 16  εd td− t  t −4  t  t  −3  −2  −1  Φ  td+  ′  t  t 1  t 2  t 3  t 4  Figure 2.1: The lattice model described by the Hamiltonian of eq. (2.1). the parameter ϕ = 2πΦ/Φ0 , Φ being the magnetic flux threading the AB ring and Φ0 = h/e being the magnetic flux quantum. We assume that the magnetic field generating the flux is small enough in the vicinity of the wires so that we may neglect the Zeemen effect in the quantum dot and the leads. Although such a tight-binding model for the leads is not a very accurate description of leads in a semi-conductor heterostructure on which such geometries are often defined, we use it here as an example of a relatively simple model that contains a natural bandwidth of 4t. We now define a basis of even and odd combinations of electron operators 1 ej ≡ √ (cj + c−j ) , 2 1 oj ≡ √ (cj − c−j ) , 2  j>0  (2.6)  j>0  (2.7)  so that the Hamiltonian can be written as H0  ∞   X = −t e†j ej+1 + o†j oj+1 + h.c.  (2.8)  j=1  H−+ Htd  i h † † = −t e1 e1 − o1 o1 i o 1 nh ∗ † ∗ † √ tde e1 − tdo o1 d + h.c. = − 2 ′  17  (2.9) (2.10)  where we have defined the shorthand notation ϕ  ϕ  −i ϕ 2  iϕ 2  tde ≡ td− e−i 2 + td+ ei 2  (2.11)  tdo ≡ td− e  (2.12)  − td+ e .  Hd remains unchanged. Immediately we notice that, for the case of zero flux, ϕ = 0, and symmetric coupling td− = td+ , the model reduces to two decoupled chains, the even channel interacting with the quantum dot and having a potential scattering interaction −t′ at j = 1 and the odd channel decoupled from the dot and  with a potential scattering interaction t′ at j = 1. However, in the general case of ϕ 6= 0, we must analyse both channels together. If we remove the dot from the system, we are left with two independent  channels, even and odd, with a potential ∓t′ at j = 1. As shown in Appendix B, this potential gives rise to two scattering phase shifts δk± in the even/odd channel respectively, the form of which is given at the Fermi surface to be τ ′ sin kF a tan δ ± ≡ ± (2.13) 1 ∓ τ ′ cos kF a  where τ ′ ≡ t′ /t and δ ± ≡ δk±F . Note that, at half-filling when kF = π/(2a),  δ + = −δ − = δ where tan δ = τ ′ . These phase shifts will play an important part when we discuss the zero-temperature properties of this system in § 2.3. The dependence on kF reflects the dependence on the electron density n in √ the leads. For example, in a 2DEG, kF = 2πn [93]. Hence, the dependence of this and subsequent quantities on kF reflects dependence on the electron  density in the leads. Noting that H−+ serves as a potential scattering term, we seek to transform to the scattering basis that essentially removes these interactions from the Hamiltonian. We do this by first introducing the complete set of wave-  18  functions that solve the Schrödinger equation for H0 φj (k) =  r  2a sin(kja) π  (2.14)  with a being the lattice spacing. We can expand our operators as ej = oj =  r  r  2a π 2a π  Z  π a  dk sin(kja)ek  (2.15)  dk sin(kja)ok  (2.16)  0  Z  π a  0  n o n o so that e†k , ek′ = o†k , ok′ = δ(k − k ′ ). The Hamiltonian becomes H0 =  Z  π a  0 π a    dk εk e†k ek + o†k ok    dk dk ′ vkk′ e†k ek′ − o†k ok′ 0 r Z π i nh a a ∗ † ∗ † dk sin ka tde ek − tdo ok d = − π 0 +h.c.  H−+ = Htd  Z  (2.17) (2.18)  (2.19)  where we have defined εk ≡ −2t cos ka 2t′ a vkk′ ≡ − sin ka sin k ′ a. π  (2.20) (2.21)  Ignoring Htd for the moment, we note that the only difference between the e and o channels is the sign of the vkk′ interaction. Hence, we define the  19  scattering basis † qek † qok  ≡ ≡  Z  Z  π a  0 π a  0  ±(k)  (2.22)  −(k)  (2.23)  dk ′ φk′ o†k′  where ≡ δ(k − k ′ ) +  φk ′  +(k)  dk ′ φk′ e†k′  Tk±′ k εk − εk′ + iη  (2.24)  and η is a positive, infinitesimal parameter. It is shown in detail in Ap± pendix A that Tkk ′ is given by ± Tkk ′ = ±  vkk′ . 1 ± τ ′ e−ik′ a  (2.25)  Thus, the Hamiltonian greatly simplifies in the qek , qok basis to H0 =  π a  Z  0  H−+ = 0 Htd    † † dk εk qek qek + qok qok  r Z π i nh a a ∗ † ∗ −∗ † dk t∗de Γ+ q − t Γ q = − do k k ek ok d π 0  +h.c.  where Γ± k  ≡  Z  0  π a  ±(k)  dk ′ sin k ′ a φk′  =  sin ka . 1 ± τ ′ e−ika  (2.26) (2.27)  (2.28)  (2.29)  The last equality is proven in Appendix A. With the potential scattering Hamiltonian H−+ vanishing due to the  transformation to the scattering basis, we are now free to rotate the basis once more to the channel that couples directly to the impurity and its orthogonal complement. In this way, anticipating our discussion on the Kondo  20  effect, we define the screening channel − tde Γ+ k qek − tdo Γk qok q Ψscr ≡ p k − 2 2 t2d− + t2d+ |Γ+ k | (1 + γ cos ϕ) + |Γk | (1 − γ cos ϕ)  (2.30)  where we have defined the asymmetry parameter γ≡  2td− td+ . 2 td− + t2d+  (2.31)  In this way the dot coupling Hamiltonian can be written as Htd =  Z  0  π a  † dk Ṽdk Ψscr k d + h.c.    (2.32)  where Ṽdk  r q q a 2 − 2 2 2 t + td+ |Γ+ ≡ − k | (1 + γ cos ϕ) + |Γk | (1 − γ cos ϕ) (2.33) π d− s r  2a 2 1 + τ ′ 2 − 2γτ ′ cos ϕ cos ka . (2.34) = − sin ka td− + t2d+ π (1 + τ ′ 2 )2 − 4τ ′ 2 cos2 ka  This form of the hybridization was first found in ref. [79]. We are interested primarily in the Kondo effect which involves only the screening channel since it is the only one that couples to the quantum dot. Hence, one can perform a Schrieffer-Wolff transformation [2, 34] so that the screening channel Hamiltonian assumes the form H =  Z  π a  † scr dk εk Ψscr k Ψk  0  +  Z  0  π a    † scr † scr scr ~ ′ Ψ Ψ · S + K ~ σ Ψ dk dk ′ Jkk′ Ψscr d kk k k′ k k′  21  (2.35)  where we have defined the effective dot spin operator ~d ≡ d† ~σ d S 2  (2.36)  with ~σ being the three Pauli matrices (recall that Ψscr k and d are spinors). The coupling parameters are given by Jkk′ = Ṽdk Ṽdk′ Kkk′ =  Ṽdk Ṽdk′ 2     1 1 + εk − εF − εd U + εd − εk ′ + εF   1 1 − εk − εF − εd U + εd − εk ′ + εF  (2.37) (2.38)  At this point, one can obtain a low-energy effective theory by integrating out high-energy modes in the usual way. The potential scattering term Kkk′ is marginal and does not renormalize. We will discuss this term in more detail in § 2.2.1 and neglect it for now. The exchange interaction is relevant ~d by the screenand diverges, giving rise to the usual Kondo screening of S ing channel Fermions for temperatures T below the Kondo temperature TK . There are, however, physical consequences due to the presence of the ABI that arise from the dependence of Jkk′ and Kkk′ on the flux ϕ and inter-lead coupling t′ that will be determined in § 2.3. To summarize the analysis thus far, through a series of basis rotations we have cast the inter-lead Hamiltonian into a potential scattering form. By transforming to the scattering basis, we have eliminated this potential scattering term and identified the operator that couples directly to the quantum dot. It is this combination that will participate in the Kondo screening of the dot. Nevertheless, there are additional potential scattering terms that can arise in the screening channel and it is this subject that we next discuss.  22  2.2  Additional Potential Scattering  Our goal is to derive an effective theory of our system that is valid at low temperatures, keeping the leading order contributions in the effective strength of the Kondo coupling J ≡ JkF kF |t′ =0     2a t2d− + t2d+ sin2 kF a U = π −εd (U + εd )  (2.39)  which we take to be a small parameter. The effective theory can be derived, to a first approximation, by linearizing the dispersion εk in a region −Q < k − kF < Q and approximating the  coupling constants Jkk′ and Kkk′ by their values at the Fermi energy JkF kF and KkF kF . However, it will be shown that when the dot level is tuned to the value εd = −U/2, KkF kF vanishes to second order in VdkF . In this case, a more careful derivation of the low-energy Hamiltonian reveals that there is still an additional potential scattering generated by the renormalization of Jkk′ . This is higher order in J than the leading order contribution to KkF kF written in eq. (2.38) but contributes to the leading order term in the additional potential scattering when eq. (2.38) vanishes at εd = −U/2. We address each of these cases separately below.  2.2.1  Asymmetric dot εd 6= −U/2  Restricting excitations to a small region about the Fermi energy as described above, the potential scattering term in the Hamiltonian generated by the Schrieffer-Wolff transformation assumes the form H R = KR  Z  Q −Q  † scr dk dk ′ Ψscr k Ψk ′  23  (2.40)  where KR ≡ Kk F k F    U + 2εd a 2 2 2 sin kF a td− + td+ − = π εd (U + εd ) ×  1 + τ ′ 2 − 2γτ ′ cos ϕ cos kF a . (1 + τ ′ 2 )2 − 4τ ′ 2 cos2 kF a  (2.41)  In order to observe the Kondo effect, we require that εd ≈ −U/2 so as to favor the formation of a local moment rather than a doubly occupied or unoccupied dot level. In this case, we see that KR is of order J. However, for the precise value of εd = −U/2, KR vanishes and there is no potential scattering generated directly by the Schrieffer-Wolff transformation at low energies to linear order in J. The presence of this potential scattering term will give rise to a phase shift δR at the Fermi surface in the screening channel. As shown in Appendix B, this is given by tan δR = −πνKR  (2.42)  for small KR where ν is the density of states at the Fermi energy. We will show in § 2.3.2 how this additional potential scattering contributes to the T = 0 conductance of the ABI.  2.2.2  Symmetric dot εd = −U/2  As discussed above, integrating out the high-energy modes to obtain a lowenergy Hamiltonian leaves the marginal interaction Kkk′ unchanged and so one obtains the term discussed in the above section. However, one can ask the question as to whether or not an additional potential scattering term is generated by the Kondo interaction Jkk′ . Normally this is not the case for one often considers a Kondo interaction that is particle-hole symmetric which cannot generate a potential scattering term since such a term breaks 24  kF , µ  kF , µ  kF , µ  kF , µ k, ν  k, ν JkkF  Jk F k  Jk F k  JkkF  Figure 2.2: The second order Feynman diagrams that contribute to VR . The solid line labelled by k, µ represents a single particle excitation of Ψscr kµ while the dashed line represents the quantum dot spin degree of freedom. particle-hole symmetry. From the form of Jkk′ of eq. (2.37), it is clear that our Kondo interaction breaks particle-hole symmetry. This is a consequence of a non-zero t′ which necessarily breaks particle-hole symmetry. Although we have transformed away the explicit t′ interaction, the particle-hole symmetry breaking is manifest in this more complicated Kondo interaction. As a result, there is no symmetry forbidding this Kondo interaction from generating an additional potential scattering term and it is to the calculation of this that we now turn our attention. Consider a renormalization group scaling by integrating out all of the wave vectors down to the Fermi energy. Although it is difficult to perform such a transformation exactly, one can make progress through a perturbative expansion in J. The leading order contribution is of order J 2 which will be much smaller than KR , eq. (2.41), which is of order J. However, KR vanishes when εd = −U/2 so that the J 2 term calculated below will contribute to the leading order term in the potential scattering. Hence, in this section, we assume that εd = −U/2. Evaluating the Feynman diagrams to second order in the Kondo interaction Jkk′ in eq. (2.35) as depicted in Figure 2.2, one finds a potential scattering term generated of the form HR = VR  Z  Q −Q  † scr dk dk ′ Ψscr k Ψk ′  25  (2.43)  where the region of integration is restricted to small momentum about the Fermi momentum kF and VR is given by 3 VR = 16  Z  0  π a  dk  JkF k JkkF . εF − εk + iηsgn(εF − εk )  (2.44)  The factor of 3/16 comes from the trace over spin degrees of freedom and the denominator is simply the time-ordered propagator of the intermediate Ψscr k Fermion. Substituting in the definition of Jkk′ of eq. (2.37) together with the definition of Ṽdk from eq. (2.34) and J from eq. (2.39), VR can be written as  3J 2 2 − 2 |Γ+ kF | (1 + γ cos ϕ) + |ΓkF | (1 − γ cos ϕ) 4 128at sin kF a  × IR+ (1 + γ cos ϕ) + IR− (1 − γ cos ϕ) . (2.45)  VR =  The factors of IR± are dimensionless integrals given by IR±  ≡  Z  π  dy  0  1 sin2 y 2 ′ ′ 1 ± 2τ cos y + τ cos y − cos kF a + iηsgn(cos y − cos kF a) u2 − (cos y − cos kF )2 × 2 (2.46) u − 4(cos y − cos kF )2  with  U . 