DOCSLIB.ORG

Multichannel Kondo Effect in an Interacting Electron System: Exact Results for the Low-Temperature Thermodynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

PHYSICAL REVIEW B VOLUME 57, NUMBER 2 1 JANUARY 1998-II Multichannel Kondo effect in an interacting electron system: Exact results for the low-temperature thermodynamics Mats Granath and Henrik Johannesson Institute of Theoretical Physics, Chalmers University of Technology and Go¨teborg University, S-412 96 Go¨teborg, Sweden ~Received 2 July 1997! We study the low-temperature thermodynamics of a spin-S magnetic impurity coupled to m>2 degenerate bands of interacting electrons in one dimension. By exploiting boundary-conformal-field-theory techniques, we derive exact results for the possible impurity thermal and magnetic response. The leading behavior of the impurity magnetic susceptibility is shown to be insensitive to the electron-electron interaction. In contrast, there are two types of scaling behavior of the impurity specific heat consistent with the symmetries of the problem: Either it remains the same as for the ordinary multichannel Kondo problem for noninteracting 21 electrons or it acquires a new leading term governed by a critical exponent am5(gm 21)/m, where gm<1is a generalized ~m channel! Luttinger-liquid charge parameter measuring the strength of the repulsive electron- electron interaction. We conjecture that the latter behavior is indeed realized when the impurity is exactly screened (m52S). @S0163-1829~98!05002-4# I. INTRODUCTION We study the problem using boundary conformal field theory ~BCFT!,12 assuming that at low temperatures the The violation of Fermi-liquid behavior in the normal state impurity-electron interaction renormalizes onto a scale- invariant boundary condition on the bulk theory. This ap- of the high-Tc superconductors has turned the study of non- Fermi-liquid phenomena into a major theme in condensed- proach, suggested by Affleck and Ludwig for the ordinary multichannel Kondo problem,8,13 has successfully been em- matter physics.1 Additional motivation comes from a grow- ployed for a single channel of interacting electrons coupled ing number of experimental realizations of low-dimensional to a spin-1/2 impurity.14,15 In the present case, with an arbi- electron structures—such as quasi-one-dimensional ~1D! or- 2 trary number of electron channels m>2, a BCFT analysis ganic conductors, or point-contact tunneling in fractional allows for a complete classification of all possible critical 3 quantum Hall devices, where electron correlations are seen behaviors of the impurity-electron composite. Being exact, to produce manifest non-Fermi-liquid behavior. this information should prove useful as a guide to, and a test Two prominent model problems, serving as paradigms for of the validity of, other, more direct approaches to the prob- the study of non-Fermi-liquids, are the Luttinger liquid4,5 and lem ~yet to be carried out!. Specifically, we shall show that the ~overscreened! multichannel Kondo effect.6–9 ‘‘Luttinger conformal invariance together with the internal symmetries liquid’’ is the code name for the universal low-energy behav- of the problem restrict the possible types of critical behavior ior of interacting electrons in 1D, whereas the multichannel to only two: Either the theory is the same as for noninteract- Kondo model describes a magnetic impurity coupled via a ing electrons or the electron interaction generates a specific spin exchange to several degenerate bands of noninteracting boundary operator that produces a new leading behavior of electrons in 3D. Both problems have yielded to exact solu- the impurity thermal response. In both cases the leading im- tions, exhibiting a wealth of properties not contained in the purity magnetic response is insensitive to the electron- standard Fermi-liquid picture of the metallic state. electron interaction, implying that the screening of the impu- Here we consider a spin-S magnetic impurity coupled to rity is realized in the same way as in the noninteracting m>2 degenerate bands of interacting electrons in 1D, thus problem, with over-, exact, or underscreening depending on extending the ordinary multichannel Kondo model to the the number of channels and the magnitude of the impurity case of interacting electrons. Specifically, we address the spin. While our method cannot pinpoint which of the two scenarios is actually realized, we conjecture, guided by question about the influence of electron-electron interaction 16,17 on the low-temperature thermal and magnetic response of the analogous results for the one-channel problem, that elec- impurity-electron ~screening-cloud! composite. This problem tron interactions in the bulk do induce an anomalous term in is particularly interesting as it involves the interplay between the impurity specific heat in the case of exact screening (m two kinds of electron correlations; one induced by the spin 52S). exchange interaction with the impurity, the other coming II. THE MODEL from the direct electron-electron interaction. Moreover, the study of a magnetic impurity in the presence of interacting As microscopic bulk model we take a multiband Hubbard electrons may shed new light on possible experimental real- chain with repulsive on-site interaction U.0, izations of the multichannel Kondo effect, such as certain 10 Uranium-containing heavy-fermion materials, or the cou- † Hel52t ~cn,iscn11,is1H.c.!1U nn,isnn,jm , pling of conduction electrons to structural defects in metal n(,i,s n,(ij,ms point contacts.11 ~1! 