Konstantin Kikoin Mikhail Kiselev Yshai Avishai Dynamical Symmetries for Nanostructures Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules Ph.D Konstantin Kikoin School of Physics and Astronomy Tel Aviv University 69978 Tel Aviv Israel konstk@post.tau.ac.il Ph.D Mikhail Kiselev The Abdus Salam Intl Center for Theoretical Physics Strada Costiera 11 34151 Trieste Italy mkiselev@ictp.it Ph.D Yshai Avishai Ben Gurion University 84105 Beer Sheva Israel yshai@bgumail.bgu.ac.il 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: 12590339 Library of Congress Control Number: 2011943752 e-ISBN 978-3-211-99724-6 ISBN 978-3-211-99723-9 DOI 10.1007/978-3-211-99724-6 SpringerWienNewYork 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 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 ˆ generators of the symmetry group of the Hamiltonian H encode certain integrals ˆ These operators induce all transformations of the motion, which commute with H which conserve the symmetry of the Hamiltonian, and may have non-diagonal maˆ trix elements only within a given irreducible representation space of H On the other vii viii Preface ˆ hand, dynamical symmetry of H 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 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 336 – in Kondo effect, 253–255, 260, 261, 263, 264, 266–268, 286 – related to dynamical symmetries, 259 Dicke model, 234 dynamical symmetries, 1, 49, 55 – definition, 5, 329, 330 – in 1D and 2D nanosystems, 332, 333 – in Kondo effect, 107, 108, 136, 330 – in Landau-Zener effect, 293, 294, 296–298 – in non-equilibrium systems, 4, 21, 104, 245, 248–255, 257, 331 – noise induced, 4, 261, 263, 264, 266–270, 272, 274, 275 dynamical symmetry breaking, – Anderson – Nambu mechanism, – Higgs – Anderson mechanism, – electroweak interaction, 31 dynamical symmetry groups, – definition, – point C2n group, 212, 216 – point C3v group, 82, 88, 180, 211 – point D2d group, 83, 85, 185 – SO(1,1) subgroup, 313 – SO(12) group, 89, 91, 183 – SO(2,1) group, 32, 34, 63, 104, 217, 233, 245 – SO(3) subgroup, 10–12, 19, 142, 183, 268–270, 272, 310 – SO(3,1) group, 9, 313, 314 – SO(4) group, 2, 9, 11, 12, 20, 21, 29, 40, 41, 50, 60, 63, 77, 79, 80, 88, 89, 91, 104, 125, 127–133, 142, 146, 149, 161, 173, 183, 222, 224, 249–255, 285, 287, 291, 292, 295, 311–313 – SO(4,1) group, 314 – SO(4,2) conformal group, 2, 13, 14, 41, 314 – SO(5) group, 20, 21, 29, 60, 78, 80, 87, 142–144, 239, 257, 261, 263, 264, 266–268 – SO(6) group, 19, 21, 60, 76, 126, 132 – SO(7) group, 21, 142–144 – SO(8) group, 21, 89, 139, 161, 177, 178, 239 – SO(9) group, 139 – SO(n) group, 3, 14, 15, 21, 23, 59, 91, 117, 160, 245, 325, 330 – SO(n,1) group, 2, 14 – SO(n,2) group, 14 – SO(p,n-p) group, 313, 314 – SU(10) group, 123 – SU(2) subgroup, 15, 17, 26, 34, 57, 64, 68, 75, 76, 85, 86, 108, 112, 113, 120, 121, 124, 127, 128, 131, 132, 138, 180, 216, Index 228, 229, 231, 276, 295, 309, 310, 312, 316–319 – SU(3) group, 2, 10, 23, 27, 29, 31, 42, 43, 59, 111, 112, 116, 145, 208, 259, 285, 299–301, 315, 316, 320, 322 – SU(4) group, 17, 22, 27, 56, 58, 65, 73–75, 84–86, 91, 110, 118, 120, 123, 137, 138, 147, 182, 183, 185, 187, 201, 208, 227–229, 231, 258, 261, 263, 264, 272, 274, 275, 285, 289, 294, 317–319, 322 – SU(5) group, 208 – SU(6) group, 65, 85, 86, 91, 123, 180, 208, 210, 214, 222 – SU(n) group, 3, 15, 23, 73, 108, 117, 119, 121, 175, 196, 