2t To evaluate these integrals, we break them up into two regions u≡  IR±  (2.47)   Z π 1 1 + dy = dy cos y − cos kF a + iη cos y − cos kF a − iη kF a 0 2 2 sin y u − (cos y − cos kF )2 × . (2.48) 1 ± 2τ ′ cos y + τ ′ 2 u2 − 4(cos y − cos kF )2 Z  kF a  The imaginary parts from each integral cancel each other. Upon evaluation  26  of the principle part of each integral, one obtains     3u2 τ ′ 2 π 1 − τ ′2 1− −1 . =± ′ 8τ 1 ± 2τ ′ cos kF a + τ ′ 2 (1 ± 2τ ′ cos kF a + τ ′ 2 )2 − u2 τ ′ 2 (2.49) Substituting this back into the above expression gives us our final result for IR±  VR : 3π 2 (νJ)2 1 + τ ′ 2 − 2γτ ′ cos kF a cos ϕ (2.50) 64τ ′ sin kF a (1 + τ ′ 2 )2 − 4τ ′ 2 cos2 kF a  2τ ′ (1 − τ ′ 2 ) cos kF a − (1 − τ ′ 4 )γ cos ϕ × γ cos ϕ + (1 + τ ′ 2 )2 − 4τ ′ 2 cos2 kF a   3u2γ cos ϕ τ ′ 2 (1 − τ ′ 2 ) (1 + τ ′ 2 )2 + 4τ ′ 2 cos2 kF a − u2 τ ′ 2 + 2 (1 + τ ′ 2 )2 + 4τ ′ 2 cos2 kF a − u2 τ ′ 2 − 16τ ′ 2 (1 + τ ′ 2 )2 cos2 kF a ) 12u2τ ′ 3 (1 − τ ′ 4 ) cos kF a − 2 (1 + τ ′ 2 )2 + 4τ ′ 2 cos2 kF a − u2 τ ′ 2 − 16τ ′ 2 (1 + τ ′ 2 )2 cos2 kF a  νVR = −  where ν is the density of states at the Fermi energy. Just as with the potential  scattering term KR , this VR term will give rise to a phase shift in the screening channel given by tan δR = −πνVR (2.51) as shown in Appendix B. In conclusion, the transformation analysis of section 2 provides a simple, generic way to account for the presence of inter-lead coupling which takes the form of a potential scattering interaction. Such a transformation effectively removes the potential scattering explicitly from the Hamiltonian in favor of a more complicated, particle-hole asymetric Kondo interaction when the dot is tuned to the Kondo regime. We have further shown that additional potential scattering terms are generated in the screening channel. The leading order contribution to this additional potential scattering is given by KR , eq. (2.41), in the case that εd 6= −U/2. When εd = −U/2, VR of eq. (2.50) contributes to 27  the leading order term in the potential scattering Hamiltonian though there may be other terms to this order not considered here. In the next section, we analyse the physical consequences of this low-energy model.  2.3 2.3.1  Physical Properties Kondo Temperature  One of the primary insights of the scattering transformation analysis is in revealing how the ABI influences the coupling between the quantum dot and the screening channel of electrons. That is, it allows us to obtain an expression for the dot-lead coupling in the Hamiltonian of eq. (2.32), given by ṼdkF (in the long wavelength limit), showing the dependence of the coupling on t′ , ϕ, and kF . We then determine the t′ , ϕ, and kF dependence of the effective Kondo coupling via the Schrieffer-Wolff transformation, eq. (2.37). This, in turn, gives rise to a t′ , ϕ, and kF dependent Kondo temperature, the precise expression of which is easy to derive. Using the low-energy effective Hamiltonian, we determine the effective Kondo coupling by evaluating eq. (2.37) at the Fermi energy −U εd (U + εd ) 2 1 + τ ′ − 2γτ ′ cos ϕ cos kF a = J (1 + τ ′ 2 )2 − 4τ ′ 2 cos2 kF a  2 J eff ≡ JkF kF = Ṽdk F  (2.52) (2.53)  where J is defined in eq. (2.39). The leading order RG definition of the Kondo temperature [2] is TK = De−1/(2νJ  28  eff )  (2.54)  1  0  0  ln( TK/TK )  0.5  -0.5  kFa = π/2 π/3 π/4 π/6 π/12  -1  -1.5 0  0.5  1  1.5 Flux φ  2  3  2.5  Figure 2.3: The flux dependence of the Kondo temperature for a value of τ ′ = 0.4, νJ = 0.3, and γ = 1. Here we see an increase in the flux dependence as the electron density is lowered. and dividing by the t′ = 0 Kondo temperature TK0 = De−1/(2νJ) , we get ln  TK τ ′ 2γ cos ϕ cos kF a + τ ′ (1 − 4 cos2 kF a) + τ ′ 3 . = − TK0 2νJ 1 − 2γτ ′ cos ϕ cos kF a + τ ′ 2  (2.55)  Although the denominator is always positive, we see that the Kondo temperature can be raised or lowered by the presence of the ABI depending on the values of τ ′ , ϕ and kF . This is shown in Figures 2.3, 2.4, and 2.5 which show the flux, t′ , and kF dependence for various values of the other parameters. For the special case of half-filled leads, kF = π/(2a), the result is partic-  29  0  0  ln( TK/TK )  -2  -4  φ=0 π/4 π/2 3π/4 π  -6  -8 0  0.5  τ’  1  1.5  Figure 2.4: The τ ′ dependence of the Kondo temperature for a value of kF = π/(6a), νJ = 0.3, and γ = 1. This exhibits the variety of behaviour that can be seen for different values of the flux and that the Kondo temperature always vanishes as τ ′ → ∞. ularly simple TK ln 0 TK  π kF = 2a  τ ′2 =− 2νJ  (2.56)  showing that the Kondo temperature is independent of flux in this case. This limiting form of the Kondo temperature is verified by the NRG as discussed in § 3.2.3.  30  1  φ=0 π/4 π/2 3π/4 π  0  0  ln( TK/TK )  0.5  -0.5  -1  -1.5 0  0.5  1  1.5 kF  2  2.5  3  Figure 2.5: The kF dependence of the Kondo temperature for a value of τ ′ = 0.4, νJ = 0.3, γ = 1, and various values of the magnetic flux.  2.3.2  S-Matrix and Conductance  The strong-coupling fixed point of the Aharanov-Bohm model under consideration can be described by a two-channel Fermi liquid. In this way, the fixed point model is fully described by a 2 × 2 S-matrix describing how the quasi-particle excitations of the two channels are scattered at the Fermi energy. In this section, we derive this S-matrix and relate it to the conductance between the two leads. The analysis of sections 2.1 and 2.2 provide the following simple picture of the strong-coupling fixed point. The direct coupling between the two leads, t′ , gives rise to a phase shift δ ± in the qek and qok channels respectively. The form of these phase shifts is presented in eq. (2.13) as computed in Appendix B. 31  By transforming to the scattering basis and removing the t′ interaction from the Hamiltonian, we were able to identify the screening channel of eq. (2.30). scr Defining the orthogonal complement, Ψ̃scr k , to Ψk and evaluating both at the Fermi energy (relevant here since we are talking about T = 0 properties), we can write the relation between the screening/non-screening basis and the even odd basis in terms of the above phase shifts as Ψscr k Ψ̃scr k  !  =U  qek qok  !  (2.57)  where ϕ  +  U =N  ϕ  ϕ  −  ϕ  −e−iδ sin δ + (td− e−i 2 + td+ ei 2 ) −e−iδ sin δ − (td− e−i 2 − td+ ei 2 ) −  ϕ  ϕ  eiδ sin δ − (td− ei 2 − td+ e−i 2 )  +  ϕ  ϕ  !  −eiδ sin δ + (td− ei 2 + td+ e−i 2 ) (2.58)  with normalization  − 1 N ≡ (t2d− + t2d+ )(sin2 δ + (1 + γ cos ϕ) + sin2 δ − (1 − γ cos ϕ)) 2 .  (2.59)  In the screening channel, there will be a phase shift with two contributions. The first is the usual π/2 Kondo phase shift. The second is the phase shift δR generated by the additional potential scattering, the leading order contribution to which will either be KR or VR 1 depending on the value of εd . Since the additional potential scattering was obtained by integrating out the high-energy modes, the generated Hamiltonian term of eq. (2.43) must be considered as a low energy, long wavelength continuum model where the influence of the lattice is inconsequential. The phase shift for such a model is derived in Appendix B and shown to be either that of eq. (2.42) or eq. (2.51). This is all of the information we require to write down the S-matrix in 1  However, there may be other contributions to the potential scattering that are of the same order as VR as discussed in chapter 4.  32  the even/odd basis: S = U†  −e2iδR 0 0 1  !  U  e2iδ 0  +  0 e2iδ  −  !  .  (2.60)  The far right matrix describes the potential scattering phase shifts due to t′ in the qek and qok channels, U rotates the basis to the screening channel and the matrix between U and U † describes the phase shift δR due to VR or KR π as well as the π/2 Kondo phase shift giving rise to the factor of −1 = e2i 2 . Multiplying the matrices, we can write S as S=  See Seo Soe Soo  !  (2.61)  with See = −Me2iδ  +  Seo = −2Mei(δ   e2iδR (1 + γ cos ϕ) sin2 δ + − (1 − γ cos ϕ) sin2 δ − (2.62)  − +δ + +δ ) R  i(δ− +δ+ +δR )  Soe = −2Me  Soo = Me2iδ  −  M≡  (2.63)  (β + iγ sin ϕ) sin δ − sin δ + cos δR  (1 + γ cos ϕ) sin2 δ + − e2iδR (1 − γ cos ϕ) sin2 δ  where we have defined  and  (β − iγ sin ϕ) sin δ − sin δ + cos δR  t2d− − t2d+ β≡ 2 td− + t2d+  1 . (1 + γ cos ϕ) sin δ + + (1 − γ cos ϕ) sin2 δ − 2  (2.64)  −  (2.65)  (2.66)  (2.67)  To relate this S-matrix to the conductance, we first construct general scattering wave functions between the even and odd channels. Consider an incoming plane wave in the even channel that is then scattered into the even and odd outgoing channel according to the above S-matrix. Such a wave  33  function takes the form ψe = e−ik|x| + See eik|x| + Soe sgn(x)eik|x|  (2.68)  where the first term is the incoming wave in the even channel, the second term the scattered even wave and the last term the scattered odd wave. Similarly, considering an incoming wave in the odd channel gives the wave function ψo = sgn(x)e−ik|x| + Seo eik|x| + Soo sgn(x)eik|x| .  (2.69)  Next, we wish to form a combination of ψe and ψo that corresponds to a right-moving wave incoming from the left. That is, we wish to form a superposition of the above two wave functions that has no left-moving component for x > 0. To this end, we form 1 (ψe − ψo ) 2 ( 1 (See + Soe − Seo − Soo ) eikx ,x > 0 2 = eikx + 12 (See + Soo − Seo − Soe ) e−ikx , x < 0  ψ ≡  (2.70)  where, indeed, we find no e−ikx component in ψ for x > 0. Looking at the x > 0 portion of ψ, we recognize the coefficient of the plane wave as the transmission probability amplitude for transmission from the left lead to the right lead T =  1 (See + Soe − Seo − Soo ) . 2  (2.71)  Using the Landauer-Büttiker formula [94, 95, 58, 59], we obtain an expression  34  for the conductance 2e2 2 |T | h  2e2 (1 + γ cos ϕ)2 sin4 δ + cos2 (δ + − δ − + δR ) = h +(1 − γ cos ϕ)2 sin4 δ − cos2 (δ + − δ − − δR )  + sin2 δ − sin2 δ + 4 cos2 δR sin2 ϕ     − 1 − γ 2 cos2 ϕ cos 2(δ + − δ − ) + cos 2δR   2 (1 + γ cos ϕ) sin2 δ + + (1 − γ cos ϕ) sin2 δ − (2.72)  G =  This is the most general expression for the conductance expressed in terms of the phase shifts δ ± generated by the inter-lead coupling t′ , the additional potential scattering KR or VR via δR , and in terms of the flux ϕ. The latter includes the explicit ϕ dependence written above as well as the dependence implicit in δR via the flux dependence of KR or VR written in eqs. (2.41) or (2.50). Although the equation is rather complicated, we see that the conductance satisfies the necessary Onsager symmetry relation G(ϕ) = G(−ϕ) [55, 56, 57, 58]. We now turn our attention to special limiting cases. For the case of kF = π/(2a) and td− = td+ considered in most previous studies, δ + = −δ − = δ with tan δ = τ ′ and the conductance simplifies to G|kF = π  2a  =  ϕ ϕ 2e2 h 2 cos (2δ − δR ) cos4 + cos2 (2δ + δR ) sin4 h 2 2 1 + cos2 δR sin2 ϕ − (cos 4δ + cos 2δR ) sin2 ϕ .(2.73) 4  It is interesting to compare this with the numerical results of [73]. For the case of εd 6= −U/2, when KR is the leading order contribution to δR , we are able to qualitatively reproduce the Fano-Kondo behaviour seen in [73] in the region εd ≈ −U/2 for which our analysis is valid. An example of this is given 35  1  0.6  2  G [2e /h]  0.8  0.4 Tb = 0 0.1 0.3 0.6 1.0  0.2  U/t = 0.5 0 -0.4  -0.3  εd / t  -0.2  -0.1  Figure 2.6: The conductance as a function of dot level εd for various values of the direct interlead transmission probability given by Tb = sin2 2δ. Here, we assume the particle-hole symmetric value of half-filling, kF = π/(2a) and γ = 1. in Figure 2.6. For the symmetric value εd = −U/2 when KR vanishes, we can view the  δR generated by VR as a small correction to the results of ref. [73]. Indeed, in the limit of δR → 0 and kF = π/(2a), our result reduces to G|δR =0,kF = π = 2a   2e2 1 − Tb cos2 ϕ h  (2.74)  where Tb = sin2 2δ is the transmission probability through the lower arm of the ABI in the absence of the upper arm. This is precisely the form reported  36  in [73] for the case of a singly-occupied quantum dot. In this way, eq. (2.73) can be viewed as an analytic description of the results of ref. [73], the latter of which required numerical input from the NRG. Such an analytic description is only valid for values of εd close to −U/2 so as to strongly favor a local moment on the quantum dot whereas the results of ref. [73] are valid for all εd . On the other hand, our complete expression for the conductance, eq. (2.72), extends previous results to cases where the Fermi energy is not situated in a particle-hole symmetric manner relative to the band edges (e.g. kF 6= π/(2a)) as well as taking into account  the additional potential scattering VR discussed in § 2.2. To further examine the correction due to VR , we look at the flux depen-  dence of the conductance in Figure 2.7 for the case that εd = −U/2 and hence VR contributes to the leading order behaviour of δR . There, each of the different coloured lines indicates a different value of the direct inter-lead coupling t′ as encoded by δ. It is seen that the conductance contrast (the difference between the minimum and maximum conductance) reaches a maximum for an intermediate value of the inter-lead coupling τ ′ = 1 (t′ = t). Furthermore, it is shown that for τ ′ < 1, the effect of the additional potential scattering VR is to decrease the conductance whereas for values τ ′ > 1, the additional potential scattering serves to increase the conductance. This fact is made more evident in Figure 2.8 where the conductance is plotted versus τ ′ for ϕ = 0. There, one can clearly see the crossover from reduced to enhanced conductance around τ ′ = 1. Given that VR offers only a small correction, we look at the δR = 0 limit of the conductance