Dynamical Symmetries for Nanostructures

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3 Konstantin Kikoin Mikhail Kiselev Yshai Avishai Dynamical Symmetries for Nanostructures Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules

4 Ph.D. Konstantin Kikoin School of Physics and Astronomy Tel Aviv University Tel Aviv Israel Ph.D. Mikhail Kiselev The Abdus Salam Intl. Center for Theoretical Physics Strada Costiera Trieste Italy Ph.D. Yshai Avishai Ben Gurion University Beer Sheva Israel This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means, and storage in data banks. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. # 2012 Springer-Verlag/Wien SpringerWienNewYork is part of Springer Science+Business Media springer.at Typesetting: SPi Publisher Services, Pondicherry, India Printed on acid-free and chlorine-free bleached paper SPIN: Library of Congress Control Number: ISBN DOI / SpringerWienNewYork e-isbn

5 Dedicated to the memory of Yuval Ne eman and John Hubbard, two great physicists whose ideas are the corner stones of the theories presented in this book.

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7 Preface The main goal of this monograph is to demonstrate the relevance of dynamical symmetry and its breaking to the rapidly growing field of nanophysics in general, and nanoelectronics in particular. It is intended to amalgamate seemingly highly abstract concepts of Group theory with the physics of recently fabricated nanoobjects such as single electron transistors. In all these systems, dynamical symmetries are shown to be intimately related with many-body physics, and in particular, the ubiquitous Kondo effect and other hallmarks of quantum impurity problems. Thereby, we expose yet another facet of the existing deep and profound relations between quantum field theory and condensed matter physics. The concept of symmetry in quantum mechanics has had its golden age in the middle of the last century. In that period, the beauty, elegance and efficiency of group theoretical physics has been exposed in numerous remarkable revelations, from classification of hadron multiplets, isospin in nuclear reactions, the orbital symmetry in Rydberg atoms, point-groups in crystallography, translational symmetry in solid state physics, and so on. At the focus of all these studies stands the symmetry group of the underlying Hamiltonian. Using the powerful formalism of group theory, the energy spectrum of the physical system possessing the pertinent symmetry could be extracted within an elegant and time saving formalism. Exploiting the properties of discrete and infinitesimal rotation and translation operators, general statements about the basic properties of quantum mechanical systems could be formulated in a form of theorems (Wigner theorem, Bloch theorem, Goldstone theorem, Adler principle, etc). The intimate relation between group theory and quantum mechanics is therefore well established and has been exposed in numerous excellent handbooks. A somewhat more subtle aspect featuring group theory and quantum mechanics emerged and was formulated later on, that is, the concept of dynamical symmetry. The notion of dynamical symmetry group is distinct from that of the familiar symmetry group. To understand this distinction in an heuristic way let us recall that all generators of the symmetry group of the Hamiltonian Ĥ encode certain integrals of the motion, which commute with Ĥ. These operators induce all transformations which conserve the symmetry of the Hamiltonian, and may have non-diagonal matrix elements only within a given irreducible representation space of Ĥ. On the other vii

8 viii Preface hand, dynamical symmetry of Ĥ is realized by transformations implementing transitions between states belonging to different irreducible representations of the symmetry group. One may then say that the generators of dynamical symmetry group of a quantum mechanical system are in fact the generators of the energy spectrum or some part of it. Special examples of dynamical symmetries in quantum mechanics emerge as hidden symmetries, where additional degeneracy exists due to an implicit symmetry of the interaction. Another example is supersymmetry, where the group algebra includes both commutation and anticommutation relations. The starting point in most of our analysis is a generalized Anderson Hamiltonian which, under certain conditions can be approximated by a generalized spin Hamiltonian encoding a myriad of exchange interactions between localized electrons in nano-objects (such as quantum states in complex quantum dots) and itinerant electrons in the reservoirs made in contact with the localized electrons. These exchange interactions may be due to spin as well as to orbital degrees of freedom. They lead to effective exchange Hamiltonians that display a rich pattern of dynamical symmetries. Mathematically, these symmetries are exposed as the pertinent exchange Hamiltonian includes, in addition to the standard spin operators, new sets of vector operators which form the basis for the representation of irreducible tensor operators entering the effective Hamiltonian. These operators induce transitions between different spin multiplets and generate dynamical symmetry groups (such as SU(n) and SO(n)) that are not exposed within the bare Anderson Hamiltonian. Like in quantum field theory, the most dramatic aspects of dynamical symmetry in the present context is not its relation with the spectrum but, rather, the manner in which it is broken. An indispensable tool for manipulating the pertinent mathematics required for identifying the relevant dynamical symmetry groups is the superalgebra of Hubbard operators, upon which we will heavily rely. The role of dynamical symmetries and their manifestations will be reviewed and analyzed in several systems such as complex quantum dots (planar, vertical and self-assembled), molecular complexes adsorbed on metallic surfaces and attached to quantum wires, cold gases confined in magnetic traps. It will be shown how these dynamical symmetries are activated by Coulomb and exchange interactions with itinerant electrons in the macroscopic Fermi or Bose reservoirs (metallic leads and substrates in various nanodevices). We will then develop the concept within numerous physical situations, including the Kondo cotunnelling in various environments. The notion of dynamical symmetry is meaningful also for the systems out of equilibrium, in presence of electromagnetic field and stochastic noise and in timedependent problems like Landau Zener effect. Thus, the main goal of this book is to generalize the principles of dynamical symmetries formulated for the integrable systems to the many-body systems, for which only the low-energy part of the excitation spectrum is known. Tel Aviv - Trieste - Beer Sheva, October 31, 2011 Konstantin Kikoin Mikhail Kiselev Yshai Avishai