208, 212, 245, 293, 315, 320, 322, 330 eightfold way, 2, 29, 31, 258 electron shuttling, 4, 104 Euler angles, evolution operator, 296, 297 exchange interaction, 17, 51, 59, 63 – double (Zener) exchange, 87 – Dzyaloshinskii – Moriya exchange, 203–205 – indirect exchange, 17, 28, 58, 59, 157, 171, 182, 193, 212, 220, 224–226, 235, 236, 250–253 – RKKY exchange, 99, 125, 126, 139 excitons in quantum dots, 18, 59, 61, 238–240 – charge transfer excitons, 18, 20, 28, 77–79, 87–89, 165, 177, 265 – exciton-electron complexes, 240–242 – in optical response, 239–244 – many-exciton states, 69, 243, 244 – singlet excitons, 59, 60, 69, 90, 139, 222, 257 – singlet-triplet excitons, 19, 20, 28, 59, 63, 77–79, 87–89, 98, 104, 125, 127, 128, 130–134, 139, 146, 148, 173, 177, 222, 257, 265, 268, 287 Fano effect, 149, 150, 207 – Fano – Cooper factor, 151, 208 – interplay with Kondo effect, 150–153, 159, 190, 191, 195 Floquet – Bloch theorem, 45, 47, 233, 292 Fock – Darwin model, 31, 33, 34, 61, 64, 135, 177 – energy levels, 32, 34, 39, 40, 61, 68 – two-band, 67, 69, 240, 241 – wave functions, 33 Friedel sum rule, 114, 134 frustration, 185, 187, 210 Index Gaussian distribution function, 270, 274, 275, 278, 279, 302–306 Gell-Mann matrices, 22 – of 3rd rank, 25, 29, 111, 116, 299–301, 315, 316, 320, 322 – of 4th rank, 22, 25, 26, 56, 110, 145, 317–319, 322 Glazman – Raikh rotation, 55, 74, 81, 84, 86, 164, 167 Green function, 118, 119, 150, 151, 159, 190, 207, 238, 242, 249, 250, 254, 255, 268–270, 272, 274, 275, 278, 279 – for nonequilibrium systems, 247 harmonic oscillator, 2, 32, 44, 49 hidden symmetry, 2, 12, 14, 41, 214 Huang – Rhys factor, 103, 220, 226, 229 Hubbard atom, 23, 27, 29, 31, 41, 43, 102, 227, 228, 319, 320 – energy levels, 24, 27, 42, 55, 56, 110, 265 – – time-dependent, 293, 299 Hubbard model, 7, 23, 44 – Hubbard chain, 136 Hubbard molecule, 72 – dimer, 72, 73 – – energy levels, 76–79 – – even occupation, 76–80 – – odd occupation, 73–75 Hubbard operators, 6, 7, 16, 17, 21, 24, 55, 75, 176, 178, 187, 214, 238, 241 – Bose-type, 7, 25, 42, 57, 116 – commutation relations, 6, 25, 26 – Fermi-type, 7, 25, 42, 55, 116 – relation to SO(n) group generators, 19, 21, 311, 325 – relation to SU(n) group generators, 17, 22, 25, 26, 56, 58, 110, 187, 258, 299, 311, 319, 320, 324, 328 Hubbard parabola, 53, 55, 56, 227, 228 Hund’s rules, 124, 200 hydrogen atom, 2, 3, 7, 10, 14, 20, 40, 49 – n-dimensional, 14, 41 hydrogen molecule, 18, 92 hyperspherical harmonics, integrable systems, 2, 4, 5, 15, 50, 68, 163 – Bethe ansatz, 114 irreducible representations, 5–7, 321 – for SU(3) group, 29, 30 irreducible tensor operators, 7, 9, 10, 22, 23, 26, 317 Keldysh contour, 277 Kolokolov representation, 297 337 Kondo effect, 28, 50, 65, 69, 108 – in molecular complexes, 199, 208, 209, 211–216 – – even occupation, 199, 200, 229, 231 – – for pair tunneling, 229, 231 – – in presence of TR precession, 205 – – odd occupation, 199–201, 210 – – phonon assisted, 218–220 – – phonon induced, 222–226 – in nonequilibrium systems, 245–247, 260, 261 – – even occupation, 249–255, 257 – in SET, 92 – – even occupation, 126–128, 130–135, 138–140, 142–145, 148, 149, 160, 169, 171–173, 175, 177, 178, 183, 185, 196, 236–239 – – odd occupation, 109–113, 115, 117–119, 121, 123, 124, 137, 138, 145–148, 156–158, 165–167, 169, 176, 180, 182–185, 187, 189, 196 – – photon assisted, 236 – – photon induced, 