for general kF which takes the form G|δR =0    2e2 cos2 (δ + − δ − ) sin4 δ + (1 + γ cos ϕ)2 + sin4 δ − (1 − γ cos ϕ)2 = h   + sin2 δ − sin2 δ + 4 sin2 ϕ − 2 cos2 (δ + − δ − ) 1 − γ 2 cos2 ϕ   2 + 2 sin δ (1 + γ cos ϕ) + sin2 δ − (1 − γ cos ϕ) . (2.75) 37  0.8  G [2e /h]  0.6 2  τ’=0.16 VR=0 0.16 0.63 VR=0 0.63 0.94 VR=0 0.94 2.04 VR=0 2.04 3.14 VR=0 3.14  0.4  0.2  1  2  3 Flux φ  4  5  6  Figure 2.7: The conductance plotted as a function of magnetic flux for νJ = 0.287, kF = π/2a, εd = −U/2 and γ = 1. Each of the different coloured lines indicates a different value of the inter-lead coupling τ ′ . The solid lines are the prediction with VR = 0 (equivalently δR = 0) with the dotted lines showing the finite VR correction. Even without including the small correction due to VR , this is a generalization of the conductance reported in ref. [73] which, like most similar studies, only considered the case where the leads exhibit particle-hole symmetry (kF = π/(2a) for our tight-binding leads). The dependence of the conductance on kF is demonstrated in Figure 2.9. For quantum dots constructed on semiconductor heterostructures where the two-dimensional electron gas has very low density, the Fermi energy will be very close to the bottom of the energy band and so exhibit strong particle-hole asymmetry. Hence, the generalized  38  0.8  2  G [2e /h]  0.6  0.4 VR=0 VR finite 0.2  1  2  3 τ’  4  5  Figure 2.8: The conductance plotted as a function of inter-lead coupling τ ′ when kF = π/(2a) and εd = −U/2. Here, νJ = 0.287 and ϕ = 0. forms for the conductance reported above seem to be more applicable to such devices than those reported in previous studies. The description of the conductance that emerges from this analysis is quite interesting. In the limit that τ ′ → 0, we recover the well-studied model of a single quantum dot embedded between two leads where one obtains unitary conductance at zero temperature (when γ = 1). As one increases τ ′ , interference effects play a stronger role until one obtains maximal interference at τ ′ = 1 (Tb = 1) where one is able to obtain total destructive interference in the form of zero conductance for certain values of the parameters (e.g. kF = π/(2a) and ϕ = 0). As one further increases τ ′ , the transmission Tb through the lower arm decreases and interference effects are diminished. 39  1  0.9  2  G [2e /h]  0.8  0.7 φ=0 π/6 π/3 π/2 2π/3 5π/6  0.6  0.5  0  0.5  1  1.5 kFa  2  2.5  3  Figure 2.9: The conductance plotted as a function of kF for τ ′ = 0.4, γ = 1 and for multiple values of the magnetic flux. We do not consider the small correction due to VR so that the conductance is given by eq. (2.75).  40  Chapter 3 Numerical Renormalization Group Analysis 3.1  Derivation of the Numerical Renormalization Group  The NRG is a powerful numerical algorithm developed to study quantum impurity models such as the one under consideration in this thesis. In this section, we describe how to derive and implement this algorithm for our model of a quantum dot in an ABI. This is a generalization of the derivation for the single-channel model described in detail in [18]. The development is very similar to the application of the NRG to other models that can be transformed into a two-channel model such as the two-impurity Kondo model [96, 97, 98]. We begin with a long wavelength version of the Hamiltonian described in eqs. (2.17)–(2.19) with a dispersion relation εk = vF k linearized about the Fermi energy where vF = 2at sin kF a is the Fermi velocity and where k is measured with respect to kF . We introduce the momentum cutoff Q such that this dispersion is valid for −Q < k < Q. We further assume 41  td− = td+ = td and εd = −U/2.  The resulting Hamiltonian is H = vF  Z  +Vd  Q −Q  Z  Z   † † dk k ek ek + ok ok − Vp  −Q  Q  −Q  Q  dk  h    dk dk ′ e†k ek′ − o†k ok′  i U 2 ϕ † ϕ † d† d − 1 (3.1) cos ek + sin ok d + h.c. + 2 2 2   where we have simplified our notation by defining the potential scattering p term Vp ≡ −vkF kF , Vd ≡ −2td a/π sin kF a, and redefining the phase of ok so as to make all coefficients real. Note that this version of the Hamiltonian does not involve a transformation to scattering states. In this way, agreement between the NRG and results inferred from the transformations of section 2 will serve as support for the scattering transformation analysis. However, it should be observed that such a linear dispersion necessarily exhibits particle-hole symmetry whereas the tight-binding model discussed in section 2 generally breaks particle-hole symmetry except for the special case of kF = π/(2a) that occurs when there is one electron per site. For this reason, the NRG as formulated here strictly serves only to support our scattering transformation analysis for the particle-hole symmetric case of kF = π/(2a). Nevertheless, we trust that our analytic results hold true for arbitrary kF . To cast the Hamiltonian of eq. (3.1) into a dimensionless form, we introduce the energy cutoff D = vF Q, dimensionless energy ε = vF k/D, and density of states ν (which is constant for our one-dimensional linear disper√ √ sion) and define ẽε ≡ Dνek and õε ≡ Dνok . In this way, we can write  the Hamiltonian as Z 1 Z 1   H † † = dε ε ẽε ẽε + õε õε − νVp dε dε′ ẽ†ε ẽε′ − õ†ε õε′ D −1 −1 r Z 1 h i 2 Γ ϕ  U ϕ + d† d − 1 (3.2) dε cos ẽ†ε + sin õ†ε d + h.c. + 2πD −1 2 2 2D 42  where Γ ≡ 2πνVd2 .  (3.3)  The key to the success of the NRG is the division of the full range of states labelled by ε into intervals with logarithmically decreasing size as one approaches ε = 0. That is, we introduce a dimensionless number Λ > 1 and divide the range −1 < ε < 1 into an infinite number of sub-intervals  Λ−n−1 < |ε| < Λ−n labelled by n = 0, 1, 2, . . .. Within each of these intervals ± we introduce a complete set of states ψnp (ε) labelled by the integer p and defined as ( Λn/2 e±iωn pε , Λ−n−1 < ±ε < Λ−n −1 )1/2 ± (1−Λ ψnp (ε) = (3.4) 0 , otherwise with  2πΛn . (3.5) 1 − Λ−1 We can now use this complete set of states to expand ẽ and õ in a new operator basis X  + − b̃ε = ψnp (ε)Anpb + ψnp (ε)Bnpb (3.6) ωn =  np  for b = e, o.  If one expands the Hamiltonian of eq. (3.2) in terms of Anpb and Bnpb , one can show that the operator that couples directly to the impurity is f0b     1 = 1 − Λ−1 2 Z 1 1 = √ dε b̃ε . 2 −1   12 X ∞  n  Λ− 2 (An0b + Bn0b )  (3.7)  n=0  (3.8)  That is, only those operators Anpb and Bnpb with p = 0 couple directly to the impurity. The p 6= 0 operators occur in the kinetic energy term which takes  43  the form [18] H0 =  Z  1  −1  dε ε ẽ†ε ẽε + õ†ε õε      X −n  † 1 † −1 Anpb Anpb − Bnpb Bnpb 1+Λ Λ = 2 npb    1 − Λ−1 X X Λ−n 2πi(p′ − p)  † † ′ ′ + A A − B B exp npb np b npb np b . ′ −p −1 2πi p 1 − Λ ′ nb p6=p  It is only in the last term that p 6= 0 operators couple to the p = 0 ones which couple directly to the impurity. Yet these terms are suppressed by a factor of (1 − Λ−1 )/(2πp). Hence, for Λ close to 1, one can write an approximate Hamiltonian by neglecting the states with p 6= 0. Indeed, it has been shown that this is an excellent approximation for values of Λ as large as 3 [17, 18, 99]. We make such an approximation here, discarding all Anpb and Bnpb operators with p 6= 0. A value of Λ = 2.5 is used for all numerical calculations presented  except where noted otherwise. We next define the operator basis {fnb } in which the renormalization group calculations will take place. We do this by defining a unitary trans-  formation from the basis of An0b and Bn0b such that f0b is given by eq. (3.7) and the kinetic energy only involves coupling between fnb and f(n+1)b . These two conditions are sufficient to uniquely define the unitary transformation. The details of this transformation are given in [17] and are not of importance here so we simply write the resulting Hamiltonian in this basis ∞   1 X X H 1 † −n −1 2 2 Λ ξn fnb f(n+1)b + h.c. 1+Λ = D 2 b=e,o n=0   2 U † † + d† d − 1 − 2νVp f0e f0e − f0o f0o 2D r i ϕ † ϕ † 2Γ h cos f0e d + h.c. . + sin f0o + πD 2 2  44  (3.9)  where ξn = (1 − Λ−n−1 )(1 − Λ−2n−1 )−1/2 (1 − Λ−2n−3 )−1/2  (3.10)  which tends to unity for n ≫ 1. In this way, we have transformed our original continuum Hamiltonian (3.1) into one in which the impurity is coupled to two semi-infinite discrete chains called “Wilson chains” with nearest-neighbour hopping between sites n and n + 1 that decreases roughly as Λ−n/2 . The renormalization group is realized by truncating the infinite chain to N sites and rescaling the Hamiltonian such that the eigenvalues are of order unity HN ≡ Λ  (N −1)/2   XN −1 X b=e,o n=0    n † f(n+1)b + h.c. Λ− 2 ξn fnb    2 † † +Ũ d† d − 1 − Ṽp f0e f0e − f0o f0o i h 1 ϕ † ϕ † +Γ̃ 2 cos f0e + sin f0o d + h.c. 2 2  (3.11)  where we have defined dimensionless coupling constants 2 2νVp 1 + Λ−1  2 2 2Γ Γ̃ ≡ −1 1+Λ πD U 2 . Ũ ≡ 1 + Λ−1 2D  Ṽp ≡  (3.12) (3.13) (3.14)  The renormalization group transformation then takes the form of the recursion relation 1  HN +1 = Λ 2 HN + ξN  X  b=e,o  fN† b f(N +1)b + h.c.    (3.15)  and is realized by iterative diagonalization, using the eigenvalues and eigenvectors of HN to define HN +1 via eq. (3.15). In practice, the eigenvalues 45  are shifted so that the lowest one is zero. Since the Hilbert space for each Hamiltonian increases exponentially with N, one keeps only the lowest M eigenvalues of HN to define HN +1 . The details of this iterative diagonalization procedure are given in appendix C, including the numerical parameters used and how one can use the symmetries of the Hamiltonian to reduce the size of the matrices that need to be diagonalized. The finite Hamiltonian HN can be related to the Hamiltonian of eq. (3.9) by   H 1 1 + Λ−1 Λ−(N −1)/2 HN . (3.16) = lim D N →∞ 2 Since the dimensionless scale of HN is of order unity by definition, this indicates that the spectrum of HN describes the spectrum of the physical Hamiltonian at an energy scale given by EN ≈   1 1 + Λ−1 Λ−(N −1)/2 D. 2  (3.17)  In this way, we can associate HN with the effective Hamiltonian at the renormalization group energy scale EN . Fixed points can be identified as regions of N over which the energy spectrum of the associated HN changes very little (for unstable fixed points) or not at all (for stable fixed points).1 These fixed point NRG spectra can then be compared with that predicted by the scattering transformation analysis described above to test the validity of said analysis as described in the subsequent section. 1  More specifically, fixed points are identified as regions when the spectrum of HN , HN +2 , HN +4 , . . . remains unchanged. The fixed point spectrum oscillates between two spectra, one for even N and one for odd N .  46  Fixed Point Free Orbital (FO) Local Moment (LM) Strong Coupling (SC)  Vd U Stability 0 0 Unstable 0 ∞ Unstable ∞ ∞ Stable  Table 3.1: Summary of fixed points for the single-channel Anderson impurity model.  3.2  Renormalization Group Flow & Kondo Temperature  3.2.1  Fixed points of the single-channel Anderson impurity model  We begin by reviewing the various fixed points present in the single-channel Anderson model [18] before describing the influence of the ABI. When investigating low-energy, long-wavelength properties, it is customary to define a model in terms of continuous fields with a linearized dispersion relation characterized by a Fermi velocity vF . In this way, as in § 3.1, we can write  the single-channel Anderson model in terms of right-moving 1D electron annihilation operators ψ(x) as H = vF  Z  ∞  dx ψ † (x) (−i∂x ) ψ(x) + Vd ψ † (0)d + h.c.  −∞  2 U U † d d−1 − + 2 2    (3.18)  where we have set the dot level to εd = −U/2 (assumed throughout this  section). This model has three fixed points summarized in Table 3.1. The free orbital (FO) fixed point occurs when Vd = U = 0. This describes free ψ Fermions with a decoupled free dot level d. The spectrum of such a model is that of free Fermions plus the four degenerate, zero-energy states of the dot. 47  The FO fixed point is unstable and flows towards the local moment (LM) fixed point as the energy scale is lowered. The LM fixed point is characterized by a diverging U → ∞ and Vd = 0. This LM fixed point is the same as the FO except that two of the four dot levels are energetically forbidden, namely, those for which d† d = 0 and d† d = 2. In other words, the quantum dot can only be singly occupied with either a spin up or spin down electron. Hence, the spectrum will be that of free Fermions plus two degenerate, zero-energy states of the dot. The LM fixed point is also unstable and eventually flows to the strong coupling (SC) fixed point described by a diverging |Vd |2 /U → ∞. The nature of this fixed point can most easily be understood by first considering a Hamiltonian close to the LM fixed point with a small |Vd | ≪ U. In this case, one can perform a Schrieffer-Wolff transformation [34] perturbatively in Vd to obtain a dot interaction Htd + Hdot ≈ Jψ † (0)~σ ψ(0) · S~d  (3.19)  ~d ≡ d† (~σ /2)d is the effective where ~σ is a vector of the three Pauli matrices, S  spin of the singly-occupied dot level, and the coupling strength J is proportional to |Vd |2 /U. This is the Kondo interaction between the localized spin  of the quantum dot and the electrons in the leads. The SC fixed point of the Anderson model is essentially the same as the strong-coupling fixed point of the Kondo model wherein J → ∞ and the dot spin is screened by forming a  singlet with the lead electrons.  