9 Acknowledgements We acknowledge fruitful discussions with our colleagues Boris Altshuler, Jan von Delft, Peter Fulde, Yuri Galperin, Yuval Gefen, Leonid Glazman, Vladimir Gritsev, David Khmelnitskii, Il ya Krive, Tetiana Kuzmenko, Stefan Ludwig, Laurens W. Molenkamp, Florina Onufrieva, Michael Pustilnik, Jean Richert, Robert Shekhter, Maarten Wegewijs. ix

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11 Contents 1 INTRODUCTION HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES RigidRotator Hydrogen atom and Runge-Lenz vector Dynamicalsymmetriesforspinsystems Hubbard atom and Fulde molecule Three-fold way for Hubbard atom Fock Darwinatom Dynamicalsymmetryandsupersymmetry Manifestationsofsupersymmetryinatomicmodels Quasienergy spectrum for periodical time-dependent problems NANOSTRUCTURES AS ARTIFICIAL ATOMS AND MOLECULES Introductory remarks Planarquantumdots Verticalquantumdots Self-assembledquantumdots Complexquantumdots Double quantum dots Triplequantumdots Moleculesandmolecularcomplexes Fullerenemoleculesasquantumdots Nanotubesasquantumdots Single electron tunneling through metal organic complexes Vibrational degrees of freedom in single molecular tunneling 101 xi

12 xii Contents 4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT Kondo mapping and beyond (surplus symmetries) Kondo effect in quantum dots with even occupation Kondo physics for short chains Serialgeometry Side geometry, Fano Kondo effect Crossgeometry Parallelgeometry Multichannel Kondo tunneling Kondo physics for small rings Kondo tunneling and Aharonov Bohm interference DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS Kondo effect in molecular environment Chiral symmetry of orbitals and Kondo tunneling Kondo effect in the presence of Thomas-Rashba precession Scanning tunneling spectroscopy via Kondo impurities Kondo effect in molecular magnets Phonon assisted tunneling Two-electron tunneling at strong electron-phonon coupling DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF QUANTUM DOTS Kondo effect in the presence of electromagnetic field Excitonicspectroscopyofquantumdots DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM ELECTRON TRANSPORT Dynamically induced finite bias anomalies in tunneling spectra Dephasing and decoherence in quantum tunneling VectorKeldyshmodelinthetimedomain TUNNELING THROUGH MOVING NANOOBJECTS Conversion of coherent charge input into the Kondo response Single-electron shuttling Time-dependentLandau-Zenereffect MATHEMATICAL INSTRUMENTATION SU(2) groupforarbitraryspin Kinematical constraints for systems with SO(n) and SU(n) symmetries SO(4) group Noncompact groups SO(p,n p) Groups of conformal transformations From SU(2) to SU(n) Bosonizationandfermionizationforarbitraryspins...320

13 Contents xiii Schwinger boson representation for the SU(2) group Holstein Primakoff boson representation for the SU(2) group Dyson Maleev representation for the SU(2) group Pomeranchuk Abrikosov spin fermion representation for the SU(2) group Spin-fermion representations for the SO(n) groups Popov Fedotovsemi-fermionrepresentation Majoranafermionization Mixedfermion-bosonrepresentations CONCLUSIONS AND PROSPECTS Index References...341

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