0163-1829/98/57~2!/987~6!/$15.0057 987 © 1998 The American Physical Society 988 MATS GRANATH AND HENRIK JOHANNESSON 57 where cn,is is the electron operator at site n, with i, j moving currents with a second species ~labeled by ‘‘2’’! of 2 51,....,m and m,s5 , band and spin indices, respec- left-moving currents: jL/R(x)[ jR/L(2x) for x.0 @with † ↑ ↓ 1 tively, and nn,is5cn,iscn,is is the number operator. This jL/R(x)[ jL/R(x)#, and analogously for spin and flavor cur- model, and its variants, has been extensively studied in the rents. This amounts to folding the system onto the positive x literature, most recently in Ref. 18 where it was argued that axis, with a boundary condition enhanced superconducting fluctuations may result when the 1/2 2/1 degeneracy of the on-site interband coupling U is properly jL ~0!5 jR ~0! ~7! lifted. At large wavelengths we can perform a continuum at the origin, and then analytically continuing the currents limit c Aa/2pC (na) ~with a the lattice spacing!, n,is→ is back to the full x axis. Here Eq. ~7! simulates the continuity which, for small U and away from half-filling, takes Eq. ~1! at x50 of the original bulk theory ~with the analogous onto boundary conditions in spin and flavor sectors!. The Sug- awara form of H* in Eq. ~6! implies invariance under inde- 1 d el † pendent U(1)i, SU(2)i , and SU(m)i transformations ~with Hel 5 dx vF :cL,is~x!i cL,is~x!: m 2 2p E H F dx i51,2 labeling the two species!, reflecting the chiral sym- d metry of the critical bulk theory. 2:c† ~x!i c ~x!: We now insert a local spin S at x50, and couple it to the R,is dx R,is G electrons by an antiferromagnetic (l.0) spin-exchange in- teraction g † † 1 :cr,is~x!cr,is~x!cs,jm~x!cs,jm~x!: 2 † † 1 HK5l:@cL,is~0!1cR,is~0!# 2 ssm @cL,im~0! † † 1g: x x x x : . 2 1cR,im~0!# S:. ~8! cL,is~ !cR,is~ !cR,jm~ !cL,jm~ ! J ~ ! • In the low-energy limit the impurity is expected13 to renor- Here cL/R,is(x) are the left/right moving components of the malize to a conformally invariant boundary condition on the electron field Cis(x), expanded about the Fermi points bulk theory, changing Eq. ~7! into a new nontrivial boundary 6k : C (x)5e2ikFxc (x)1eikFxc (x). Summation F is L,is R,is condition on Hel* . By using BCFT to extract the set of over repeated indices for band, spin, and chirality r,s5R,L boundary operators present for this boundary condition, is implied. The normal ordering is defined with respect to the finite-temperature effects due to the impurity can be accessed filled Dirac sea, and vF and g are given by vF via standard finite-size scaling by treating ~Euclidean! time 52at sin(akF) and g5Ua/p, respectively. as an inverse temperature. For the purpose of implementing BCFT techniques in the presence of a Kondo impurity ~yet to be added!, we decouple III. IMPURITY CRITICAL BEHAVIOR charge, spin, and band (flavor) degrees of freedom in Eq. ~2! by a Sugawara construction,19 using the U~1!~charge!, The set of boundary operators j and the corresponding boundary scaling dimensions D Ocan be derived from the SU(2)m ~level m, spin!, and SU(m)2 ~level 2, flavor! Kac- j Moody currents: finite-size spectrum of Hel* through a conformal mapping from the half-plane to a semi-infinite strip. The mapping is † Jr~x!5:cr,is~x!cr,is~x!:, ~3! such that the boundary condition corresponding to the impu- rity on the half-plane is mapped to both sides of the strip, 1 with the boundary operators in the plane in one-to-one cor- J ~x!5:c† ~x! s c ~x!:, ~4! r r,is 2 sm r,im respondence to the eigenstates on the strip. In particular, the boundary dimensions are related to the energy spectrum A † A through the relation E5E 1 v / , with E the ground- J ~x!5:c T c :, ~5! 0 p D l 0 r r,is ij r,js state energy, and l the width of the strip.12 A 2 where s are the Pauli matrices, and T , AP$1,...,m 21%, A complete set of eigenstates of Hel* are given by the are generators of the defining representation of SU(m) with charge, spin, and flavor conformal towers, each tower de- normalization trTATB51/2dAB.
Recommended publications
  • Many-Body Effects in Optical Excitations of Transition Metal Dichalcogenides

