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Inelastic scattering and “dephasing” from quantum impurities Gergely Zaránd (BUTE, TU Karlsruhe) Collaborators: László Borda (BUTE) Natan Andrei (Rutgers) Jan von Delft (LMU) Gergely Zaránd, László Borda, Jan von Delft, Natan Andrei Phys. Rev. Lett. 93, (2004) Pankaj Mehta, László Borda, Gergely Zarand, Natan Andrei, P. Coleman, Phys. Rev. B 72, 014430 (2005) 27/4/2006 Theory seminar, Grenoble Outline • Motivation, experimental relevance scattering off magnetic impurities is important in many experiments • How to define / compute inelastic scattering ? use of reduction formulas • Application to the Kondo problem: Non-perturbative results using numerical renormalization group • Conclusions 27/4/2006 Theory seminar, Grenoble Motivation, some conceptual questions, and experimental relevance 27/4/2006 Theory seminar, Grenoble (My) Definition of Inelastic scattering: A scattering process where the electron (or other particle) scatters by changing the quantum state of the “environment” Inelastic scattering destroys quantum interference (AB interference, localization, UCF etc.) : the typical time scale of inelastic scattering Example of AB interferometer (goes back to Einstein): electron photon (phonon, electron-hole pair) 27/4/2006 Theory seminar, Grenoble Some (unanswered) conceptual questions: Is separation of “particle” and “environment” possible? • Fermions, e.g. • Quasiparticles are always “dressed” Is one measuring quasiparticles or particles ? Can one always describe “nature” in terms of quasiparticles as T goes to 0 ? Yes ! They are sometimes Very different from electrons… • Luttinger liquid • some NFL impurity models • Quantum critical points • 1D disordered interacting electrons 27/4/2006 ? Theory seminar, Grenoble Sources of inelastic scattering • Electron-electron interaction • Magnetic impurities • Tow-level systems • Phonons • Magnons • … 27/4/2006 Expectation: diverges as Theory seminar, Grenoble T 0 1 from weak localization Pierre et al [PRB 2003] Mohanty, Jariwala, & Webb [PRL 1997] Saturation of 27/4/2006 Theory seminar, Grenoble Theoretical proposals • Electron-electron interaction: Intrinsic dephasing is suppressed Aleiner, Altshuler, Gershenson [1999] Altshuler, Aronov, Kmelnitsky [J. Phys. C 1982] Golubev & Zaikin [PRB 1999 & PRB 2000] Saturation of ~ T 2 / 3 • Magnetic impurities: Magnetic impurities mediate inelastic scattering Kaminski & Glazman [PRL 2001]; (Sólyom and Zawadowski [Z. Phys. ,1969]) Göppert, Galperin, Altshuler, Grabert [PRB, 2002] Kroha & Zawadowski [PRL 2002] • Tow-level systems: Imry, Fukuyama, Schwab [EPL, 1999] Zawadowski, von Delft, Ralph [PRL 1999] 27/4/2006 Theory seminar, Grenoble Experiments Experiments measuring • Weak localization experiments on wires: Mohanty, Jariwala, & Webb [PRL 1997]; Mohanty, & Webb [PRL 2003] Saturation of importance of magnetic scattering Pierre et al [PRB 2003] Schopfer, Bauerle, Rabaud, Saminadayar [PRL 2003] Bauerle, et al [PRL 2005] • Energy distribution measurements: Pothier, Gueron, Birge, Esteve, Devoret [PRL 1997]; 27/4/2006 1 Theory seminar, Grenoble Systematic study of magnetic scattering Out-of equilibrium measurements SC wires L ~ L Pothier, Gueron, Birge, Esteve, Devoret [PRL 1997]; 27/4/2006 Theory seminar, Grenoble Importance of impurities Ag(6N) ~ T 2 / 3 Au(6N) • No saturation in high purity samples • Mn doping similar to low purity Ag(5N) Cu(6N) Pierre et al [PRB 2003] 27/4/2006 Theory seminar, Grenoble 1 from Kondo impurities ~ 1/ T 2 Schopfer, Bauerle, Rabaud, Saminadayar [PRL 2003] ~ 1/ T Mohanty & Webb [PRL 2003] Experiments 27/4/2006 1 ~ T for T TK 1 ~ cst for T TK Theory seminar, Grenoble How to define / compute inelastic scattering from a quantum impurity ? 