3.2.2  Fixed points of the Aharanov-Bohm quantum dot model  The low-energy transformations of section 2 reveals that the renormalization group flow for the ABI model under consideration will be very similar to that of the single-channel Anderson model just described. Indeed, we have 48  learned that a single, independent combination of the lead electrons, Ψscr k , couples directly to the dot just as in the single-channel Anderson model. The precise nature of this screening channel will depend on both flux ϕ and the inter-lead coupling t′ but the point is that there is a single channel available to screen the spin of the quantum dot. For simplicity, we consider only the symmetric case where td− = td+ = td and εd = −U/2. The primary difference with the Anderson model discussed in the previous section is the addition of two potential scattering phase shifts δ ± depending on t′ and the modification of the dot-lead coupling VdkF → ṼdkF as discussed  in § 2.1. We find that the FO and LM fixed points, with ṼdkF = 0, will be the same as in the single-channel Anderson model with the addition of the phase shifts δ ± arising from the direct tunneling between the leads which were incorporated into the definition of qek and qok . The SC fixed point of the ABI model will be one in which the dot spin is fully screened by the Ψscr k combination of lead electrons. Just as in the Kondo model, this will give rise ± to a π/2 phase shift in the Ψscr k channel in addition to the phase shifts δ arising from the direct tunneling t′ . Furthermore, the FO and LM fixed points occur for ṼdkF = 0 and, since  ṼdkF encodes the t′ dependence of the model, we predict that the cross-over scale of these fixed points will be unaffected by the presence of the ABI (i.e. in the region of these fixed points, the t′ and ϕ dependence of ṼdkF is inconsequential). However, the cross-over energy scale to the SC fixed point, that is, the Kondo temperature TK , will be influenced by the direct tunneling t′ and flux ϕ as discussed in § 2.3.1. Our analysis of the fixed points follows that of [18, 19]. Let us first consider the FO fixed point which, in terms of the NRG formalism, is defined  49  by Γ̃ = 0 and Ũ = 0, resulting in HN,FO = Λ  (N −1)/2  X N −1 X b=e,o n=0    n † Λ− 2 ξn fnb f(n+1)b + h.c.    † † ˜ −Vp f0e f0e − f0o f0o .  (3.20)  This has the form of two decoupled Wilson chains, each with a potential scattering term at the origin. Such chains were analyzed in [19] where the Ṽp dependence of the single-particle energies was described in detail. Extending their analysis to two decoupled channels as described in eq. (3.20), one can diagonalize the non-interacting fixed point Hamiltonian and write it in terms of the single-particle and hole excitations  HN,FO   P  P(N +1)/2  + † † −  η ( Ṽ )g g + η ( Ṽ )h h , N odd p nb nb nb p nb nb b=e,o hPn=1  nb  i = P N/2 † † + − + †  , N even. b=e,o n=1 η̂nb (Ṽp )gnb gnb + η̂nb (Ṽp )hnb hnb + η̂0b g0b g0b (3.21)  Here, gnb destroys a quasiparticle while hnb destroys a quasihole. The corresponding single particle/hole excitations are N-dependent in general but, for N > 10 (approximately), they are found to only depend on whether N is  even or odd, in which case one obtains Ṽp -dependent energy levels η̂ ± (Ṽp ) or η ± (Ṽp ) respectively. The precise numerical values of these energy levels depend on Λ and Ṽp . The Ṽp dependence is described in [19] where it was found that + − ηnb (Ṽp ) = ηnb (−Ṽp )  (3.22)  and similarly for η̂. Furthermore, since the potential scattering in the e channel is equal in magnitude but opposite in sign to that in the o channel, the above relation can be written as ± ∓ ηne (Ṽp ) = ηno (Ṽp )  50  (3.23)  and similarly for η̂. In this way, we recover a form of particle-hole symmetry even at finite Ṽp where the energy spectrum of particles in the e channel are equivalent to the spectrum of holes in the o channel and vice versa. We can now combine these single-particle/hole excitations in multi-particle/hole combinations (being sure to respect the Pauli exclusion principle), together with the four degenerate zero-energy states of the dot level and so construct the FO fixed point spectrum. The lowest such energy levels are given in Table 3.2 along with the corresponding total charge Q and total spin S quantum numbers. The spectrum for the LM fixed point is closely related to that of the FO. The corresponding NRG Wilson-chain Hamiltonian for the LM fixed point is HN,LM = Λ  (N −1)/2   XN −1 X b=e,o n=0    n † f(n+1)b + h.c. Λ− 2 ξn fnb    † † ˜ −Vp f0e f0e − f0o f0o  2 † . + lim Ũ d d − 1  (3.24)  Ũ →∞  which is identical to that for the FO fixed point with the addition of an infinite U Coulomb repulsion on the dot level. The corresponding spectrum of the LM fixed point will be the same as that for the FO fixed point with the exclusion of all of those states for which the dot level is empty or doublyoccupied as these now have an infinite energy cost. The lowest energy levels of the LM fixed point are listed in Table 3.3. To determine the spectrum of the SC fixed point, we must first identify the linear combination of electrons that screens the local moment on the quantum dot. However, as discussed, we do not transform to scattering states in the NRG and so we simply use the combination in eq. (3.1) that couples directly to the quantum dot as the screening channel, keeping the potential scattering terms in the Hamiltonian, allowing the numerics to account for those terms directly. That is, we transform the original Hamiltonian, 51  Energy 0  Num. Value 0.0000  + η1e  0.1495  − η1o  + 2η1e  − 2η1o  + − η1e + η1o  0.2990  Q 2S -1 0 0 1 1 0 0 1 1 0 1 2 2 1 0 1 -1 0 -1 2 -2 1 1 0 2 1 3 0 -1 0 -2 1 -3 0 -1 0 -1 2 0 1 0 1 0 3 1 0 1 2  Energy + − 2η1e + η1o  Num. Value 0.4485  + − η1e + 2η1o  + − 2η1e + 2η1o  0.5980  + η1o  1.3580  − η1e  Q 2S 0 1 1 0 1 2 2 1 0 1 -1 0 -1 2 -2 1 -1 0 0 1 1 0 0 1 1 0 1 2 2 1 0 1 -1 0 -1 2 -2 1  Table 3.2: The lowest energies and associated total charge Q and total spin S quantum numbers of the FO NRG fixed point of the ABI model for odd N. The numerical values for the single-particle excitation energies were obtained by diagonalizing the Hamiltonian of eq. (3.20) using a value of Ṽp = 3.0 and Λ = 2.5. All energies within a particular box are equal by eq. (3.23).  52  Energy 0 + η1e  Num. Value 0.0000 0.1495  − η1o + 2η1e − 2η1o + − η1e + η1o  0.2990  Q 2S 0 1 1 0 1 2 -1 0 -1 2 2 1 -2 1 0 1 0 1 0 3  Energy + − 2η1e + η1o  Num. Value 0.4485  + − η1e + 2η1o + − 2η1e + 2η1o + η1o − η1e  0.5980 1.3580  Q 2S 1 0 1 2 -1 0 -1 2 0 1 1 0 1 2 -1 0 -1 2  Table 3.3: The lowest energies and associated total charge Q and total spin S quantum numbers of the LM NRG fixed point of the ABI model for N odd. The single-particle energy levels are the same as in Table 3.2 using Ṽp = 3.0 and Λ = 2.5. All energies within a particular box are equal by eq. (3.23). eq. (3.1), by rotating to a basis ϕ ϕ ek + sin ok 2 2 ϕ ϕ = sin ek − cos ok 2 2  ψ1k = cos  (3.25)  ψ2k  (3.26)  so that H = vF  XZ  b=1,2  −Vp  Z  dk k  † ψbk ψbk  + Vd  Z  h  † dk dk ′ cos ϕ ψ1k ψ1k′    U 2 † dk ψ1k d + h.c. + d† d − 1 (3.27) 2 i   † † − ψ2k ψ2k′ + sin ϕ ψ1k ψ2k′ + h.c.  and take ψ1k as the screening channel. The strong-coupling fixed point involves the ψ1 (x = 0) electrons forming a singlet with the dot local moment, effectively removing the ψ1 (0) and d degrees of freedom from the dynamics and giving rise to a π/2 phase shift in the ψ1 channel. One can then apply the standard NRG transformations 53  and approximations described in § 3.1 to the resulting model in order to  obtain a Wilson chain NRG form of the SC fixed point Hamiltonian. The π/2 phase shift is implemented by shrinking the length of the ψ1 Wilson chain by one site representing the removal of the site that is entangled in the Kondo singlet. The result for the NRG SC fixed point Hamiltonian is HN,SC = Λ  (N −1)/2   N −2 X  Λ  −(n+1)/2  n=0 N −1 X  +  n=0  1    † ξn fn,1 fn+1,1 + h.c.    n † Λ− 2 ξn fn,2 fn+1,2 + h.c.  † † f0,1 + Ṽp cos ϕf0,2 f0,2 −Λ− 2 Ṽp′ cos ϕf0,1  q   † − 41 ′ −Λ Ṽp Ṽp sin ϕ f0,1 f0,2 + h.c. . (3.28)  Here, fn,1 and fn,2 are the NRG Wilson chain operators derived from ψ1 and ψ2 respectively. The differing Λ prefactors are due to the normalizations required for the two different length chains. We have also added an additional factor, Ṽp′ , which arises from the additional potential scattering term in the screening channel discussed in section 2.2. For now we simply take it as a single fitting parameter and return to its precise analysis in § 3.3. Since the  ψ2 channel does not participate in the screening of the quantum dot, we do † not expect any additional potential scattering term proportional to f0,2 f0,2 . † For the cross-term involving f0,1 f0,2 +h.c., we simply take the geometric mean of the two potential scattering terms of the two channels and find that this provides a good fit to the NRG data. To obtain the SC fixed point spectrum, we first find the single-particle energy levels by numerically diagonalizing eq. (3.28) for a finite value of N. As before, we find that for N > 10 (approximately), the energy levels depend only on the parity of N and not on its precise value. Unlike the FO and LM fixed point spectra, the resulting energy levels will depend on the flux ϕ in 54  Energy 0 ν1− 2ν1− ν2− ν2− + ν1−  Num. HN,SC 0.000 0.0709 0.1416 0.6737 0.7445  Value NRG Q 2S 0.000 1 0 0.0711 0 1 0.1422 -1 0 0.6736 0 1 0.7447 -1 0 0.7447 -1 2  Energy ν2− + 2ν1− ν1+ + ν1 + ν1− ν1+ + 2ν1− 2ν2−  Num. HN,SC 0.8153 0.8201 0.8910  Value NRG Q 2S 0.8158 -2 1 0.8203 2 1 0.8914 1 0 0.8914 1 2 0.9618 0.9625 0 1 1.3474 1.3472 -1 0  Table 3.4: The lowest energies and associated total charge Q and total spin S quantum numbers of the SC NRG fixed point of the ABI model for odd N. The NRG parameters used are Ṽp = 3.0 and ϕ = 1.047. The same parameters were used to determine the energy levels of HN,SC where a value of Ṽp′ = 2.885 was found to reproduce the NRG data. addition to the Ṽp dependence. Similar to eq. (3.21), we can write the SC fixed point Hamiltonian in terms of the single particle and hole excitations  HN,SC   P    (2N +1)/2 ν + (Ṽp , Ṽ ′ , ϕ)g † gn + ν − (Ṽp , Ṽ ′ , ϕ)h† hn n p n n p n n=1  = P(2N +1)/2  + ′ † − ′ †  ν̂ ( Ṽ , Ṽ , ϕ)g g + ν̂ ( Ṽ , Ṽ , ϕ)h h p n p n n p n n p n n=1  , N odd , N even. (3.29)  Because of the coupling of the 1 and 2 channels in eq. (3.28), the quasiparticle excitations cannot be labelled by a channel index since it is no longer a good quantum number. The full many-body spectrum is constructed by combining these singleparticle excitations in such a way as to respect Fermi statistics. The effect of the Kondo singlet, in addition to the π/2 phase shift already implemented in eq. (3.28), is simply to add an additional charge to the quantum numbers of the quasiparticle excitations due to the Fermion doing the screening that was not included in the fixed point Hamiltonian. The lowest such energies are listed in Table 3.4. Guided by the results of the transformations of section 2, we have now  55  identified the three fixed points of the Aharanov-Bohm quantum dot model and written the corresponding Hamiltonians in a Wilson chain form, eqs. (3.20), (3.24), and (3.28). This allows us to determine the fixed point spectra, the lowest values of which have been listed in Tables 3.2, 3.3, and 3.4. We are now prepared to test these predictions by comparing these spectra with the actual energy levels that are computed in the NRG. This comparison is achieved by looking at the flow of the energy levels of each HN (as defined in eq. (3.11)) for increasing N. An example is shown in Figure 3.1 where we have plotted the lowest few energy levels of the Q = 1, S = 0 subspace as a function of odd N. It is shown that the fixed point spectra predicted above are indeed approached in the appropriate regime. For example, for 5 < N < 10, all of the energies of the Q = 1, S = 0 subspace of the unstable FO fixed point are approached with the proper numerical value as given in Table 3.2. Similarly, for 19 < N < 33, the predicted energy levels of the LM fixed point (Table 3.3) are approached. The same is true for the SC fixed point where, in Table 3.4, Ṽp′ is fit in order to produce the fixed point spectrum produced by the NRG algorithm (for the parameters used to generate the NRG data, a value of Ṽp′ = 2.885 was found to give the best fit). In Figure 3.2, we show a similar plot of a single energy level as a function of odd N in the Q = 1, S = 0 subspace where the different lines indicate energies produced from different values of the flux ϕ. Here we see that, as predicted, the FO and LM fixed point energy levels that are approached are independent of ϕ whereas those of the SC fixed point are strongly flux dependent. The slight flux dependence that appears in the LM region is probably due to the fact that Ṽd is not quite zero (i.e. the LM fixed point is approached but never reached). Indeed, the flux dependence of the energy levels in this region decreases the closer the LM fixed point is approached. For a more quantitative analysis of the flux dependence of the SC fixed point, we plot the lowest NRG energy levels of the final, stable fixed point  56  Energy  2  1  0 0  20  40 N (odd)  60  80  Figure 3.1: The lowest energy levels with quantum numbers Q = 1, S = 0 as produced by the NRG as a function of odd N. The values of the predicted fixed point energies from Tables 3.2 and 3.3 are indicated by arrows on the left side and energies from Table 3.4 are indicated on the right. The parameters used to generate this plot are Γ/D = 0.0003142, U/D = 0.001, νVp = 1.05, and ϕ = 1.047. Here we see the unstable FO fixed point is approached for 5 < N < 10, the unstable LM fixed point for 19 < N < 33, and the stable SC fixed point for N > 60. with those predicted by diagonalizing the Hamiltonian of eq. (3.28) as a function of ϕ in Figure 3.3. The fact that a single parameter fit of Ṽp′ perfectly reproduces the flux dependence of the entire NRG fixed point spectrum strongly supports the validity of the above RG analysis. Indeed, because the SC fixed point is stable, we can explicitly compare the fixed point spectrum produced by the NRG with that predicted by eq. (3.28) as we have done in Table 3.4 for the first few levels. 