nanophysics. coherence and transport, 2005, p.641

Quantum field theory II 1952 Quantum mechanics.. Nuclear physics III 1953 Quantum mechanics.. Elementary particle physics IV 1954 Quantum mechanics.. Experiments in high energy physics V

Université Joseph FourierLes Houches

Session LXXXI

2004

Nanophysics: Coherence and Transport

Contributors to this volume

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É COLE DE P HYSIQUE DES H OUCHES

Service inter-universitaire commun

à l’Université Joseph Fourier de Grenoble

et à l’Institut National Polytechnique de Grenoble

Subventionné par le Ministère de l’Éducation Nationale,

de l’Enseignement Supérieur et de la Recherche,

le Centre National de la Recherche Scientifique,

le Commissariat à l’Énergie Atomique

Membres du conseil d’administration :

Yannick Vallée (président), Paul Jacquet (vice-président), Cécile DeWitt, ThérèseEncrenaz, Bertrand Fourcade, Luc Frappat, Jean-François Joanny, Michèle Leduc,Jean-Yves Marzin, Giorgio Parisi, Eva Pebay-Peyroula, Michel Peyrard, LucPoggioli, Jean-Paul Poirier, Michel Schlenker, François Weiss, Jean Zinn-Justin

Directeur :

Jean Dalibard, Laboratoire Kastler Brossel, Paris, France

Directeurs scientifiques de la session LXXXI:

Hélène Bouchiat, Laboratoire de Physique des Solides, Orsay, France

Yuval Gefen, Weizmann Institute of Science, Rehovot, Israel

Sophie Guéron, Laboratoire de Physique des Solides, Orsay, France

Gilles Montambaux, Laboratoire de Physique des Solides, Orsay, France

Previous sessions

I 1951 Quantum mechanics Quantum field theory

II 1952 Quantum mechanics Statistical mechanics Nuclear physics

III 1953 Quantum mechanics Solid state physics Statistical mechanics.

Elementary particle physics

IV 1954 Quantum mechanics Collision theory Nucleon–nucleon interaction.

Quantum electrodynamics

V 1955 Quantum mechanics Non equilibrium phenomena Nuclear reactions.

Interaction of a nucleus with atomic and molecular fields

VI 1956 Quantum perturbation theory Low temperature physics.

Quantum theory of solids Ferromagnetism VII 1957 Scattering theory Recent developments in field theory.

Nuclear and strong interactions Experiments in high energy physics VIII 1958 The many body problem