    Many-Body Effects in Optical Excitations of Transition Metal Dichalcogenides

    Research Collection Doctoral Thesis Many-Body Effects in Optical Excitations of Transition Metal Dichalcogenides Author(s): Sidler, Meinrad Publication Date: 2018-02 Permanent Link: https://doi.org/10.3929/ethz-b-000290909 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library DISS. ETH NO. 24988 Many-Body Effects in Optical Excitations of Transition Metal Dichalcogenides A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by MEINRAD SIDLER MSc ETH Physics, ETH Zurich born on 21.06.1989 citizen of Grosswangen LU accepted on the recommendation of: Prof. Ata¸cImamo˘glu,examiner Prof. Bernhard Urbaszek, co-examiner 2018 c Summary This dissertation treats a quantum impurity problem in a semiconductor system. Quantum impurity problems describe the interaction between a single quantum object and a complex environment. They are ubiquitous in physical systems and represent a fundamental field of research in many-body physics. Prominent examples are the Anderson orthogonality catastrophe and the Kondo effect. In both cases, the impurity is much heavier than the constituents of the interacting environment. If the mass of the impurity is comparable to the surrounding particles, we have a mobile impurity. These systems are usually harder to solve, as evident in the case of lattice polarons, which were first proposed in 1933 by Lev Landau. A complex but accurate description was found years later in 1955 by Richard Feynman. In recent years, strong coupling between single, mobile quantum impurities and a fermionic bath was realized in cold atoms.
  • A New View on the Origin of Zero-Bias Anomalies of Co Atoms Atop Noble Metal Surfaces ✉ ✉ Juba Bouaziz 1 , Filipe Souza Mendes Guimarães 1 & Samir Lounis 1

    A New View on the Origin of Zero-Bias Anomalies of Co Atoms Atop Noble Metal Surfaces ✉ ✉ Juba Bouaziz 1 , Filipe Souza Mendes Guimarães 1 & Samir Lounis 1

    ARTICLE https://doi.org/10.1038/s41467-020-19746-1 OPEN A new view on the origin of zero-bias anomalies of Co atoms atop noble metal surfaces ✉ ✉ Juba Bouaziz 1 , Filipe Souza Mendes Guimarães 1 & Samir Lounis 1 Many-body phenomena are paramount in physics. In condensed matter, their hallmark is considerable on a wide range of material characteristics spanning electronic, magnetic, thermodynamic and transport properties. They potentially imprint non-trivial signatures in 1234567890():,; spectroscopic measurements, such as those assigned to Kondo, excitonic and polaronic features, whose emergence depends on the involved degrees of freedom. Here, we address systematically zero-bias anomalies detected by scanning tunneling spectroscopy on Co atoms deposited on Cu, Ag and Au(111) substrates, which remarkably are almost identical to those obtained from first-principles. These features originate from gaped spin-excitations induced by a finite magnetic anisotropy energy, in contrast to the usual widespread inter- pretation relating them to Kondo resonances. Resting on relativistic time-dependent density functional and many-body perturbation theories, we furthermore unveil a new many-body feature, the spinaron, resulting from the interaction of electrons and spin-excitations loca- lizing electronic states in a well defined energy. ✉ 1 Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, Jülich 52425, Germany. email: [email protected]; [email protected] NATURE COMMUNICATIONS | (2020) 11:6112 | https://doi.org/10.1038/s41467-020-19746-1 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-19746-1 ignatures of many-body phenomena in solid state physics theoretical transport spectra are nearly identical to the experi- Sare diverse1–5.
  • Kondo Effect in Mesoscopic Quantum Dots