57  1  0 π/6 π/3 π/2  Energy  0.8  0.6  0.4  0.2  20  40 N (odd)  60  80  Figure 3.2: The lowest non-zero energy level with quantum numbers Q = 1, S = 0 as produced by the NRG as a function of odd N. The different lines correspond to different values of the flux ϕ. The parameters used to generate this plot are the same as in Figure 3.1. The value of NK , related to the Kondo temperature via eq. (3.33), is indicated by the arrow and is the same for all values of ϕ.  3.2.3  Kondo Temperature from the NRG  In § 2.3.1, we derived an expression for the Kondo temperature in terms of the  inter-lead tunneling t′ , the flux ϕ, and the Fermi momentum kF , eq. (2.55). For the particle-hole symmetric value of kF = π/(2a), this expression takes the simple form of eq. (2.56). It is this latter form that can be compared to the NRG which was derived from a model with a particle-hole symmetric linear dispersion. 58  NRG data HN,SC  Energy  1.5  1  0.5  0 1  2  3 Flux φ  4  5  6  Figure 3.3: The lowest NRG energy levels of the final stable fixed point (circles) are plotted as a function of flux ϕ and compared with those given by the fixed point Hamiltonian of eq. (3.28) (solid lines). The parameters used to generate this plot are the same as in Figure 3.1 except here we use a value of νVp = 0.525. The single value of Ṽp′ was tuned in order to fit the NRG fixed point energy levels for ϕ = 0. This single parameter is able to reproduce the predicted flux dependence. To do this, we must write eq. (2.56) in terms of the Anderson model parameters appearing in the NRG Wilson chain form of the Hamiltonian, eq. (3.9), that serve as input to the NRG. First, we define an effective Kondo coupling for the continuum model of eq. (3.1) J =2  4Vd2 . U  59  (3.30)  The factor of 2 is included because eq. (3.1) involves coupling to both the even and odd channels whereas, in eq. (2.39), we defined J for the screening √ channel only. In transforming to the screening channel, a factor of 2 appears in Ṽdk resulting in the derived J acquiring a factor of 2 which we account for here explicitly so that we can compare the NRG results with those derived analytically. Using the definition of Γ, (3.3), we get νJ = 4Γ/(πU).  (3.31)  Next, we recall that Vp = −vkF kF = 2atτ ′ /π so that we can write τ ′ = πνVp . The resulting expression is ln  π 2 (νVp )2 TK = − . TK0 2νJ  (3.32)  The right hand side of this equation, together with eq. (3.31), now contains parameters related directly to the input parameters of the NRG. We now must extract the Kondo temperature from the NRG data for multiple values of Vp and J in order to confirm the validity of (3.32). The Kondo temperature is defined as the energy scale at which the screening of the local moment takes place and the Hamiltonian crosses over to the stable SC fixed point. In the NRG, TK will be related to the value of N at which the energy levels cross over from that of the LM or FO fixed point to those of the SC as described in the previous section. This value of N, which we denote NK , at which the crossover takes place can be related to a corresponding energy scale using eq. (3.17), namely kB TK ≈   1 1 + Λ−1 Λ−(NK −1)/2 D. 2  (3.33)  One simply has to extract the value of NK from the NRG energy level data in order to obtain TK . In practice, we measure NK for the lowest 20 NRG 60  2  νVp = 0.35 νVp = 0.70 νVp = 1.05  Energy  1.5  1  0.5  10  20  30 N (odd)  40  50  60  Figure 3.4: Select energy levels with quantum numbers Q = 1, S = 0 as produced by the NRG as a function of odd N. The different lines correspond to different values of the inter-lead coupling Vp and the arrows of the same line type indicate the value of NK at which the Kondo crossover takes place for each case. The parameters used to generate this plot are the same as in Figure 3.1. energy levels and use the mean value hNK i to determine TK .  In Figure 3.4, we have plotted select NRG energy levels as a function of N for different values of Vp . There is clearly a trend of increasing NK with increasing Vp which, from eq. (3.33), indicates a decrease in TK as a function of Vp as predicted in eq. (3.32). Furthermore, if one looks at Figure 3.2, there is clearly no change in the value of NK for the different values of flux ϕ. Indeed, we find no numerical evidence for variation of TK as a function of ϕ for the particle-hole symmetric value of kF = π/(2a). 61  0  -1  0  (2νJ/π ) ln(TK/TK )  -0.5  νJ = 0.400 0.255 0.127 0.072 0.051 0.025 Analytic Prediction  2  -1.5  -2  -2.5  -3 0  0.5  νVp  1  1.5  Figure 3.5: A plot of TK as extracted from the NRG (symbols) as a function of the input value of νVp . The different symbols describe data with different input values of νJ giving rise to different effective Kondo couplings. The solid line indicates the prediction described in eq. (3.32). For a more quantitive comparison, we have plotted the value of TK extracted from the NRG as a function of νVp in Figure 3.5 for multiple values of the Kondo coupling J. The analytic form predicted in eq. (3.32) provides an excellent, parameter-free fit to the numerical data.  3.3  Phase Shifts and VR  As discussed in § 2.3.2, the SC fixed point is comprised of two independent  Fermi liquids characterized by two phase shifts. These phase shifts are de62  termined by the eigenvalues of the S-matrix of eq. (2.61). In this section, we wish to compare these two predicted phase shifts with those derived from the NRG. Once again, given the particle-hole symmetric formulation of the NRG, we can only make this comparison at the special value of kF = π/(2a). In this special limit, one can see from eq. (2.13) that δ + = −δ − ≡ δ where  tan δ = τ ′ . We further simplify to the symmetric case td− = td+ = td . In this case, the two eigenvalues of the S-matrix are   √ λ± = −eiδR iA ± 1 − A2  (3.34)  A ≡ cos 2δ sin δR + sin 2δ cos δR cos ϕ.  (3.35)  with  Writing these as pure phases λ± = e2iα± , the phase shifts are given by cos 2α± = A sin δR ∓  √  1 − A2 cos δR .  (3.36)  In the special case of ϕ = 0 when the two channels fundamentally decouple, one obtains cos 2α± = ∓ cos(2δ + δR ± δR ),  (ϕ = 0)  (3.37)  or π − δ − δR , 2 = δ,  α+ =  (ϕ = 0)  (3.38)  α−  (ϕ = 0).  (3.39)  The two phase shifts α± fully define the strong-coupling fixed point spectrum.  3.3.1  Phase shifts from the NRG  First, we consider a system of two independent Fermi liquids on a finite line of length L and with linear dispersion relations. The energy levels will then 63  take the form [100] εn(i)  2πvF = L    δi q− π    (3.40)  where q ∈ Z and δi are the phase shifts in the ith channel.  The situation with the NRG is not quite so simple due to the non-uniform hopping in the Wilson chain that goes like Λ−n/2 at the nth site. However, one can still extract a sensible phase shift describing the overall shift of the (non-uniform) energy spectrum. We present a method for extracting these phase shifts from the NRG data that is similar to that used in [100] though we are much more modest about the claimed analogy between the non-uniform NRG spectrum and that of eq. (3.40). As discussed in § 3.2.2, the many-body spectrum of the strong-coupling  fixed point is built up of two channels of single-particle excitations, both of which we denoted together as νn± where the ± superscripts indicate whether the excitation is that of a particle (+) or a hole (-). With knowledge of only  the total charge Q and total spin S quantum numbers of each many-body energy from the NRG, one can identify the single-particle energy levels for ± ± each of the two channels which we denote as νn+ and νn− . It is from these that we estimate the phase shift in each channel. For clarity, let us assume that the NRG chain length N is even 2 and that − − + + the four lowest energy levels are ordered such that ν1+ < ν1− < ν1− < ν1+ , as depicted in Figure 3.6. The phase shift in each of the channels is going to be  proportional to the lowest single-particle energy level in each channel, in this − − case, ν1+ and ν1− . However, because of the non-uniform Λ-dependent spacing of the energy levels, we normalize each phase shift by the lowest energy level 2  The odd N phase shifts are related by a shift of π/2.  64  + ν1+  + ν1−  − ν1+  − ν1−  + channel  - channel  Figure 3.6: Energy level diagram of the single-particle NRG energy levels of the two channels. The shift of each relative to the Fermi energy (here indicated by the dotted line) defines the phase shift in each channel. spacing in their respective channels. That is, we define the phase shifts as α+ α−  − ν1+ = − + π ν1+ + ν1+ ν− = − 1− + π. ν1− + ν1−  (3.41) (3.42)  If the channels are shifted in the other direction relative to the Fermi energy (that is, if the lowest single-particle excitation is that of a particle instead of  65  + + − − a hole: ν1+ < ν1− < ν1− < ν1+ ), the phase shifts are taken to be + ν1+ − + π ν1+ + ν1+ ν+ = − − 1− + π. ν1− + ν1−  α+ = −  (3.43)  α−  (3.44)  ± One can now extract the values of ν1i from the many-body NRG energy spectrum obtained by diagonalizing HN as described in § 3.2.2. Assume that N is sufficiently high such that the RG has reached the strong-coupling fixed  point. The ground state, describing no particles or holes and set arbitrarily to E0 = 0, will have total spin quantum number S = 0 and a charge quantum number of Q0 = +1 or Q0 = −1 depending on whether the lowest singleparticle energy is a hole or a particle respectively. Let us assume that Q0 = − − +1 for clarity. Then, the values of ν1+ and ν1− are given by the two lowest many-body energies with a charge quantum number of Q = 0 and spin + + quantum number S = 1/2 (these lowest energies would be ν1+ and ν1− if + + Q0 = −1). The values of ν1+ and ν1− are given by the lowest many-body energies with charge quantum number Q = +2 and spin quantum number − − S = 1/2 (in the case of Q0 = −1, ν1+ and ν1− would be given by the lowest  Q = −2, S = 1/2 many-body energies). In this way, one can extract the single-particle/hole energies and estimate the phase shifts from the NRG data. As an illustration of the Λ dependence of these phase shifts, we consider the simple case of zero-flux, ϕ = 0. In this case, the original Hamiltonian can be completely decoupled into two separate channels and so the two channels operate completely independently. The channel coupled to the quantum dot is the screening channel and so obtains a π/2 phase shift in addition to that given by −δ whereas the other channel is non-interacting with only a potential scattering phase shift δ. This can be seen clearly in Figure 3.7 where we have plotted the two phase shifts as a function of Vp . 66  1.5  Phase Shift α  Non-screening channel Λ=2.0 Λ=2.5 Λ=3.0 Λ=3.5 NRG Λ=2.0 NRG Λ=2.5 NRG Λ=3.0 NRG Λ=3.5  1  0.5 Screening channel  0 0  0.2  0.4  0.6 νVp  0.8  1  Figure 3.7: The phase shift of the two channels as a function of Vp from the NRG data with an effective Kondo coupling νJ = 0.255 and zero flux. In this case, the two channels are independent with the screening channel phase shift starting at π/2 for νVp = 0 and the non-screening channel phase shift starting at 0 for νVp = 0. The symbols are the phase shifts obtained from the full NRG many-body energy levels. The ascending solid lines are the phase shifts obtained from the single-particle energy levels of a single noninteracting Wilson chain with potential scattering Vp and the descending solid lines are π/2 minus the ascending lines. The solid lines do not take into account the small correction due to the additional potential scattering VR that occurs in the screening channel.  