27/4/2006 Theory seminar, Grenoble Inelastic scattering for Kondo model: T=0 Interaction of a S = ½ magnetic impurity with one band of itinerant electrons Kondo temperature: Ground state is a singlet ~ Fermi liquid [Nozières, 1974] 1 / ~ T / TK 2 Quasiparticles at T=0 DO NOT RELAX electrons DO Electric field couples to ELECTRONS (not quasiparticles) 27/4/2006 Theory seminar, Grenoble Inelastic scattering for Kondo model: T=0 Consider impurity in ground state + electron wave packet far away Elastic scattering energy E Inelastic Scattering electron leaves behind excitations 27/4/2006 Theory seminar, Grenoble Definition of the S-matrix b, in Sˆ a, in b, out a, in in the interaction representation: Sˆ T exp T-matrix: Sˆ 1̂ i Tˆ i H int (t ) dt Many-body operator Note: Ŝ describes the scattering of ELECTRONS, not quasiparticles! 27/4/2006 Theory seminar, Grenoble ! Scattering of single electron states single electron states: i.e. eigenstates of H0, but as or, in terms of wave packets: Electrons far away when the interaction is off Electron is still far away when the interaction is on Note: we send in ELECTRONS and watch 27/4/2006 Theory seminar, Grenoble outgoing electrons, not quasiparticles! Connection to scattering cross sections T-matrix: Sˆ 1̂ i Tˆ Many-body operator Total cross section (optical theorem): forward scattering of single particles Elastic cross section: elastic cross section is also related to ! total cross section Tˆ : Sum over all final states with precisely one outgoing electron Inelastic scattering cross section: inel total el 27/4/2006 Theory seminar, Grenoble How to compute p, Tˆ p' , ' Reduction formulas relate the time-ordered Green’s function with p, Tˆ p' , ' Follow, e.g., Itzikson & Zuber to obtain: Full time-ordered Green’s function 27/4/2006 Theory seminar, Grenoble Application to Kondo problem 27/4/2006 Theory seminar, Grenoble Case of Kondo model Impurity spin Conduction electron ~J S ~ J 2 F ; F ( ) [Costi PRL2000] 27/4/2006 Theory seminar, Grenoble Method: Wilson’s numerical renormalization [Wilson75] group one defines a sequence of discretized Hamiltonians diagonalize iteratively 1 H lim (1 1) (N 1) / 2H N N 2 • • • • 27/4/2006 at N-th iteration one can calculate physical quantities at energy scale N~-N/2 Spectral function of ANY local operator Hilbert transform real part too High precision data needed (symmetries) Proper normalization is crucial Theory seminar, Grenoble Results obtained by numerical renormalization group: inel ( ) G.Z., László Borda, Jan von Delft, Natan Andrei, Phys. Rev. Lett. 93, (2004) • • inel roughly linear for 0.05TK 0.5TK inel ~ 2 for 0.05 TK • Similar behavior is expected as a function of temperature [ as indeed found by T. Micklitz et al, PRL 2005] 27/4/2006 Theory seminar, Grenoble Results obtained by numerical renormalization group: [C. Bauerle, F. Mallet, F. D. Mailly, G. Eska, and L. Saminadayar, PRL 95, 266805 (2005)] 27/4/2006 Theory seminar, Grenoble High energy scattering rate: At very high frequencies all the scattering is inelastic ! Inelastic scattering Elastic scattering ~ J2 ~ J4 See also M. Garst, P. Wölfle, L. Borda, J. von Delft, L. I. Glazman, cond-mat/0507431 27/4/2006 Theory seminar, Grenoble Magnetic field dependence already a very small field results in a strong spindependence 27/4/2006 Theory seminar, Grenoble 2CK model ~ anisotropy inel ( 0) el ( 0) 27/4/2006 Theory seminar, Grenoble Conclusions: • inel ( , B) can be computed by exploiting reduction formulas and using NRG • the quadratically vanishing inelastic rate appears only well below T K • even a very small of the inelastic rate • For TK inel ( ) ~ B results in a strong spin asymmetry we obtain 1 ln 2 ( / TK ) el ( ) ~ 1 ln 4 ( / TK ) [confirmed in M. Garst, et al., cond-mat/0507431] • Our formalism carries over to other quantum impurity models • The finite T version of our formula describes the dephasing from magnetic impurities in weak localization experiments [T. Micklitz et al, cond-mat/0509583] 27/4/2006 Theory seminar, Grenoble