IX 1959 The theory of neutral and ionized gases

X 1960 Elementary particles and dispersion relations

XI 1961 Low temperature physics

XII 1962 Geophysics; the earth’s environment

XIII 1963 Relativity groups and topology

XIV 1964 Quantum optics and electronics

XV 1965 High energy physics

XVI 1966 High energy astrophysics

XVII 1967 Many body physics

XVIII 1968 Nuclear physics

XIX 1969 Physical problems in biological systems

XX 1970 Statistical mechanics and quantum field theory

XXI 1971 Particle physics

XXII 1972 Plasma physics

XXIII 1972 Black holes

XXIV 1973 Fluids dynamics

XXV 1973 Molecular fluids

XXVI 1974 Atomic and molecular physics and the interstellar matter

XXVII 1975 Frontiers in laser spectroscopy

XXVIII 1975 Methods in field theory

XXIX 1976 Weak and electromagnetic interactions at high energy

XXX 1977 Nuclear physics with heavy ions and mesons

XXXI 1978 Ill condensed matter

XXXII 1979 Membranes and intercellular communication

XXXIII 1979 Physical cosmology

XXXIV 1980 Laser plasma interaction

XXXV 1980 Physics of defects

XXXVI 1981 Chaotic behaviour of deterministic systems

XXXVII 1981 Gauge theories in high energy physics

XXXVIII 1982 New trends in atomic physics

XXXIX 1982 Recent advances in field theory and statistical mechanics

XL 1983 Relativity, groups and topology

XLI 1983 Birth and infancy of stars

XLII 1984 Cellular and molecular aspects of developmental biology

XLIII 1984 Critical phenomena, random systems, gauge theories

XLIV 1985 Architecture of fundamental interactions at short distances

XLV 1985 Signal processing

XLVI 1986 Chance and matter

XLVII 1986 Astrophysical fluid dynamics

XLVIII 1988 Liquids at interfaces

XLIX 1988 Fields, strings and critical phenomena

L 1988 Oceanographic and geophysical tomography

LI 1989 Liquids, freezing and glass transition

LII 1989 Chaos and quantum physics

LIII 1990 Fundamental systems in quantum optics

LIV 1990 Supernovae

LV 1991 Particles in the nineties

LVI 1991 Strongly interacting fermions and high T c superconductivity

LVII 1992 Gravitation and quantizations

LVIII 1992 Progress in picture processing

LIX 1993 Computational fluid dynamics

LX 1993 Cosmology and large scale structure

LXI 1994 Mesoscopic quantum physics

LXII 1994 Fluctuating geometries in statistical mechanics and quantum field theory LXIII 1995 Quantum fluctuations

LXIV 1995 Quantum symmetries

LXV 1996 From cell to brain

LXVI 1996 Trends in nuclear physics, 100 years later

LXVII 1997 Modeling the Earth’s Climate and its Variability

LXVIII 1997 Probing the Standard Model of Particle Interactions

LXIX 1998 Topological aspects of low dimensional systems

LXX 1998 Infrared space astronomy, today and tomorrow

LXXI 1999 The primordial universe

LXXII 1999 Coherent atomic matter waves

LXXIII 2000 Atomic clusters and nanoparticles

LXXIV 2000 New trends in turbulence

LXXV 2001 Physics of bio-molecules and cells

LXXVI 2001 Unity from duality: Gravity, gauge theory and strings

LXXVII 2002 Slow relaxations and nonequilibrium dynamics in condensed matter LXXVIII 2002 Accretion discs, jets and high energy phenomena in astrophysics LXXIX 2003 Quantum Entanglement and Information Processing

LXXX 2003 Methods and Models in Neurophysics

Publishers:

- Session VIII: Dunod, Wiley, Methuen

- Sessions IX and X: Herman, Wiley

- Session XI: Gordon and Breach, Presses Universitaires

- Sessions XII–XXV: Gordon and Breach

- Sessions XXVI–LXVIII: North Holland

- Session LXIX–LXXVIII: EDP Sciences, Springer

- Session LXXIX-LXXX: Elsevier

MASLOV Dmitrii, Department of Physics, University of Florida, PO Box

118440, Gainesville, FL 32611-8440, USA

RALPH Dan, 536 Clark Hall, Cornell University, Ithaca, NY 14853, USAVAN RUITENBEEK Jan, Atomic and Molecular Conductors, Leiden Institute ofPhysics, Leiden University, Niels Borhweg 2, 2333 CA Leiden, The Netherlands

xi

Seminar speakers

APRILI Marco, ESPCI, 10 rue Vauquelin, 75231 Paris cedex 05, FranceEGGER Reinhold, Institute für Theoretische Physik, Heinrich-Heine Univ., Uni-versitaetsstrasse 1, D-40225 Duesseldorf, Germany

ENSSLIN Klaus, Solid State Physics Laboratory, ETH Zurich, CH 8093, zerland

Swit-FAZIO Rosario, Scuola Normale Superiore, Classe di Scienze, Piazza dei lieri 7, 56126 Pisa, Italy

Cava-FEIGELMAN Mikhail, Russian Academy of Sciences, Landau Institute for oretical Physics, Kosiginstr.2, 119334 Moscow, Russian Federation

The-IMRY Yoseph, Weizmann Institute of Science, Department of Condensed Matter,

Cen-MAKHLIN Yuriy, Institute für Theoretische Festörperphysik, Univ Karlsruhe,D-76128 Karlsruhe, Germany

MEIR Yigal, Ben Gurion University, Department of Physics, 87109 Beer Sheva,Israel

REULET Bertrand, Applied Physics Dept, BCT 411, Yale Univ., New Haven,

STERN Ady, Weizmann Institute of Science, Department of Condensed Matter,

76100 Rehovot, Israel

YACOBY Amir, The Joseph and Belle Centre for Submicron Research, mann Institute, 76100 Rehovot, Israel

Weiz-xiv

BRAIG Stephan, Cornell University, Lab of Atomic and Solid State Physics,

117 Clark Hall, Ithaca, NY 14853, USA

CATELANI Gianluigi, Physics dept., Mail Code 5262, Columbia Univ., NewYork, NY 10027, USA