236–240 – interplay with Aharonov – Bohm effect, 189–193, 206 – magnetic field induced, 126–128, 130, 138, 145, 169, 171–173, 175, 180, 182, 183, 216, 225, 226, 287, 289 – multichannel, 162–167 – orbital, 90, 175, 177, 178, 201 – overscreened spin, 124, 162, 240 – spinons and holons, 116–118, 163 – three-channel, 153, 167, 181 – two-channel, 90, 132, 153, 167, 169, 171, 176–178, 188, 221 – – non-Fermi liquid regime, 154, 163–167, 169, 171–173, 175, 187, 189, 210 – – orbital anisotropy, 124, 125, 164–167 – two-site Kondo model, 124, 126, 139, 155 – underscreened spin, 124, 196, 240 Kugel – Khomskii Hamiltonian, 75, 84, 187, 210 Landau – Zener effect, 4, 99 – in n-level systems, 293, 294, 296–301, 307 – in presence of noise, 301–307 – in two-level systems, 293, 301–306 Landau levels, 39 Landauer formula, 151 Lang – Firsov canonical transformation, 102, 219, 227 Larmor (diamagnetic) shift, 63–65, 68, 69, 177, 185, 200 338 line-shape function, 103, 220, 226, 234–236 Luttinger liquid, 155 magnetic flux, 180–185, 189–193, 287, 289 molecular complex, 3–5, 91, 96–101, 197 – in single-electron transistors, 198 – in STM spectroscopy, 201, 206–211 – molecular grid, 98, 99, 199 – rare-earth metal organic complexes (REMOC), 96, 101 – transition metal organic complexes (TMOC), 96, 101, 212–216, 222–226 molecular excitons, 94, 101 molecular magnets, 44, 99, 211–216 molecule – as quantum dot, 92, 95, 108, 128, 132, 197, 198 – carbon peapod, 96 – fullerene, 92–94, 100, 132, 199 – – dimetallofullerene, 94 – – endofullerene, 94 – in break junction geometry, 91, 93, 96 – in STM geometry, 91 – metallocene, 94, 96, 97 – nanotube, 92, 94, 95, 100, 108, 128, 199, 255 multiplets, 1, 2, 9, 15–18, 20, 21, 23–25, 29, 42, 56, 60, 73, 93, 125, 156, 157, 165, 177, 180, 257 – hadron multiplets, 1, 29 – – isospin and hypercharge, 29, 31, 258 nanoelectromechanical single-electron tunneling transistor (NEM-SET), 105, 283, 284, 290 – shuttling in Kondo regime, 283, 291, 292 noise, 263 – in quantum dots, 264–268 – in TLS, 276–280 – in ultracold gases, 280 – Keldysh model, 269 – – scalar, 270, 272, 279, 280, 304–306 – – vector, 272, 274, 275, 277–280, 302–306 Onsager relations, 194 optical traps, 300, 330 orthogonality catastrophe, 107, 330 – shake-up effects, 239, 241, 243, 244 paramagnetic susceptibility, 247, 272, 275 path integral method, 277–279 Pauli matrices, 16, 17, 75, 309, 310 polaron shift, 220, 226, 227 Index projection operator, 6, 9, 50 pseudospin operator, 17, 26, 75, 84, 90, 175, 178, 182, 183, 187, 189, 201, 213, 272, 274, 275, 293, 299, 318, 319, 330 quantum box, 175 quantum criticality, 3, 173, 175 quantum dot, 3, 5, 91, 330 – complex quantum dots, 4, 5, 70, 71, 91 – – double quantum dots (DQD), 70–73, 75–80, 125, 137, 138, 147–149, 154, 160, 167, 169, 171–173, 175, 249–255, 286, 287, 295–298 – – Fulde molecule, 28, 29, 60, 72, 94 – – triple quantum dots (TQD), 70, 71, 80, 81, 83, 85–90, 139, 140, 142–146, 152, 153, 155–158, 160, 165–167, 177, 189–193, 195, 289 – geometry, 71, 136 – – Δ -shape, 71, 83, 185, 187, 189, 287, 289 – – ∇-shape, 71, 83, 185, 187, 188, 190 – – cross, 71, 81, 155–158 – – parallel, 71, 74, 81, 159, 160, 165–167, 177, 189 – – ring-shaped, 179 – – serial, 71, 74, 81, 137–140, 142–146 – – T-shape, 71, 74, 81, 96, 147–149, 152–155, 167, 169, 171–173, 175, 249–255, 261, 263, 264, 266–268, 286 – – triangular, 71, 82, 83, 179, 180, 182–185, 187, 