67  The most striking feature of Figure 3.7 is the different Λ dependence in the phase shift of the screening and non-screening channels obtained from the NRG data. To help understand this, we have plotted as solid lines the phase shifts that one would expect in a non-screening and screening Wilson chain (we ignore the effects of the small correction due to VR for now). For the non-screening channel, one can diagonalize directly the Wilson chain Hamiltonian with a potential scattering Vp at the first site using different values of Λ and so obtain the single-particle energy spectra directly without having to perform the NRG. From this direct single-particle spectra one can define the phase shift as described above and these are plotted as the solid ascending lines. As can be seen, these match perfectly the phase shifts in the non-screening channel obtained from the NRG data, as they must. To leading order (again, neglecting VR ), one might expect the phase shift in the screening channel to be simply π/2 minus the above Λ-dependent phase shifts since the potential scattering in the screening channel is equal in magnitude but opposite in sign to that in the non-screening channel. We have plotted this expectation as the descending solid lines in the Figure. On the contrary, the phase shifts obtained directly from the NRG data show very little Λ dependence compared with the non-screening channel. The precise reason for this is unknown though it may be due to a similar Λ dependence in the additional potential scattering VR as shown in Figure 3.12 that is compensating for the Λ dependence of the bare δ phase shift. Despite this, we nevertheless obtain good support for our prediction of the phase shifts from the tight-binding model. In the remainder of our analysis, we will use the Λ dependent phase shift obtained from diagonalizing the potential scattering Wilson chain discussed above for the bare phase shift δ generated by Vp that appears in eqs. (3.35), (3.38), and (3.39). See [19] for more information on the Vp dependence of the NRG spectrum.  68  3.3.2  NRG evidence for VR  We now turn our attention back to the additional potential scattering VR that was derived in section 2.2. Having shown that the phase shifts can be extracted from the NRG, we can now compare the predicted phase shifts in eq. (3.36) with those of the NRG. For simplicity, we continue to assume td− = td+ = td and εd = −U/2. To compare our analytic results with those of the NRG, we use the same correspondence as was used in section 3.2.3, namely τ ′ = πνVp and νJ = 4Γ/(πU). We focus first on the case of zero flux, ϕ = 0, where the phase shifts take an especially simple form given in eqs. (3.38) and (3.39). These two phase shifts are plotted in Figure 3.8 as a function of Vp where the symbols indicate those values derived from the NRG data while the lines are the analytic prediction from the tight-binding model. Here we see that, indeed, only the phase shift of the screening channel (the one that obtains π/2 when Vp = 0) deviates from the VR = 0 prediction, indicating that an additional phase shift is generated in the screening channel only. However, VR provides only a small correction so it is easier to extract VR from the NRG phase shifts and compare its functional form directly with that of eq. (2.50). To extract VR , we take the arctan of the derived NRG phase shift and subtract from that the π/2 contribution arising from the Kondo screening as well as the bare phase shift δ due to Vp . This latter phase shift will be Λ dependent and can be calculated numerically as described in [19]. In Figure 3.9, we compare directly the predicted dependence of VR on Vp with that derived from the NRG phase shifts for various values of J. We find that both analytic and numeric calculations of VR share the same qualitative behaviour, peaking around νVp ≈ 0.3 (corresponding to t′ ≈ t in  the original tight-binding model), but that precise quantitative agreement is not obtained. The nature of this disagreement is discussed in section 4.  We next turn our attention to testing the (νJ)2 dependence in eq. (2.50) by plotting the value of VR as determined from the NRG phase shifts versus 69  Phase Shifts α  1.5  VR = 0 νJ = 0.191 NRG νJ = 0.191 0.255 NRG 0.255 0.318 NRG 0.318  1  0.5  0 0  0.2  0.4  0.6 νVp  0.8  1  1.2  Figure 3.8: The phase shifts of the two channels as a function of Vp . The symbols denote phase shifts derived from the NRG data while the lines are the analytic predictions. The solid black line is the curve expected if there is no additional potential scattering (i.e. VR = 0). We have set ϕ = 0 to generate this plot. νJ in Figure 3.10. The most striking characteristic is the apparent deviation from universal behaviour as U/D approaches unity. We see that this trend is captured by the U dependence in eq. (2.50) but that precise quantitative agreement is elusive, perhaps because of the presence of a cubic term which we do not consider. A complete analysis of VR with an Anderson impurity rather than reducing, via the Schrieffer-Wolff transformation, to one with a spin impurity may elucidate the nature of this behaviour. For further analysis, we fit the largest data set with U/D = 0.001 to a 70  0  νVR  -0.01  νJ=0.0955 NRG νJ=0.0955 0.127 NRG 0.127 0.191 NRG 0.191 0.254 NRG 0.254 0.318 NRG 0.318  -0.02  -0.03  0  0.2  0.4  0.6 νVp  0.8  1  Figure 3.9: The additional potential scattering VR as derived from the NRG phase shifts (symbols) and from the analytic tight-binding model (lines) for various values of the effective Kondo coupling J. We have set ϕ = 0 to generate this plot. third degree polynomial of the form VR = a0 + a1 (νJ) + a2 (νJ)2 + a3 (νJ)3 .  (3.45)  A third degree polynomial was chosen instead of a second degree function because the data goes to quite large values of νJ where we expect our second order analysis to break down. The values of the parameters are tabulated in Table 3.5. It is seen that the coefficients a0 and a1 , which we predict to vanish, are indeed at least an order of magnitude lower than the quadratic and cubic coefficients. Doing another fit neglecting these first two terms, 71  0  νVR  -0.05  -0.1  U/D = 0.0001 0.001 0.01 0.1 1.0 0.001 [full cubic fit] 0.001 [quadratic/cubic only fit] 2  0.1 [(νJ) prediction] 2  1.0 [(νJ) prediction] 0.1  0.2  0.3  νJ  0.4  0.5  0.6  Figure 3.10: The additional potential scattering as determined from the NRG phase shifts as a function of effective Kondo coupling J. Each different symbol uses a fixed value of the dot Coulomb repulsion U while varying Γ such that the range of J, given in eq. (3.31), is roughly the same for each iteration. The solid line presents the best fit third degree polynomial to the U/D = 0.001 points. In this data, νVp = 0.3 and ϕ = 0. that is, to a form VR = b2 (νJ)2 + b3 (νJ)3  (3.46)  gives b2 = −0.42 which is the same order of magnitude as the value of −0.24 predicted by eq. (2.50). Up until this point, we have been focussing primarily on the form of the additional potential scattering VR and so, for simplicity, have taken the flux ϕ = 0. In Figure 3.11, we have plotted the phase shifts α± versus the flux ϕ  as derived from the NRG with comparison to the predicted form described 72  Coefficient a0 a1 a2 a3 b2 b3  Value -0.00081 0.045 -0.62 0.50 -0.42 0.29  Table 3.5: The parameters for the best fit of eqs. (3.45) and (3.46) to the U/D = 0.001 data in Figure 3.10.  Phase Shift α  1.5  α+ NRG α- NRG  1  0.5  1  2  3 Flux φ  4  5  6  Figure 3.11: The phase shifts of the strong-coupling fixed point as determined from the NRG (symbols) and compared with that predicted in eq. (3.36) (lines). Here, the effective Kondo coupling is νJ = 0.191 and νVp = 0.25. 73  0  νVR  -0.005  -0.01  -0.015  Λ=2.5 Λ=3.0 Λ=3.5  -0.02  0  0.2  0.4  0.6 νVp  0.8  1  1.2  Figure 3.12: The value of the additional potential scattering VR as derived from the NRG for various values of Λ, all using a value of νJ = 0.254. in eq. (3.36). There we find the agreement to be quite good and suggests that our predicted flux dependence is robust. Finally, we note that, although it seems that the Λ dependence of the screening channel phase shift is suppressed (see Figure 3.7), there does appear to be some systematic Λ dependence in VR itself as seen in Figure 3.12. In order to take this effect into account, one would need to derive an expression for VR from the Wilson chain Hamiltonian, eq. (3.9), as opposed to the much simpler tight-binding model as was done in eq. (2.50). Nevertheless, we still find convincing agreement of the behaviour of VR between that derived analytically and from the NRG despite this apparent Λ dependence.  74  Chapter 4 Discussion We have presented a systematic study of a minimal model of an AharanovBohm interferometer with an embedded quantum dot connected to two conducting leads. Although aspects of such a model and ones similar to it have been studied by other groups in the past [72, 73, 74, 75, 76, 77, 78, 79], our work provides a complete picture of the physics of such a system when the quantum dot chemical potential ǫd is near −U/2 and the system is in the Kondo regime, including new effects not discussed previously. In particular, we have elucidated precisely how the Kondo effect arises in such a system by identifying the screening channel; we have completely mapped out the renormalization group flow of the system and its dependence on flux ϕ and inter-lead tunneling t′ ; we have calculated the dependence of the Kondo temperature and conductance on the same parameters as well as, for the first time, the electron density in the leads (via the factors of kF appearing throughout); we have calculated the effects of additional potential scattering that arises from the breaking of particle-hole symmetry; we have provided wide numerical support from the NRG for many of our findings that goes beyond simply computing the occupancy of the quantum dot as in [73] or the dot density of states [63]. Although our work is quantitatively precise, the physical picture that arises has been stated in simple physical terms that 75  fully describes the zero-temperature properties. It is interesting to compare our results for the Kondo temperature TK with those of [86], [84], and [79]. The first reference, which calculates the Kondo temperature in a manner similar to that described in § 2.3.1, finds  much different behaviour for TK compared to our eq. (2.55). However, the authors of [86] consider a much different model involving an ABI with a quantum dot embedded in a ring much larger than the Kondo screening cloud of the dot. Hence, there is no discrepancy with our calculation. In the second reference [84], the authors use a slave boson mean field theory to estimate the Kondo temperature for variable sized rings. For the smallest configuration with only one site in the ring in addition to the quantum dot, they find a flux dependent Kondo temperature assuming particlehole symmetric leads, kF = π/(2a). Although the calculation of [84] was for a different model than that considered here, the two models are quite close and the nature of this apparent discrepancy is not clear. It is interesting to note that the authors find very similar behaviour at kF = π/(2a) to that found by us for electron densities less than half-filled, kF < π/(2a) (see Figure 2.3). It may be that the particle-hole symmetry breaking caused by moving away from half-filling in our calculation mimics the particle-hole symmetry breaking caused by the negative on-site energy of the additional site in the ring used in [84]. Perhaps it is this type of particle-hole symmetry breaking that leads to a flux dependent Kondo temperature. This is speculation and further analysis of both methods would be required to resolve this apparent discrepency. Reference [79] follows a very similar procedure to that used here, transforming to the scattering basis and identifying the screening channel. However, they mainly consider the U → ∞ limit with finite dot energy level εd . Their subsequent scaling analysis, assuming half-filled leads with particlehole symmetric Fermi energy εF = 0, produces a flux dependent Kondo temperature. Although this seems to contradict our conclusion that the Kondo  76  temperature is flux independent at half-filling, our result was obtained in a much different limit, with εd ≈ −U/2. The authors do claim that, for finite U, the flux dependence is suppressed (though still present) when εd = −U/2. However, we find no evidence of any flux dependence in the Kondo temperature when kF = π/(2a). We now turn to the topic of the apparent discrepancies presented in the NRG evidence for the additional potential scattering VR . As discussed in the text, we expect there to be cubic and higher order contributions to