CHAUVIN Martin, Quantronics Group, SPEC, CEA Saclay, 91191 Yvette cedex, France

Gif-sur-CHOLASCINSKI Mateusz, Inst für Theoretische Festkorperphysik, Univ ruhe, W Gaede strasse 1, D-76128 Karlsruhe, Germany

Karls-CHTCHELKATCHEV Nikolai, Landau Inst for Theoretical Physics, Kosyginastr 2, 119334 Moscow, Russia

CSONKA Szabolcs, Budapest Univ of Technology, Dept Physics, ElectronTransport Research Group, Budafoki ut 8, 1111 Budapest, Hungary

DE MARTINO Alessandro, Inst Theoretical Physics IV, Heinrich Heine versity, Universitaetsstrasse 1, Gebaeude 25.32, D-40225 Duesseldorf, GermanyDIMITROVA Olga, Landau Institute for Theoretical physics, Chernogolovka,Moscow region, 142432, Russia

Uni-DUBI Jonathan, Physics dept., Ben-Gurion Univ of the Negev, Beer-Sheva,

84105, Israel

DUFOULEUR Joseph, LPN, route de Nozay, 91460 Marcoussis, France

xv

FERRIER Meydi, Laboratoire de Physique des Solides, Université Paris-Sud,

GOREN Liliach, Weizmann Inst of Science, 76100 Rehovot, Israel

GRISHIN Alexander, School of Physics and Astronomy, Univ Birmingham,Edgbaston, Birmingham B15 2 TT, UK

HAGELAAR Joris, Eindhoven Univ Technology, Dept Applied Physics, trum 1.80, Den Dolech 2, PO Box 513, 5612 AZ Eindhoven, The NetherlandsHALBRITTER Andras, Budapest Univ of Technology, Dept Physics, ElectronTransport Research Group, Budafoki ut 8, 1111 Budapest, Hungary

Spec-HOUZET Manuel, CEA DRFMC/SPSMS Bat C1, 17 avenue des Martyrs, 38054Grenoble cedex 9, France

HUARD Benjamin, Groupe Quantronique, SPEC, CEA Saclay, 91191 Gif surYvette cedex, France

KOCH Jens, Freie Univ Berlin, Inst Theoretische Physik, Arnimallee 14,

Uniw-MEIDAN Dganit, Weizmann Inst of Science, 76100 Rehovot, Israel

MICHAELIS Björn, Inst Lorentz, PO Box 9506, Nl-2300 RA Leiden, TheNetherlands

MORTEN Jan Petter, Norwegian Univ of Science and Technology, Dept ics, 7491 Trondheim, Norway

Phys-xvi

OSSIPOV Alexander, ICTP, Condensed Matter Section, Strada Costiera 11,I-3014 Trieste, Italy

PORTIER Fabien, SPEC, DRECAM, CEA Saclay, 91191 Gif sur Yvette, FranceRATCHOV Alexandre, LPM2C, Maison des Magistères, 25 avenue des martyrs,

BP 166, 38042 Grenoble cedex, France

ROMITO Alessandro, Scuola Normale Superiore and NEST-INFM, piazza deiCavalieri, 7, I-56126 Pisa, Italy

RYTCHKOV Valentin, Univ Geneva, Dept Theoretical Physics, 24 quai E.Ansermet, 1211 Geneva, Switzerland

SAHA Ronojaoy, Dept Physics, Univ Florida, Gainesville, FL 32611, USASALOMEZ Julien, LPM2C, Maison des Magistères, 25 avenue des martyrs, BP

166, 38042 Grenoble cedex, France

SEDLMAYR Nicholas, School of Physics and Astronomy, Univ Birmingham,Edgbaston, Birmingham B15 2TT, UK

SEGALA Julien, SPEC, DRECAM, CEA Saclay, 91191 Gif sur Yvette, FranceSEOANEZ ERKELL Cesar, Oscar ICMM-CSIC, Cantoblanco, 28049 Madrid,Spain

SIM Heung-Sun, School of Physics, Korea Inst for Advanced Study, 207-43Cheongryangri-dong, Dongdaemun-gu, Seoul 130-722, Korea