189–195, 210, 261, 263, 264, 272 – – V-shape, 71, 83, 84 – lateral(planar), 51, 52, 72, 108, 153 – – level spacing, 51, 66, 169 – self-assembled, 65, 67, 69, 238, 240–244 – – level spacing, 67 – – wetting layer, 67, 69, 240–244 – vertical (disk-like), 61–65, 71, 160, 176, 185, 192 quantum wire, 155 quasienergy levels, 46, 47, 292 quasienergy states, 45, 46, 233, 245, 292 Rayleigh distribution function, 274, 275 renormalization group (RG), 110–113, 115, 146 – Anderson procedure, 113, 122, 140, 228, 229, 231, 266 – – for nonequilibrium systems, 246 – fixed point, 114, 124, 126, 163, 164, 166, 214 Index – flow trajectories, 113, 142, 157, 161, 164, 166, 171, 214 – Jefferson – Haldane procedure, 110, 112, 121, 140, 146, 149, 157, 160, 177, 191, 228, 266 – multistage renormalization, 110, 112, 116, 121, 125, 130–134, 137, 138, 144, 148, 149, 154, 158, 178, 180, 191, 222, 228, 229, 231, 266, 330 – numerical RG, 114, 115, 119, 134, 176, 187, 190, 192, 201, 209, 215 – scaling equations, 113, 114, 129, 130, 133, 134, 137, 146, 158, 161–164, 166, 171, 174, 182, 193, 214, 231, 252, 253 – scaling invariants, 113, 140 Riccati equation, 297, 300 rigid rotator, 2, 7–9, 14, 20 – n-dimensional, 14 rotation operators, 11, 311–313 Runge – Lenz vector, 2, 3, 10, 11, 14, 20, 40 scattering phase shift, 114, 134, 159, 163, 190, 244 Schră dinger equation, 2, 5–7, 10, 15, 16, 20, o 45–47, 301 Schrieffer – Wolff (SW) Hamiltonian, 58, 73, 75, 81, 83, 89, 93, 108, 148, 153, 156, 164, 166, 182, 187, 193, 220 – Coqblin – Schrieffer model, 120, 122 – Cornut – Coqblin model, 121, 122 – for negative U Anderson model, 228, 229 – in presence of dynamical symmetries, 61, 78, 84, 87, 88, 90, 91, 107, 128, 129, 133, 143, 144, 146, 158, 171, 178, 188, 222–226, 228, 229, 250, 266, 267, 287, 289, 291, 292 – in presence of TR precession, 202–205 – magnetic anisotropy, 171, 172, 208, 212–214, 229 – three-site exchange Hamiltonian, 185 – time-dependent, 219, 235, 236, 260, 266, 267, 287, 289, 291, 292 – two-site exchange Hamiltonian, 124, 125 selection rules, 2, 10, 209 single electron tunneling (SET), 3, 52, 54 – electron cotunneling, 57, 59, 73, 177 – – inelastic, 108, 109, 258 – – pair tunneling, 227–229, 231, 258 – – phonon assisted, 217, 218, 220, 222–226 – – photon assisted, 233–236, 260, 261 – in non-equilibrium systems, 236 – through molecules, 92, 197, 199 – – nanotubes, 95 – – phonon assisted, 92, 100, 102 339 spin filters and valves, 196 spin relaxation, 254, 255 spin-orbit interaction, 121, 123 spirality, 95, 96, 206, 255 strongly correlated electron systems (SCES), 7, 24, 26, 41, 54, 107 superalgebra, 6, 25, 35–37, 43 – Z(2) graded, 36, 37, 40 supersymmetry, 3, 14, 35–42 – in mesoscopic systems, 44 – supercharge operators, 35–38, 40–43 – supersymmetric Hamiltonian, 37, 43 – – energy spectrum, 38, 41 – – spinor representation, 38 Thomas – Rashba (TR) precession, 201 – TR Hamiltonian, 202–206 three-level systems, 23, 81, 316 – even occupation, 80, 145 – – energy levels, 80, 86, 88, 177, 183 – – wave functions, 86, 87 – odd occupation, 81, 157, 158 – – energy levels, 81, 82, 84, 85, 156, 165, 180 – – wave functions, 82, 146, 156, 165 transition matrix (T -matrix), 115, 119, 150, 221 trimers, 86, 210 tunneling conductance, 54, 164, 167, 169 – finite bias anomaly (FBA), 109, 201, 237, 245, 248–255, 257, 331 – unitarity limit, 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