    Kondo Effect in Mesoscopic Quantum Dots

    1 Kondo Effect in Mesoscopic Quantum Dots M. Grobis,1 I. G. Rau,2 R. M. Potok,1,3 and D. Goldhaber-Gordon1 1 Department of Physics, Stanford University, Stanford, CA 94305 2 Department of Applied Physics, Stanford University, Stanford, CA 94305 3 Department of Physics, Harvard University, Cambridge, MA 02138 Abstract A dilute concentration of magnetic impurities can dramatically affect the transport properties of an otherwise pure metal. This phenomenon, known as the Kondo effect, originates from the interactions of individual magnetic impurities with the conduction electrons. Nearly a decade ago, the Kondo effect was observed in a new system, in which the magnetic moment stems from a single unpaired spin in a lithographically defined quantum dot, or artificial atom. The discovery of the Kondo effect in artificial atoms spurred a revival in the study of Kondo physics, due in part to the unprecedented control of relevant parameters in these systems. In this review we discuss the physics, origins, and phenomenology of the Kondo effect in the context of recent quantum dot experiments. After a brief historical introduction (Sec. 1), we first discuss the spin-½ Kondo effect (Sec. 2) and how it is modified by various parameters, external couplings, and non-ideal conduction reservoirs (Sec. 3). Next, we discuss measurements of more exotic Kondo systems (Sec. 4) and conclude with some possible future directions (Sec. 5). Keywords Kondo effect, quantum dot, single-electron transistor, mesoscopic physics, Anderson model, Coulomb blockade, tunneling transport, magnetism, experimental physics 2 1 Introduction At low temperatures, a small concentration of magnetic impurities — atoms or ions with a non-zero magnetic moment — can dramatically affect the behavior of conduction electrons in an otherwise pure metal.
  • Bosonization of 1-Dimensional Luttinger Liquid

    Bosonization of 1-Dimensional Luttinger Liquid

    Bosonization of 1-Dimensional Luttinger Liquid Di Zhou December 13, 2011 Abstract Due to the special dimensionality of one-dimensional Fermi system, Fermi Liquid Theory breaks down and we must find a new way to solve this problem. Here I will have a brief review of one-dimensional Luttinger Liquid, and introduce the process of Bosonization for 1- Dimensional Fermionic system. At the end of this paper, I will discuss the emergent state of charge density wave (CDW) and spin density wave (SDW) separation phenomena. 1 Introduction 3-Dimensional Fermi liquid theory is mostly related to a picture of quasi-particles when we adiabatically switch on interactions, to obtain particle-hole excitations. These quasi-particles are directly related to the original fermions. Of course they also obey the Fermi-Dirac Statis- tics. Based on the free Fermi gas picture, the interaction term: (i) it renormalizes the free Hamiltonians of the quasi-particles such as the effective mass, and the thermodynamic properties; (ii) it introduces new collective modes. The existence of quasi-particles results in a fi- nite jump of the momentum distribution function n(k) at the Fermi surface, corresponding to a finite residue of the quasi-particle pole in the electrons' Green function. 1-Dimensional Fermi liquids are very special because they keep a Fermi surface (by definition of the points where the momentum distribution or its derivatives have singularities) enclosing the same k-space vol- ume as that of free fermions, in agreement with Luttingers theorem. 1 1-Dimensional electrons spontaneously open a gap at the Fermi surface when they are coupled adiabatically to phonons with wave vector 2kF .
  • A Glimpse of a Luttinger Liquid IGOR A. ZALIZNYAK Physics Department