SLOGGETT Clare, School of Physics, Univ New South Wales, Sidney NSW

TANASKOVIC Darko, Nat High Magnetic Field Lab., Florida State Univ., 1800

E Paul Dirac dr, Tallahassee, Florida 32310-3706, USA

TEXIER Christophe, LPTMS, Univ Paris Sud, Bât 100, 91405 Orsay cedex,France

TSYPLYATYEV Oleksandr, Lancaster Univ., Lancaster, LA1 4YB, UK

TUREK Marko, Univ Regensburg, Inst Theoretical Physics, D-93040 burg, Germany

Regens-xvii

UTSUMI Yasuhiro, MPI of Microstructure Physics, Weinberg 2, 06120 Halle,Germany

VITUSHINSKY Pavel, CEA DRFMC/SPSMS, 17 rue des martyrs, 38054 ble, France

Greno-WEI Tzu-Chieh, Dept Physics, Univ Illinois at Urbana-Champaign, 1110 WestGreen Street, Urbana, IL 61801-3080, USA

ZITKO Rok, Jozef Stefan Inst., Jamova 39, 1000 Ljubljana, Slovenia

ZVONAREV Mikhail, Orsted Lab., Niels Bohr Inst., Universitetsparken 5,

DK-2100 Copenhagen, Denmark

xviii

fol-The developments of nanofabrication in the past years have enabled the design

of electronic systems, that exhibit spectacular signatures of quantum coherence.Experimental results gave rise to numerous initially unexpected (and at timestheoretically unresolved) surprises

Nanofabricated quantum wires and dots containing a small number of trons are ideal experimental playgrounds for probing electron-electron interac-tions and their interplay with disorder Going down to even smaller scales, mole-cules such as carbon nanotubes, fullerenes or hydrogen molecules can now beinserted in nanocircuits Measurements of transport through a single chain ofatoms have been performed as well Much progress has also been made in thedesign and fabrication of superconducting and hybrid nanostructures, be theynormal/superconductor or ferromagnetic/superconductor Quantum coherence isthen no longer that of individual electronic states, but rather that of a supercon-ducting wavefunction of a macroscopic number of Cooper pairs condensed in thesame quantum mechanical state Beyond the study of linear response regime, thephysics of non-equilibrium transport (including non-linear transport, rectification

elec-of a high frequency electric field as well as shot noise) has received much tion, with significant experimental and theoretical insights All these quantitiesexhibit very specific signatures of the quantum nature of transport which cannot

atten-be obtained from basic conductance measurements

Participants in the School were exposed to the basic concepts and analyticaltools needed to understand this new physics This was presented in a series oftheoretical fundamental courses, in parallel with more phenomenological oneswhere physics was discussed in a less formal way and illustrated by numerousexperimental examples

xix

Dmitri Maslov gave a long series of lectures emphasizing the importance ofelectron-electron interactions in one-dimensional quantum transport Startingfrom the Fermi liquid description of transport, he showed that electron-electroninteractions, whose signature shows up already at higher dimensions, dominatethe physics of transport in 1D Technical aspects concerning the subtleties in-volved in the re-summation of perturbative diagrammatic expansions, the renor-malisation group in 1D and bosonisation were all discussed in great detail Em-phasis was given to the underlying physics Particular attention was given to thevarious signatures of interactions in the physical world, including the density ofenergy states as well as the electric and thermal conductances of 1D wires andtheir specific sensitivity to the presence of reservoirs or tunneling barriers Thislast important point was also treated in detail in the seminar of Igor Lerner Con-crete illustrations of these concepts were given in the seminars of Amir Yacoby

on cleaved-edge semiconducting quantum wires and of Reinhold Egger on bon nanotubes which various physical properties were also reviewed by ChristianSchoenenberger whereas Alik Kasumov focused on their superconducting prop-erties (both intrinsic and proximity induced.)

car-Electron-electron interactions are also responsible for the spectacular port properties of 2D electron systems confined at the interface between GaAsand GaAlAs in a doped heterostructure, giving rise in particular to the fractionalquantum Hall effect at a strong magnetic field Specifically, the course of JimEisenstein was devoted to recent experiments in double layer systems addressing(depending on the strength of the inter-layer coupling) the physics of Coulombdrag or the formation of a completely coherent state involving strong tunnel-ing between the two layers This spectacular formation of a macroscopic Bose–Einstein condensate of excitons was demonstrated and discussed in the semi-nar of Ady Stern Boris Shklovskii presented interesting extensions of the con-cepts of Coulomb interactions and screening to the physics of biological mole-cules