VR that we do not calculate so discrepencies for values of νJ that approach unity should be expected. However, discrepancies remain even for relatively small values of νJ and we offer here some possibilities for why this might be. As written at the end of section 3.3.2, the correspondence between the tight-binding model used to derive VR in eq. (2.50) and that used in the NRG is only approximate, especially for values of Λ > 1. This leads to artificial Λ dependence in many of the quantities extracted from the NRG as has been presented above. This is probably true for the value of VR extracted from the NRG, as seen in Figure 3.12, suggesting that the form of VR may be non-universal in that it may depend on the details of the band structure of the leads. To explore the universality of the form of VR , we have repeated the derivation of VR for a model with a linear dispersion in the leads rather than the tight-binding cosine dispersion presented in the text. It was found that, while qualitatively the same as the form of VR in eq. (2.50), the two forms of VR did differ in numerical details. From this we conclude that the form of VR is non-universal. In light of this fact, one would ideally repeat the calculation of VR , not for the tight-binding chain presented but for the full Λ-dependent Wilson chain and so obtain the Λ dependence of VR . However, given the non-uniform ‘tunneling amplitudes’ in the Wilson chain that go as Λ−n/2 for hopping from the nth site, such a calculation would be much more difficult. Another possible source for this discrepancy is the possibility of addi-  77  tional contributions to potential scattering arising from the Schrieffer-Wolff transformation. We have performed such a transformation to second order in Ṽdk and concluded that the potential scattering KkF kF that arises vanishes when εd = −U/2 so that, in this regime, VR contributes to the lead-  ing order term in the potential scattering. However, given the fact that a non-zero t′ breaks particle-hole symmetry, there is nothing preventing the Schrieffer-Wolff transformation from generating a potential scattering term that is fourth order in Ṽdk (equivalently, second order in J). It would be interesting though non-trivial to carry out the Schrieffer-Wolff transformation to higher orders to see if indeed such potential scattering terms are present and if they can account for the disagreement with the NRG. Despite all of these possibilities, it is clear that such a VR term is present in both the tight-binding model as well as in the NRG and that they share the same qualitative behaviour and modestly agree quantitatively. Given this, we expect such a VR term to be present in any real physical system and we expect it to share the same qualitative dependence on flux ϕ, inter-lead tunneling t′ (peaking around t′ ≈ t) and on electron density in the leads via the dependence on kF but do not claim that it will be precisely as that given in eq. (2.50) which is based on an overly simplified tight-binding model. Furthermore, although present, the contribution of VR to the conductance is very small for typical values of νJ, as seen in Figure 2.7, and so will probably be difficult to detect explicitly in any physical system. Nevertheless, the remainder of our analysis is robust and confirmed numerically and provides a framework in which to think about such quantum dot systems as well as providing new insights into the influence of interference on the Kondo effect. We conclude with a discussion of the relationship between the theoretical work presented in this thesis and various systems that are studied experimentally. As discussed in § 1.3, all of the Aharanov-Bohm interferometers studied experimentally involve conduction paths of finite length. This is in contrast to the simpler tunnel junctions assumed in the theoretical model  78  Vg  QD Right Lead  Left Lead  Vp  Figure 4.1: A schematic representation of the geometry for a proposed experiment closely related to the model studied. The shaded areas are the metallic gates on top of a two dimensional electron gas with the quantum dot indicated by QD. The horizontal pathway acts as a quantum point contact, allowing only a single channel of electrons. The gate Vg tunes the energy levels of the dot whereas Vp creates a potential barrier in the middle of the direct horizontal channel through which the electrons can tunnel. that we studied. Furthermore, the width of the conduction paths in typical experimental interferometers allows for multiple conduction channels rather than the single, one-dimensional conduction channels assumed in our study. The simpler model was chosen so as to elucidate the fundamental effects of interference on the manifestation of the Kondo effect in quantum dots without being obscured by too many details. At the same time, such a model poses an obstacle to direct quantitative comparison of our results with experimental observations of the typical interferometer devices currently studied. However, it has been proposed [101] that gate geometries could be constructed on a semiconductor heterostructure that realize the tunnel junction connections assumed in this thesis. An example of one such proposed device is shown in Figure 4.1. There, the horizontal channel acts as a quantum point contact, allowing only a single channel of electrons. 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(A.2) the definition of qαk in terms of e†k and o†k from  eqs. (2.22)–(2.23) and using the relations Z π i h a † † H, ek = εk ek + dk ′ vk′ k e†k′ 0 Z π i h a H, o†k = εk o†k − dk ′ vk′ k o†k′ 0  88  (A.3) (A.4)  one obtains (εk −  ±(k) εk ′ ) φk ′  =± ±(k)  Substituting now the definition of φk′  Z  π a  0  dq φ±(k) vk′ q . q  (A.5)  from eq. (2.24) in the above expres-  ± sion gives an integral equation for Tkk ′  ± Tkk vkk′ + ′ = ±  Z  π a  0  ± vkq Tqk ′ dq εk′ − εq + iη  !  .  (A.6)  If we now take the ansatz ± ± Tkk ′ = Tk ′ sin ka  (A.7)  and substitute this into eq. (A.6) together with the definition of vkk′ from eq. (2.21) and εk = −2t cos ka, one obtains the following equation for Tk Tk± = ∓  2t′ a sin ka ′ π 1 ± τπ Ik  (A.8)  where, as before, τ ′ ≡ t′ /t and Ik is the dimensionless integral 1 Ik ≡ 2  Z  π  −π  dy  sin2 y cos y − cos ka + iη  (A.9)  This integral can be solved in the complex plane. Making the change of variables z = eiy , one can write this as 1 Ik = − 4i  I  2  (z 2 − 1) dz 2 2 z (z − 2z (cos ka − iη) + 1)  (A.10)  where the contour of integration is the unit circle centered at the origin in the complex z plane. The integrand has poles at z = z0 = 0 (second order) and at z = z±  89  (simple), the latter given by z± = cos ka ± i| sin ka| − iη ∓ η  cos ka . | sin ka|  (A.11)  Since η is a positive infinitesimal quantity, one can show that |z+ | < 1 whereas |z− | > 1 so that only the z+ and z0 poles lie within the contour. Applying the residue theorem π Ik = − (Res(z = z0 ) + Res(z = z+ )) 2 = πe−ika .  (A.12) (A.13)  Substituting this final value back into eq. (A.8) and the resulting Tk back ± into the ansatz, eq. (A.7), produces the promised form of Tkk ′ ± Tkk ′ = ∓  2t′ a sin ka sin k ′ a π 1 ± τ ′ e−ik′ a  (A.14)  as stated in eq. (2.25). Finally, we compute the functional form of Γ± k as defined and stated in ±(k) eq. (2.29). Substituting the definition of φk′ as defined in eq. (2.24) into ± the definition of Γ± k and using the derived form of Tkk ′ , eq. (A.14), gives  Γ± k  Z  π a  ±(k)  dk ′ sin k ′ a φk′ 0   τ′ Ik = sin ka 1 ∓ π 1 ± τ ′ e−ika  ≡  (A.15) (A.16)  where Ik is the same dimensionless integral defined in eq. (A.9) and computed in eq. (A.13). Hence, the final result is obtained upon substitution Γ± k =  sin ka 1 ± τ ′ e−ika  as reported in eq. (2.29). 90  (A.17)  Appendix B Potential scattering phase shift B.1  Lattice Model  Consider a single semi-infinite tight-binding chain with an on-site potential at the first site  ∞   X e†j ej+1 + h.c. − t′ e†1 e1 . H = −t  (B.1)  j=1  This is the same as the even-channel Hamiltonian H = H0 +H−+ of eqs. (2.17)–  (2.18) in the limit td+ = td− = 0. The presence of a finite t′ will give rise to a phase shift in the single-particle wave function and it is the calculation of this phase shift that is the subject of this appendix. We write the eigenvectors of the Hamiltonian as |φi =  ∞ X j=1  φj e†j |0i  (B.2)  where {φj } are coefficients to be determined such that they satisfy the  Schrödinger equation  H |φi = εk |φi  (B.3)  with εk = −2t cos ka. The Schrödinger equation can be written as the fol91  lowing series of algebraic equations −tφ2 − t′ φ1 = εk φ1 −t (φj−1 + φj+1 ) = εk φj  (B.4) , j > 1.  (B.5)  The equations in the second line can be solved by taking coefficients of the form φj = sin (kja + δk ) (B.6) and δk is determined by eq. (B.4) to be tan δk =  τ ′ sin ka 1 − τ ′ cos ka  (B.7)  where, as before, τ ′ = t′ /t. This is the form of the phase shift δk+ that occurs in the even channel. The phase shift in the odd channel is the same but with τ ′ → −τ ′ so that  B.2  tan δk±  τ ′ sin ka . =± 1 ∓ τ ′ cos ka  (B.8)  Continuum Model  At low energies (long wavelengths), one can take the continuum limit of the tight-binding model and linearize the dispersion relation about kF . In this way, one can write an approximate real-space Hamiltonian H = vF  Z  ∞  dx ψ † (x)(−i∂x )ψ(x) + VR ψ † (0)ψ(0)  (B.9)  −∞  where vF is the Fermi velocity. We assume that νVR ≪ 1 where ν is the density of states at the Fermi energy. As in the lattice model, we introduce eigenvectors of the Hamiltonian |φk i =  Z  ∞ −∞  dxφk (x)ψ † (x) |0i 92  (B.10)  which satisfy the Schrödinger equation H |φk i = vF k |ki .  (B.11)  This puts the following condition on the functions φk (x) −ivF ∂x φk (x) + VR φk (0)δ(x) = vF kφk (x).  (B.12)  We now take the ansatz  with derivative   i(kx+δR )   e φk (x) = ei(kx−δR )   cos δR  i(kx+δR )   ike ∂x φk (x) = ikei(kx−δR )   2iδ(x) sin δR  ,x > 0 ,x < 0  (B.13)  ,x = 0  ,x > 0 ,x < 0 , x = 0.  (B.14)  In order for φk (x) to satisfy eq. (B.12), we require tan δR = −  VR = −πνVR . 2vF  93  (B.15)  Appendix C Implementation of the Numerical Renormalization Group The numerical renormalization group (NRG) procedure described in § 3.1 in-  volves iterative diagonalization of the Hamiltonians HN defined in eq. (3.11). By “iterative”, we mean that, starting with N = 0, HN is diagonalized to give a set of eigenvalues E (N ) (n) and associated eigenvectors |niN . A basis for the  larger Hilbert space of HN +1 is then defined by taking the outer product of |niN with the 16 states ΩN +1 defined by acting on the ground state |0i with all  unique combinations of the set of operators {fN† +1,e,↑, fN† +1,e,↓ , fN† +1,o,↑ , fN† +1,o,↓} (we reinstitute the spin index for the remainder of this appendix). One then N defines matrix elements of HN +1 in the expanded Hilbert space {|niN }n ΩN +1  and numerically diagonalizes the resulting matrix to produce eigenvalues E (N +1) (n) and associated eigenvalues |niN +1 and the procedure is repeated.  Under this process, the successive Hilbert spaces grow exponentially with N as 16N . In order to keep the size of the resulting matrices at a scale that can be numerically diagonalized in a reasonable amount of time, only the states of HN with the lowest M eigenvalues are used to define the expanded 94  Hilbert space of HN +1 . We used a value of M = 3000 for all of the data shown in this thesis though this represents more than 3000 states due to the symmetry reduction described below (i.e. some of these 3000 states are one representative of an n-tuple of equivalent states that are related by symmetry). In many cases, we have repeated the calculation with M = 3500–4000 and found no consequential change in the numerical results. In this way, we are confident that the approximation of truncating the Hilbert space in this way is valid. The motivation and explanation for the validity of this approximation is closely tied to the logarithmic discretization of the energy states for Λ > 1 as described in [17]. The size of the matrices needing to be diagonalized can be reduced substantially by taking advantage of the symmetries of the Hamiltonians HN . In addition to this gain in numerical efficiency, keeping track of the symmetries of the energy eigenstates allows one to label each state by their respective quantum numbers and so aid in identifying the various fixed points of the model as has been done throughout chapter 3. The procedure described below is very similar to that used for the twoimpurity Kondo problem [96, 97, 98]. The total charge and spin operators can be defined as QN =  N  X XX  µ=↑,↓ b=e,o n=0  ~N = S  N X XX    † fnbµ fnbµ − 1 + d†µ dµ − 1  † fnbµ  µ,ν=↑,↓ b=e,o n=0  ~σµν ~σµν fnbν + d†µ dν 2 2  (C.1)  (C.2)  where ~σµν are the Pauli matrices. Both h i of these operators are conserved ~ 2 , HN = [S z , HN ] = 0. As a result, the quantities in that [QN , HN ] = S N N  quantum numbers of total charge Q, total spin S, and z component of the total spin S z can be used to label the eigenvalues and eigenvectors of HN .  