    A Glimpse of a Luttinger Liquid IGOR A. ZALIZNYAK Physics Department

    1 A glimpse of a Luttinger liquid IGOR A. ZALIZNYAK Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA email: [email protected] The concept of a Luttinger liquid has recently been established as a fundamental paradigm vital to our understanding of the properties of one-dimensional quantum systems, leading to a number of theoretical breakthroughs. Now theoretical predictions have been put to test by the comprehensive experimental study. Our everyday life experience is that of living in a three-dimensional world. Phenomena in a world with only one spatial dimension may appear an esoteric subject, and for a long time it was perceived as such. This has now changed as our knowledge of matter’s inner atomic structure has evolved. It appears that in many real-life materials a chain-like pattern of overlapping atomic orbitals leaves electrons belonging on these orbitals with only one dimension where they can freely travel. The same is obviously true for polymers and long biological macro-molecules such as proteins or RNA. Finally, with the nano-patterned microchips and nano-wires heading into consumer electronics, the question “how do electrons behave in one dimension?” is no longer a theoretical playground but something that a curious mind might ask when thinking of how his or her computer works. One-dimensional problems being mathematically simpler, a number of exact solutions describing “model” one-dimensional systems were known to theorists for years. Outstanding progress, however, has only recently been achieved with the advent of conformal field theory 1,2. It unifies the bits and pieces of existing knowledge like the pieces of a jigsaw puzzle and predicts remarkable universal critical behaviour for one-dimensional systems 3,4.
  • Chapter 1 LUTTINGER LIQUIDS: the BASIC CONCEPTS

    Chapter 1 LUTTINGER LIQUIDS: the BASIC CONCEPTS

    Chapter 1 LUTTINGER LIQUIDS: THE BASIC CONCEPTS K.Schonhammer¨ Institut fur¨ Theoretische Physik, Universitat¨ Gottingen,¨ Bunsenstr. 9, D-37073 Gottingen,¨ Germany Abstract This chapter reviews the theoretical description of interacting fermions in one dimension. The Luttinger liquid concept is elucidated using the Tomonaga- Luttinger model as well as integrable lattice models. Weakly coupled chains and the attempts to experimentally verify the theoretical predictions are dis- cussed. Keywords Luttinger liquids, Tomonaga model, bosonization, anomalous power laws, break- down of Fermi liquid theory, spin-charge separation, spectral functions, coupled chains, quasi-one-dimensional conductors 1. Introduction In this chapter we attempt a simple selfcontained introduction to the main ideas and important computational tools for the description of interacting fermi- ons in one spatial dimension. The reader is expected to have some knowledge of the method of second quantization. As in section 3 we describe a constructive approach to the important concept of bosonization, no quantum field-theoretical background is required. After mainly focusing on the Tomonaga-Luttinger 1 2 model in sections 2 and 3 we present results for integrable lattice models in section 4. In order to make contact to more realistic systems the coupling of strictly ¢¡ systems as well as to the surrounding is addressed in section 5. The attempts to experimentally verify typical Luttinger liquid features like anoma- lous power laws in various correlation functions are only shortly discussed as this is treated in other chapters of this book. 2. Luttinger liquids - a short history of the ideas As an introduction the basic steps towards the general concept of Luttinger liquids are presented in historical order.
  • Pairing in Luttinger Liquids and Quantum Hall States