trans-The description of transport and noise out of equilibrium in nanoscale tronic systems has received growing attention in recent years, proving to be anefficient diagnostic tool (experimental and theoretical) in characterizing disorder,interactions and correlations in electronic systems The course of Alex Kamenevreviewed the theoretical tools (Keldysh theory combined with a non-linear sigmamodel field theory) required for an efficient treatment of this physics

elec-Following similar lines Vladimir Kravstov focused on the perturbative aspects

of real time Keldysh theory, specifically addressing non-linear ac transport inmesoscopic systems He made the connection between high frequency rectifica-tion of an Aharonov Bohm ring and dephasing in the presence of an ac electricfield or a non-equilibrium noise He also showed that energy absorption givesrise to localization in energy space (dynamic localization)

xx

The course of Frank Hekking described the physics of Andreev reflection atNormal/Superconductor interfaces which produces rich sub-gap energy depen-dent features, both in transport and noise through hybrid nanostructures Thesestrongly depend on the nature of the barrier at the NS interface and on dis-order on the normal side, giving rise to long-range coherent scattering Themore recently investigated Ferromagnetic/Superconducting hybrid systems, alsopresented in the seminar of Marco Aprili, lead to new spectacular phenomenawith oscillations of the phase of the superconducting order parameter in theferromagnetic region and the possibility to make π junctions The seminar ofMichael Feigelman also showed how Andreev scattering on small superconduct-ing particles can strongly affect quantum transport in a 2D disordered metallicplane.

Leonid Glazmann gave a review of electronic transport through quantum dots

in the Coulomb blockade regime He started from the physics of resonant ing and its treatment within the simple-minded picture of the addition spectrum.Attention was given to the role of fundamental symmetries and interactions char-acterising dots weakly coupled to reservoirs He went on to investigate moresubtle limits, first an off-resonance scenario giving rise to cotunneling (inelas-tic or elastic at very low temperature), and then, as the effective dot-lead cou-pling increases, Kondo physics comes into play Here the magnetic momentinvolving the highest occupied energy level of the dot combines into a coherentmany-body state with the electrons in the reservoirs There is a broad spectrum

tunnel-of signatures tunnel-of Kondo physics on the conductance tunnel-of the dot as a function tunnel-oftemperature and magnetic field This course was complemented by seminars

of Klaus Ensslin on spectroscopy experiments on quantum dots, Yigal Meir onKondo physics in quantum point contacts and Juergen Koenig on the possibility

to observe Fano-like resonances in quantum dots as well as relating dephasing tospin-flip processes

The course of Dan Ralph, devoted to tunneling measurements of individualquantum states in metallic nanostructures, also constituted a beautiful illustration

xxi

of these concepts with a particular emphasis on spin effects and superconductivity

in nanoparticles, going to the limit of single-molecule transistors

Atomic point contacts both in the normal and superconducting states tute the ultimate object for the investigation of quantum transport and requiresboth specific experimental techniques and theoretical concepts as presented inthe course of Jan van Ruitenbeek and Alfredo Levy Yeyati The notion of con-ductance channels for a single atom determines the physical properties of theseatomic-sized conductors out of equilibrium such as shot noise and superconduct-ing sub-gap structures

consti-Finally, the lectures of Daniel Estève were devoted to one of the great promises

of nanophysics– the possibility to realize solid state quantum bits (in the context

of quantum computation) by coherent manipulation of charge, flux or spin ables in a nanocircuit Selecting the right observables, and controlling the sources

vari-of decoherence by designing an optimum electromagnetic environment, are standing problems in the field The crucial question of dephasing by an electro-magnetic environment was also addressed in the seminar of Joe Imry Adding

out-to this broad spectrum of lectures were the seminars by Rosario Fazio and YuriyMakhlin, who addressed the possibility to investigate and control types of topo-logical Berry phase factors in qubits

These lectures were complemented by informal sessions or tutorials organized

on the spot by the lecturers, as well as by a number of seminars and poster sions given by the students They contributed by their unabated enthusiasm andcuriosity to the great atmosphere of the school

ses-The summer school and the present volume have been made possible by the nancial support of the following institutions, whose contribution is gratefully ac-knowledged:

fi The “Marie Curie Conferences and Training Course” program of the EuropeanUnion through the Consortium of European Physics School (CEPS)

-The “Advanced Scientific Institute” program of the Scientific and tal Affairs division of NATO

Environmen The “Lifelong learning” program of the Centre National de la Recherche tifique (France)