Organized in this way and carrying out the iterative diagonalization procedure described above, HN +1 will not have matrix elements between states 95  |iiN +1 |1iN +1 |2iN +1 |3iN +1 |4iN +1 |5iN +1 |6iN +1 |7iN +1  |8iN +1  |9iN +1  |10iN +1  |11iN +1 |12iN +1 |13iN +1 |14iN +1 |15iN +1 |16iN +1  Definition |0i fN† +1,e,↑ |0i fN† +1,e,↓ |0i fN† +1,o,↑ |0i fN† +1,o,↓ |0i fN† +1,e,↑ fN† +1,e,↓ |0i † fN† +1,o,↑  fN +1,o,↓ |0i  Q 2S -2 0 -1 1 -1 1 -1 1 -1 1 0 0 0 0   √1 fN† +1,e,↑ fN† +1,o,↓ − fN† +1,e,↓ fN† +1,o,↑ |0i 2 † fN† +1,e,↑   fN +1,o,↑ |0i † † † † √1 fN +1,e,↑ fN +1,o,↓ + fN +1,e,↓ fN +1,o,↑ |0i 2 † fN +1,e,↓ fN† +1,o,↓ |0i fN† +1,e,↑ fN† +1,o,↑ fN† +1,o,↓ |0i fN† +1,e,↓ fN† +1,o,↑ fN† +1,o,↓ |0i fN† +1,e,↑ fN† +1,e,↓ fN† +1,o,↑ |0i fN† +1,e,↑ fN† +1,e,↓ fN† +1,o,↓ |0i fN† +1,e,↑ fN† +1,e,↓ fN† +1,o,↑ fN† +1,o,↓ |0i  2S z 0 1 -1 1 -1 0 0  0  0  0  0  2  2  0  2  0  0 1 1 1 1 2  2 1 1 1 1 0  -2 1 -1 1 -1 0  Table C.1: Definition of the states |iiN +1 with their associated values of (Q, S, S z ). with different (Q, S, S z ) values so that the matrices in each subspace can be diagonalized independently. Furthermore, the resulting energy eigenvalues are independent of S z so that one can avoid keeping track of this index by taking advantage of the Wigner-Eckart theorem [102, 103, 104] as described below. Let |Q, S, S z ; ωiN be the eigenvector of HN in the (Q, S, S z ) subspace labelled by ω with eigenvalue E (N ) (Q, S; ω) (as mentioned earlier, the eigenvalues are independent of S z ). We can organize the 16 states spanned by the creation operators {fN† +1,b,µ }b=e,o;µ=↑,↓ into states of definite (Q, S, S z ) labelled by {|iiN +1 }16 i=1 . These are tabulated in Table C.1. Next, we take the outer product of each of these states with each of the 96  |Q, S, S z ; ωiN states and organize them into states of definite (Q, S, S z ) using  the necessary Clebsch-Gordon coefficients for adding spins S ⊗ 21 and spins S ⊗ 1 described in appendix D. The resulting states are written as |Q, S; ω, iiN +1 =  3 X j=1  aij (S) |nij iN +1 Qi , Si , Sijz ; ω  (C.3)  N  and tabulated in Tables C.2 and C.3 from which the coefficients aij (S) and index functions nij , Qi , Si and Sijz can be inferred.1 Note that, for reasons that will become clear below, we only keep those product states with S z = S and so drop the redundant S z label on the left hand side of eq. (C.3) (i.e. S z = S is implied). It is in this basis that we shall define matrix elements of HN +1 by taking matrix elements of eq. (3.15). The diagonal elements are simply given by N +1  hQ, S; ωi| HN +1 |Q, S; ωiiN +1 = Λ1/2 E (N ) (Qi , Si ; ω)  (C.5)  where we have used the fact that HN Qi , Si , Sijz ; ω  N  = E (N ) (Qi , Si ; ω) Qi , Si , Sijz ; ω  N  .  (C.6)  The off-diagonal elements can be written as N +1  hQ, S; ωi| HN +1 |Q, S; ω ′i′ iN +1 X aij (S)ai′ j ′ (S)φ(ni′ j ′ ) N +1 hnij | fN† +1,b,µ |ni′ j ′ iN +1 = 2ξN j,j ′ ,b,µ  ×N Qi , Si , Sijz ; ω fN bµ Qi′ , Si′ , Siz′ j ′ ; ω ′  N  (C.7)  where φ(n) is the sign accumulated by commuting fN,b,µ through |niN +1 p For p example, from the fourth line of Table C.2, we infer: a41 = (2S + 1)/(2S + 2), z = S + 1/2, a42 = − 1/(2S + 2), a43 = 0, n41 = 3, n42 = 2, Q4 = Q + 1, S4 = S + 1/2, S41 z and S42 = S − 1/2. 1  97  |Q, S; ω, 1iN +1 = |1iN +1 |Q + 2, S, S; ωiN |Q, S; ω, 2iN +1 = |2iN +1 |Q + 1, S − 1/2, S − 1/2; ωiN |Q, S; ω, 3iN +1 = |4iN +1 |Q + 1, S − 1/2, S − 1/2; ωiN r 2S + 1 |3i |Q + 1, S + 1/2, S + 1/2; ωiN |Q, S; ω, 4iN +1 = 2S + 2 N +1 r 1 − |2i |Q + 1, S + 1/2, S − 1/2; ωiN 2S + 2 N +1 r 2S + 1 |5i |Q + 1, S + 1/2, S + 1/2; ωiN |Q, S; ω, 5iN +1 = 2S + 2 N +1 r 1 − |4i |Q + 1, S + 1/2, S − 1/2; ωiN 2S + 2 N +1 |Q, S; ω, 6iN +1 = |9iN +1 |Q, S − 1, S − 1; ωiN |Q, S; ω, 7iN +1 = |6iN +1 |Q, S, S; ωiN |Q, S; ω, 8iN +1 = |7iN +1 |Q, S, S; ωiN Table C.2: Definition of the states |Q, S; ω, iiN +1 for i = 1 . . . 8 from which the coefficients aij (S) and index functions nij , Qi , Si and Sijz of eq. (C.3) can be inferred. from the left. The factors N +1 hnij | fN† +1,b,µ |ni′ j ′ iN +1 are easy to compute from Table C.1 so that all that remains is to compute the matrix elements of fN bµ in terms of the eigenvectors of HN . Efficiency can be gained by noting that we can write matrix elements with S 6= S in terms of those with S z = S using the Wigner-Eckart theorem [102, z  103, 104], stated here as N  Qi , Si , Sijz ; ω fN bµ Qi′ , Si′ , Siz′ j ′ ; ω ′ N D E z = ψSijz ,µ Si′ , Si′ j ′ N hQi′ , Si′ ; ω ′| | fN† b | |Qi , Si ; ωiN (C.8)  E D where ψSijz ,µ Si′ , Siz′ j ′ are Clebsch-Gordon coefficients described in appendix D 98  |Q, S; ω, 9iN +1 = |8iN +1 |Q, S, S; ωiN r S |Q, S; ω, 10iN +1 = |10iN +1 |Q, S, S; ωiN S+1 r 1 |9i |Q, S, S − 1; ωiN − S + 1 N +1 r 2S − 1 |11iN +1 |Q, S + 1, S + 1; ωiN |Q, S; ω, 11iN +1 = 2S + 1 s 2S − 1 − |10iN +1 |Q, S + 1, S; ωiN S(2S + 1) s 1 |9i |Q, S + 1, S − 1; ωiN + S(2S + 1) N +1 |Q, S; ω, 12iN +1 = |12iN +1 |Q − 1, S − 1/2, S − 1/2; ωiN |Q, S; ω, 13iN +1 = |14iN +1 |Q − 1, S − 1/2, S − 1/2; ωiN r 2S + 1 |Q, S; ω, 14iN +1 = |13iN +1 |Q − 1, S + 1/2, S + 1/2; ωiN 2S + 2 r 1 |12iN +1 |Q − 1, S + 1/2, S − 1/2; ωiN − 2S + 2 r 2S + 1 |15iN +1 |Q − 1, S + 1/2, S + 1/2; ωiN |Q, S; ω, 15iN +1 = 2S + 2 r 1 |14iN +1 |Q − 1, S + 1/2, S − 1/2; ωiN − 2S + 2 |Q, S; ω, 16iN +1 = |16iN +1 |Q − 2, S, S; ωiN (C.4) Table C.3: Definition of the states |Q, S; ω, iiN +1 for i = 9 . . . 16.  99  and  N  hQi′ , Si′ ; ω ′| | fN† b | |Qi , Si ; ωiN are invariant matrix elements.  In practice, we only require 4 such non-zero matrix elements which, from eq. (C.8), can be written as (we drop the N subscripts here for clarity) † hQ + 1, S + 1/2; ω ′| | fN† b | |Q, S; ωi = hQ + 1, S + 1/2; ω ′| fb↑ |Q, S; ωi (C.9) r 2S + 1 † hQ + 1, S − 1/2; ω ′| | fN† b | |Q, S; ωi = hQ + 1, S − 1/2; ω ′| fb↓ |Q, S; ωi 2S  where we have used the Clebsch-Gordon coefficients of eqs. (D.11) and (D.14) (recall that it is implied that S z = S for the states |Q, S; ωiN ). We recall that  the vectors |Q, S; ωiN are the eigenvectors of HN obtained by diagonalizing HN in the |Q, S; ω, iiN basis and so can be written |Q, S; ωiN =  X δ,i  (N )  UQ,S (ω; δ, i) |Q, S; δ, iiN .  (C.10)  Focusing on the matrix element in eq. (C.9), we substite eq. (C.10) on the right hand side and use eq. (C.3) for the |Q, S; δ, iiN vectors. After computing  the resulting inner products we obtain N  hQ + 1, S + 1/2; ω ′| | fN† b | |Q, S; ωiN X (N ) (N ) = aij (S)ai′ j ′ (S + 1/2)UQ,S (ω; δ, i)UQ+1,S+1/2 (ω ′; δ, i′ ) δ,i,i′ ,j,j ′  ×N hni′ j ′ | fN† b↑ |nij iN  (C.11)  Similarly, we obtain N  hQ + 1, S − 1/2; ω ′| | fN† b | |Q, S; ωiN r 2S + 1 X (N ) (N ) aij (S)ai′ j ′ (S − 1/2)UQ,S (ω; δ, i)UQ+1,S−1/2(ω ′ ; δ, i′ ) = 2S ′ ′ δ,i,i ,j,j  ×N hni′ j ′ | fN† b↓ |nij iN .  100  (C.12)  We are now equipped with everything that we need to write our final expression for the off-diagonal matrix elements. Substituting eq. (C.8) into eq. (C.7), we obtain N +1  hQ, S; ωi| HN +1 |Q, S; ω ′i′ iN +1 Xh Wiib ′ (S) N hQi′ , Si′ ; ω ′| | fN† b | |Qi , Si ; ωiN = ξN b  +Wib′ i (S) N hQi , Si ; ω| | fN† b | |Qi′ , Si′ ; ω ′iN  i  (C.13)  where Wiib ′  ≡  X  D  aij (S)ai′ j ′ (S)φ(ni′ j ′ ) ψ  j,j ′ ,µ  The matrix elements of HN +1  z ,µ Sij  Si′ , Siz′ j ′  E  N +1  hnij | fN† +1,b,µ |ni′ j ′ iN +1 .  (C.14) are now fully defined by eqs. (C.5) and (C.13)  in terms of factors that are known or easily computed. The eigenvalues E (N ) (Q, S; ω) are known from the previous iteration; the factors of aij (S) and the index functions nij , Si , and Sijz can be inferred from eq. (C.3) and Tables C.2 and C.3; the factors of φ(n) and N +1 hnij | fN† +1,b,µ |ni′ j ′ iN +1 are easily computed from Table C.1; the Clebsch-Gordon coefficients are listed in eqs. (D.11)–(D.14); and the non-zero invariant matrix elements appearing in eq. (C.13) can be computed from eqs. (C.11) and (C.12) with knowledge (N )  of the eigenvectors UQ,S (ω; δ, i) obtained from the previous iteration. Having defined the matrix elements of HN +1 in this way, one can diagonalize the matrix in each of the (Q, S) subspaces and so obtain the eigenvalues (N +1)  E (N +1) (Q, S; ω) and eigenvectors UQ,S iteration.  101  (ω; δ, i) required for the subsequent  Appendix D Clebsch-Gordon Coefficients ~1 and S ~2 of spin S1 and S2 respectively. We Consider two spin operators S define eigenvectors of these operators as ~ 2 |Sj , mj i = Sj (Sj + 1) |Sj , mj i S j Sjz |Sj , mj i = mj |Sj , mj i  (D.1) (D.2)  and the product vectors |ψm1 ,m2 i ≡ |S1 , m1 i |S2 , m2 i .  (D.3)  ~1 + S ~2 and define eigenvectors of J~ as Let J~ = S J~2 |J, mi = J(J + 1) |J, mi  (D.4)  J |J, mi = m |J, mi .  (D.5)  z  The Clebsch-Gordon coefficients are the inner products of these last two vectors hψm1 ,m2 |J, mi .  102  (D.6)  D.1  S ⊗ 12  Let S1 = S and S2 = 1/2. In this case, one can obtain a closed form expression for all of the Clebsch-Gordon coefficients as presented in detail in [104].1 The result is   1 ψm− 1 , 1 S + , m 2 2 2    =  s  S+m+ 2S + 1  1 2  s  S − m + 12 1 ψm+ 1 ,− 1 S + , m = 2 2 2 2S + 1 s   S − m + 21 1 ψm− 1 , 1 S − , m = − 2 2 2 2S + 1 s   S + m + 21 1 = . ψm+ 1 ,− 1 S − , m 2 2 2 2S + 1     (D.7) (D.8) (D.9) (D.10)  Of these coefficients, only those with the maximum value of m are needed in appendix C so we write these explicitly for reference  1 ψS, 1 S + , S + 2 2  1 ψS+1,− 1 S + , S + 2 2  1 ψS−1, 1 S − , S − 2 2  1 ψS,− 1 S − , S − 2 2   1 2  1 2  1 2  1 2  1  = 1  (D.11)  = 0  (D.12) r  1 = − 2S + 1 r 2S = . 2S + 1  (D.13) (D.14)  It should be noted that we use a different convention for defining the phase of the Clebsch-Gordon coefficients than is used in [104].  103  D.2  S⊗1  Let S1 = S and S2 = 1. In this case, a succinct analytic form for all of the Clebsch-Gordon coefficients cannot be derived as in the previous section. However, only those coefficients hψm1 ,m2 |J, J i are required for the procedure  described in appendix C and so we calculate those explicitly here. The total spin J can assume three values: S + 1, S, and S − 1. Since |ψS,1 i is the only product state with m = m1 + m2 = S + 1, it immediately  follows that  |S + 1, S + 1i = |ψS,1 i  (D.15)  so that we immediately obtain one of the needed Clebsch-Gordon coefficients hψS,1 |S + 1, S + 1i = 1.  (D.16)  To obtain the coefficients for the vector |S, Si, we take the ansatz |S, Si = α |ψS−1,1 i + β |ψS,0 i  (D.17)  since the vectors on the right hand side are the only ones with m = m1 + m2 = S. The coefficients α and β are the Clebsch-Gordon coefficients we require. To determine their value, we observe that |S, Si and |S + 1, Si are both eigenvectors of J~ with different eigenvalues and, since J~ is a Hermitian operator, must be orthogonal hS + 1, S |S, S i = 0.  (D.18)  We further demand that |S, Si have unit norm |α|2 + |β|2 = 1.  (D.19)  In order to apply the condition of eq. (D.18), we need an expression for |S + 1, Si. This is obtained by introducing the lowering operator S − of any 104  ~ as spin operators S S − = S x − iS y  (D.20)  such that S − |S, mi =  p  (S + m)(S − m + 1) |S, m − 1i .  (D.21)  We then apply the operator J − in two ways to the state |S + 1, S + 1i, first using the fact that |S + 1, S + 1i is an eigenstate of J~ so that we can apply  it directly to the state  J − |S + 1, S + 1i =  p  2(S + 1) |S + 1, Si  (D.22)  and second, writing J − = S1− + S2− and using the equivalence of eq. (D.15) so that √ √  J − |S + 1, S + 1i = S1− + S2− |ψS,1 i = 2S |ψS−1,1 i + 2 |ψS,0 i . (D.23)  Equating the right hand sides of eqs. (D.22) and (D.23), we obtain |S + 1, Si =  r  S |ψS−1,1 i + S+1  r  1 |ψS,0 i . S+1  (D.24)  Substituting eqs. (D.24) and (D.17) into eq. (D.18) and applying the condition of eq. (D.19) is enough to fully determine the values of α and β and, hence, the Clebsch-Gordon coefficients r  1 hψS−1,1 |S, S i = α = − S+1 r S hψS,0 |S, S i = β = . S+1  (D.25) (D.26)  We now apply a similar procedure to obtain the three remaining required  105  Clebsch-Gordon coefficients a, b, and c using the ansatz |S − 1, S − 1i = a |ψS−2,1 i + b |ψS−1,0 i + c |ψS,−1 i .  (D.27)  The orthonormality conditions read hS + 1, S − 1 |S − 1, S − 1 i = 0  (D.28)  hS, S − 1 |S − 1, S − 1 i = 0  (D.29)  |a|2 + |b|2 + |c|2 = 1  (D.30)  and these are enough to determine the value of the coefficients. We obtain expressions for |S + 1, S − 1i and |S, S − 1i by applying J −  in two ways to eqs. (D.24) and (D.17) respectively, similar to how eq. (D.24) was obtained. Substituting the resulting vectors into the orthonormality conditions and solving for the coefficients gives s  1 S(2S + 1) s 2S − 1 hψS−1,0 |S − 1, S − 1i = b = − S(2S + 1) r 2S − 1 hψS,−1 |S − 1, S − 1i = c = . 2S + 1  hψS−2,1 |S − 1, S − 1i = a =  (D.31) (D.32) (D.33)  The equations (D.16), (D.25)–(D.26), and (D.31)–(D.33) tabulate the ClebschGordon coefficients required for the procedure described in appendix C. More specifically, these coefficients are used for writing the basis states of Tables C.2 and C.3.  106  

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