    Pairing in Luttinger Liquids and Quantum Hall States

    PHYSICAL REVIEW X 7, 031009 (2017) Pairing in Luttinger Liquids and Quantum Hall States Charles L. Kane,1 Ady Stern,2 and Bertrand I. Halperin3 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 2Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel 3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 8 February 2017; published 18 July 2017) We study spinless electrons in a single-channel quantum wire interacting through attractive interaction, and the quantum Hall states that may be constructed by an array of such wires. For a single wire, the electrons may form two phases, the Luttinger liquid and the strongly paired phase. The Luttinger liquid is gapless to one- and two-electron excitations, while the strongly paired state is gapped to the former and gapless to the latter. In contrast to the case in which the wire is proximity coupled to an external superconductor, for an isolated wire there is no separate phase of a topological, weakly paired superconductor. Rather, this phase is adiabatically connected to the Luttinger liquid phase. The properties of the one-dimensional topological superconductor emerge when the number of channels in the wire becomes large. The quantum Hall states that may be formed by an array of single-channel wires depend on the Landau-level filling factors. For odd- denominator fillings ν ¼ 1=ð2n þ 1Þ, wires at the Luttinger phase form Laughlin states, while wires in the strongly paired phase form a bosonic fractional quantum Hall state of strongly bound pairs at a filling of 1=ð8n þ 4Þ.
  • Kondo Effect in Real Metals Ph

    Kondo Effect in Real Metals Ph

    Kondo effect in real metals Ph. Nozières, Annie Blandin To cite this version: Ph. Nozières, Annie Blandin. Kondo effect in real metals. Journal de Physique, 1980, 41 (3), pp.193- 211. 10.1051/jphys:01980004103019300. jpa-00209235 HAL Id: jpa-00209235 https://hal.archives-ouvertes.fr/jpa-00209235 Submitted on 1 Jan 1980 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Physique 41 (1980) 193-2111 MARS 1980, 193 Classification Physics Abstracts 75.20H Kondo effect in real metals Ph. Nozières Institut Laue-Langevin, 156X, 38042 Grenoble Cedex, France and A. Blandin Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France (Reçu le 21 septembre 1979, accepte le 30 octobre 1979) Résumé. 2014 Dans le cadre d’un hamiltonien d’Anderson généralisé, nous étudions le comportement d’une impureté magnétique dans un métal réel, compte tenu de la structure orbitale des électrons localisés sur l’impureté, du champ cristallin et du couplage spin-orbite. Nous nous limitons à un schéma atomique, dans lequel l’impureté a une valeur bien définie entière (une transformation de Schrieffer Wolff est alors possible). Les principales étapes d’une procédure de renormalisation sont décrites en détail.
  • Tunneling Conductance of Long-Range Coulomb Interacting Luttinger Liquid

    Tunneling Conductance of Long-Range Coulomb Interacting Luttinger Liquid

    Tunneling conductance of long-range Coulomb interacting Luttinger liquid DinhDuy Vu,1 An´ıbal Iucci,1, 2, 3 and S. Das Sarma1 1Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 2Instituto de F´ısica La Plata - CONICET, Diag 113 y 64 (1900) La Plata, Argentina 3Departamento de F´ısica, Universidad Nacional de La Plata, cc 67, 1900 La Plata, Argentina. The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long- range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one- dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale- dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid. I. INTRODUCTION This power-law tunneling behavior is considered a signa- ture of the Luttinger liquid since in Fermi liquids, G is simply a constant for small values of T and V (as long Luttinger liquids emerge from interacting one- 0 as k T; eV E , where E is the Fermi energy).
  • Heavy Fermions: Electrons at the Edge of Magnetism

    Heavy Fermions: Electrons at the Edge of Magnetism

    Heavy Fermions: Electrons at the Edge of Magnetism Piers Coleman Rutgers University, Piscataway, NJ, USA Mathur et al., 1998). The ‘quantum critical point’ (QCP) that 1 Introduction: ‘Asymptotic Freedom’ in a Cryostat 95 separates the heavy-electron ground state from the AFM rep- 2 Local Moments and the Kondo Lattice 105 resents a kind of singularity in the material phase diagram 3 Kondo Insulators 123 that profoundly modifies the metallic properties, giving them a a predisposition toward superconductivity and other novel 4 Heavy-fermion Superconductivity 126 states of matter. 5 Quantum Criticality 135 One of the goals of modern condensed matter research 6 Conclusions and Open Questions 142 is to couple magnetic and electronic properties to develop Notes 142 new classes of material behavior, such as high-temperature Acknowledgments 142 superconductivity or colossal magnetoresistance materials, References 143 spintronics, and the newly discovered multiferroic materials. Further Reading 148 Heavy-electron materials lie at the very brink of magnetic instability, in a regime where quantum fluctuations of the magnetic and electronic degrees are strongly coupled. As such, they are an important test bed for the development of 1 INTRODUCTION: ‘ASYMPTOTIC our understanding about the interaction between magnetic FREEDOM’ IN A CRYOSTAT and electronic quantum fluctuations. Heavy-fermion materials contain rare-earth or actinide The term heavy fermion was coined by Steglich et al. (1976) ions, forming a matrix of localized magnetic moments. The in the late 1970s to describe the electronic excitations in active physics of these materials results from the immersion a new class of intermetallic compound with an electronic of these magnetic moments in a quantum sea of mobile con- density of states as much as 1000 times larger than copper.
  • Kondo Effect in Quantum Dots