Scien The Université Joseph Fourier, the French Ministry of Research and the missariat à l’Energie Atomique, through their constant support to the School.The staff of the School, especially Brigitte Rousset and Isabelle Lelièvre, havebeen of great help for the preparation and development of the school, and wewould like to thank them warmly on behalf of all students and lecturers

Com-H Bouchiat, Y Gefen, S Guéron, G Montambaux and J Dalibard

xxii

Course 1 Fundamental aspects of electron correlations and

quantum transport in one-dimensional systems,

2 Non-Fermi liquid features of Fermi liquids: 1D physics in higher

4 Renormalization group for interacting fermions 45

xxiii

5.4 Comparison of bulk and edge tunneling exponents 61

7.3 Conductance of a wire attached to reservoirs 82

7.3.3 Kubo formula for a wire attached to reservoirs 86

7.5 Thermal conductance: Fabry-Perrot resonances of plasmons 92Appendix A Polarization bubble for small q in arbitrary dimensionality 94

Appendix C Some details of bosonization procedure 98

Appendix C.2.1 Commutation relations for bosonic fields ϕ and ϑ 101

xxiv

Seminar 1 Impurity in the Tomonaga-Luttinger model:

A functional integral approach, by I.V Lerner

3 The effective action for the Tomonaga-Luttinger Model 116

6 Tunnelling density of states near a single impurity 122Appendix A Jacobian of the gauge transformation 125

Course 2 Novel phenomena in double layer

two-dimensional electron systems, by J.P Eisenstein 129

3 Coulomb drag between parallel 2D electron gases 146

4 Tunneling between parallel two-dimensional electron gases 153

4.3 2D-2D tunneling in a perpendicular magnetic field 157

5 Strongly-coupled bilayer 2D electron systems and excitonic

5.3 Tunneling and interlayer phase coherence at νT = 1 165

xxv

Contents

Course 4 Non-linear quantum coherence effects in

2 Weak Anderson localization in disordered systems 251

3 Non-linear response to a time-dependent perturbation 2583.1 General structure of nonlinear response function 2593.2 Approximation of single photon absorption/emission 261

7 Weak dynamic localization and no-dephasing points 276

5 Scattering approach based on operator averages 295

5.3 Zero frequency noise in a two terminal conductor 299

5.3.2 Transition between the two noise regimes 300

xxvii

6.2 Noise correlations in a Y–shaped structure 305

8 Noise in normal metal-superconducting junctions 3108.1 Bogolubov transformation and Andreev current 3118.2 Noise in normal metal–superconductor junctions 314

8.4 Hanbury-Brown and Twiss experiment with a superconducting

9.1 Filtering spin/energy in superconducting forks 327

10.1 Edge states in the fractional quantum Hall effect 33710.2 Transport between two quantum Hall edges 340

10.5 Poissonian noise in the quantum Hall effect 346

3.1.1 dc current: dynamical Coulomb blockade 370

3.1.4 Effect of an external fluctuating voltage 371

Course 6 Electron subgap transport in hybrid systems

combining superconductors with normal or

2.1 Single particle tunnelling in a tunnel junction 388

3.4 Subgap noise of a superconductor-normal-metal tunnel interface 409

3.4.4 An explicit example: a wire out of equilibrium 413

4 Tunnelling in a three-terminal system containing ferromagnetic metals 416

4.2 Co-tunnelling and crossed Andreev tunnelling rates 419

4.2.2 Calculation of spin-dependent tunnel rates 420

dot, by Leonid I Glazman and Michael Pustilnik 427

3.2 Coulomb blockade peaks at low temperature 441

4 Activationless transport through a blockaded quantum dot 443

5 Kondo regime in transport through a quantum dot 448

xxx

5.6 Splitting of the Kondo peak in a magnetic field 4645.7 Kondo effect in quantum dots with large spin 468

Course 8 Transport at the atomic scale: Atomic and

molecular contacts, by A Levy Yeyati and

Course 9 Solid State Quantum Bit Circuits, by

1.1 From quantum mechanics to quantum machines 541

xxxi

2.1 Kane’s proposal: nuclear spins of P impurities in silicon 546

3.1.1 Hamiltonian of Josephson qubit circuits 550

3.1.3 Survey of Cooper pair box experiments 552

3.2.1 Qubit-environment coupling Hamiltonian 554

4.1 Relaxation and dephasing in the quantronium 558

5.1 Ultrafast ’DC’ pulses versus resonant microwave pulses 561

7.2 A tunable coupling element for Josephson qubits 568

7.4 Control of the interaction mediated by a fixed Hamiltonian 570

Abstracts of seminars presented at the School 577

xxxii

Course 1

FUNDAMENTAL ASPECTS OF ELECTRON

CORRELATIONS AND QUANTUM TRANSPORT IN

ONE-DIMENSIONAL SYSTEMS

Dmitrii L Maslov

Department of Physics, University of Florida, P.O Box 118441, Gainesville, FL 32611-8440, USA

H Bouchiat, Y Gefen, S Guéron, G Montambaux and J Dalibard, eds.

Les Houches, Session LXXXI, 2004

Nanophysics: Coherence and Transport

c

2005 Elsevier B.V All rights reserved

1

2 Non-Fermi liquid features of Fermi liquids: 1D physics in higher dimensions 8

3.5 Dyson equation for the Green’s function 37

4 Renormalization group for interacting fermions 45

5 Single impurity in a 1D system: scattering theory for interacting fermions 50 5.1 First-order interaction correction to the transmission coefficient 51

5.4 Comparison of bulk and edge tunneling exponents 61

7.5 Thermal conductance: Fabry-Perrot resonances of plasmons 92 Appendix A Polarization bubble for small q in arbitrary dimensionality 94

Appendix C Some details of bosonization procedure 98

Appendix C.2.1 Commutation relations for bosonic fields ϕ and ϑ 101 Appendix C.3 Problem with backscattering 102

Some aspects of physics on interacting fermions in 1D are discussed in a like manner We begin by showing that the non-analytic corrections to the Fermi-liquid forms of thermodynamic quantities result from essentially 1D collisionsembedded into a higher-dimensional phase space The role of these collisionsincreases progressively as dimensionality is reduced and, finally, they lead to abreakdown of the Fermi liquid in 1D An exact solution of the Tomonaga-Luttingermodel, based on exact Ward identities, is reviewed in the fermionic language Tun-neling in a 1D interacting systems is discussed first in terms of the scattering theoryfor interacting fermions and then via bosonization Universality of conductancequantization in disorder-free quantum wires is discussed along with the breakdown

tutorial-of this universality in the spin-incoherent case A difference between charge versal) and thermal (non-universal) conductances is explained in terms of Fabry-Perrot resonances of charge plasmons

(uni-1 Introduction

The theory of interacting fermions in one dimension (1D) has survived severalmetamorphoses From what seemed to be a purely mathematical exercise up untilthe 60s, it had evolved into a practical tool for predicting and describing phenom-

ena in conducting polymers and organic compounds–which were the 1D systems

of the 70s Beginning from the early 90s, when the progress in nanofabricationled to creation of artificial 1D structures–quantum wires and carbon nanotubes,the theory of 1D systems started its expansion into the domain of mesoscopics;this trend promises to continue in the future Given that there is already quite

a few excellent reviews and books on the subject [1–10] I should probably gin with an explanation as to what makes this review different from the others.First of all, it is not a review but–being almost a verbatim transcript of the lec-tures given at the 2004 Summer School in Les Houches–rather a tutorial on some(and definitely not all) aspects of 1D physics A typical review on the subjectstarts with describing the Fermi Liquid (FL) in higher dimensions with an aim ofemphasizing the differences between the FL and its 1D counter-part –Luttinger

be-5

6 D.L Maslov

Liquid (LL) My goal–if defined only after the manuscript was written–was rather

to highlight the similarities between higher-D and 1D systems The progress in

understanding of 1D systems has been facilitated tremendously and advanced to

a greater detail, as compared to higher dimensions, by the availability of exact orasymptotically exact methods (Bethe Ansatz, bosonization, conformal field the-ory), which typically do not work too well above 1D The downside part of thisprogress is that 1D effects, being studied by specifically 1D methods, look some-what special and not obviously related to higher dimensions Actually, this is

not true Many effects that are viewed as hallmarks of 1D physics, e.g., the

sup-pression of the tunneling conductance by the electron-electron interaction andthe infrared catastrophe, do have higher-D counter-parts and stem from essen-tially the same physics For example, scattering at Friedel oscillations caused

by tunneling barriers and impurities is responsible for zero-bias tunneling alies in all dimensions [11, 16] The difference is in the magnitude of the effectbut not in its qualitative nature Following the tradition, I also start with the FL

anom-in Sec 2, but the maanom-in message of this Section is that the difference between