    Kondo Effect in Quantum Dots

    Kondo Effect in Quantum Dots Seminar: Nanoscopic Systems 29. 7. 2005 Gesa von Bornstaedt Gesa von Bornstaedt Kondo Effect in Quantum Dots Outline 1. What is the Kondo effect? 2. Calculation of the Kondo effect for a single magnetic impurity 3. ”Poor Man’s Scaling” - A renormalization group approach 4. Investigation of the Kondo effect in quantum dots 5. Conclusion Gesa von Bornstaedt Kondo Effect in Quantum Dots 1. What is the Kondo effect? • Resistivity is dominated by phonon scattering for T 0 • In most metals the resistivity decreases monotonically for T → 0 • But: resistance minimum in some metals for T → 0 • Kondo effect: Resonant spin flip scattering at local magnetic impurities • Influences transport properties Figure 1: Minimum in the resistivity of Au (de Haas, de Boer and van den Berg, 1934) Gesa von Bornstaedt Kondo Effect in Quantum Dots 2. Calculation for a single local magnetic impurity Hamiltonian Kondo’s assumption: a local magnetic moment associated with a spin S which is coupled via an exchange interaction with the conduction band electron. † 1 + − − + H = ∑kckσ ckσ + J(ω) S s + S s + Jz(ω)Szsz kσ 2 † 1 + † − † † † = ∑kckσ ckσ + ∑ J(ω) S ck↓ck0↑ + S ck↑ck0↓ + Jz(ω)Sz ck↑ck0↑ − ck↓ck0↓ kσ 2 k,k0 S = local impurity spin s = conduction electron spin Figure 2: Spin flip scattering of a localized impurity moment by a conduction electron Gesa von Bornstaedt Kondo Effect in Quantum Dots Perturbative approach • T > 0 =⇒ expectation values can’t be taken with respect to the ground state • Appropriate: Matsubara Method Starting
  • Joaquin M. Luttinger 1923–1997

    Joaquin M. Luttinger 1923–1997

    Joaquin M. Luttinger 1923–1997 A Biographical Memoir by Walter Kohn ©2014 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. JOAQUIN MAZDAK LUTTINGER December 2, 1923–April 6, 1997 Elected to the NAS, 1976 The brilliant mathematical and theoretical physicist Joaquin M. Luttinger died at the age of 73 years in the city of his birth, New York, which he deeply loved throughout his life. He had been in good spirits a few days earlier when he said to Walter Kohn (WK), his longtime collaborator and friend, that he was dying a happy man thanks to the loving care during his last illness by his former wife, Abigail Thomas, and by his stepdaughter, Jennifer Waddell. Luttinger’s work was marked by his exceptional ability to illuminate physical properties and phenomena through Visual Archives. Emilio Segrè Photograph courtesy the use of appropriate and beautiful mathematics. His writings and lectures were widely appreciated for their clarity and fine literary quality. With Luttinger’s death, an By Walter Kohn influential voice that helped shape the scientific discourse of his time, especially in condensed-matter physics, was stilled, but many of his ideas live on. For example, his famous 1963 paper on condensed one-dimensional fermion systems, now known as Tomonaga-Luttinger liquids,1, 2 or simply Luttinger liquids, continues to have a strong influence on research on 1-D electronic dynamics. In the 1950s and ’60s, Luttinger also was one of the great figures who helped construct the present canon of classic many-body theory while at the same time laying founda- tions for present-day revisions.