D = 1 and D > 1 is not all that dramatic In particular, it is shown that the

well-known non-analytic corrections to the FL forms of thermodynamic tities (such as a venerable T3ln T -correction to the linear-in-T specific heat in3D) stem from rare events of essentially 1D collisions embedded into a higher-dimensional phase space In this approach, the difference between D = 1 and

quan-D > 1 is quantitative rather than qualitative: as the dimensionality goes down,

the phase space has difficulties suppressing the small-angle and 2kF−scattering

events, which are responsible for non-analyticities The crucial point when theseevents go out of control and start to dominate the physics happens to be in 1D.This theme is continued in Sec 5, where scattering from a single impurity em-bedded into a 1D system is analyzed in the fermionic language, following thework by Yue, Matveev, Glazman [11] The drawback of this approach–the per-turbative treatment of the interaction–is compensated by the clarity of underlyingphysics Another feature which makes these notes different from the rest of theliterature in the field is that the description goes in terms of the original fermi-ons for quite a while (Secs 2 through 5), whereas the weapon of choice of all1D studies–bosonization– is invoked only at a later stage (Sec 6 and beyond).The rationale–again, a post-factum one–is two-fold First, 1D systems in a meso-scopic environment–which are the main real-life application discussed here– areinvariably coupled to the outside world via leads, gates, etc As the outside world

is inhabited by real fermions, it is sometimes easier to think of, e.g., both the

interior and exterior a quantum wire coupled to reservoirs in terms of the sameelementary quasi-particles Second, after 40 years or so of bosonization, what

could have been studied within a model of fermions with linearized dispersion

and not too strong interaction–and this is when bosonization works–was probably

Fundamental aspects of electron correlations and quantum transport 7

studied (As all statements of this kind, this is one is also an exaggeration.) Thelast couple of years are characterized by a growing interest in either the effects

that do not occur in a model with linearized dispersion, e.g., Coulomb drag due

to small-momentum transfers [17], energy relaxation, and phase breaking [18](the last two phenomena also require three-body processes in 1D) or situationswhen strong Coulomb repulsion does not permit linearization of the spectrum atany energies [19, 20, 21] Experiment seems to indicate that the Coulomb re-pulsion is strong in most systems of interest, thus the beginning of studies of atruly strongly-coupled regime is quite timely Once the assumption of the linearspectrum is abandoned, the beauty of a bosonized description is by and large lost,and one might as well come back to original fermions Sec 6 is devoted to trans-port in quantum wires, mostly in the absence of impurities The universality ofconductance quantization is explained in some detail, and is followed by a briefdiscussion of the recent result by Matveev [19], who showed that incoherence inthe spin sector leads to a breakdown of the universality at higher temperatures(Sec 7.4) Also, a difference in charge (universal) and thermal (non-universal)transport–emphasized by Fazio, Hekking, and Khmelnitskii [22]– in addressed inSec 7.5 What is missing is a discussion of transport in a disordered (as opposed

to a single-impurity) case However, this canonically difficult subject, which volves an interplay between localization and interaction, is perhaps not readyfor a tutorial-like discussion at the moment (For a recent development on thissubject, see Ref [18].)

in-Even a brief inspection of these notes will show that the choice between ing them comprehensive or self-contained was made for the latter It is quite easy

mak-to see what is missing: there is no discussion of lattice effects, bosonization isintroduced without Klein factors, he sine-Gordon model is not treated properly,chiral Luttinger liquids are not discussed at all, and the list goes on The discus-sion of the experiment is scarce and perfunctory However, the few subjects thatare discussed are provided with quite a detailed–perhaps somewhat excessivelydetailed– treatment, so that a reader may not feel a need to consult the referencelist too often For the same reason, the notes also cover such canonical procedures

as the perturbative renormalization group in the fermionic language (Sec 4) andelementary bosonization (Sec 6), which are discussed in many other sources and

a reader already familiar with the subject is encouraged to skip them

Also, a relatively small number of references (about one per page on average)

indicates once again that this is not a review The choice of cited papers is

sub-jective and the reference list in no way pretends to represent a comprehensivebibliography to the field My apologies in advance to those whose contributions

to the field I have failed to acknowledge here

= kB = 1 through out the notes, unless specified otherwise