Inelastic scattering and “dephasing” from quantum impurities

1. 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, LászlóBorda, Gergely Zarand, Natan Andrei, P. Coleman, Phys. Rev. B 72, 014430 (2005) Theory seminar, Grenoble

2. 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 Theory seminar, Grenoble

3. Motivation, some conceptual questions, and experimental relevance Theory seminar, Grenoble

4. electron photon (phonon, electron-hole pair) (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): Theory seminar, Grenoble

5. 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 ? • Luttinger liquid • some NFL impurity models Yes ! They are sometimes Very different from electrons… • Quantum critical points • 1D disordered interacting electrons ? Theory seminar, Grenoble

6. Sources of inelastic scattering • Electron-electron interaction • Magnetic impurities • Tow-level systems • Phonons • Magnons • … Expectation: diverges as Theory seminar, Grenoble

7. from weak localization Pierre et al [PRB 2003] Mohanty, Jariwala, & Webb [PRL 1997] Saturation of Theory seminar, Grenoble

8. 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 • 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] Theory seminar, Grenoble

9. Experiments Experiments measuring • Weak localization experiments on wires: • Mohanty, Jariwala, & Webb [PRL 1997]; • Mohanty, & Webb [PRL 2003] • Pierre et al [PRB 2003] • Schopfer, Bauerle, Rabaud, Saminadayar [PRL 2003] • Bauerle, et al [PRL 2005] Saturation of importance of magnetic scattering Systematic study of magnetic scattering • Energy distribution measurements: • Pothier, Gueron, Birge, Esteve, Devoret [PRL 1997]; Theory seminar, Grenoble

10. Out-of equilibrium measurements SC wires Pothier, Gueron, Birge, Esteve, Devoret [PRL 1997]; Theory seminar, Grenoble

11. Importance of impurities Ag(6N) Au(6N) • No saturation in high purity samples • Mn doping similar to low purity Ag(5N) Cu(6N) Pierre et al [PRB 2003] Theory seminar, Grenoble

12. from Kondo impurities Schopfer, Bauerle, Rabaud, Saminadayar [PRL 2003] Mohanty & Webb [PRL 2003] Experiments Theory seminar, Grenoble

13. How to define / compute inelastic scattering from a quantum impurity ? Theory seminar, Grenoble

14. 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] Quasiparticles at T=0 DONOT RELAX electronsDO Electric field couples to ELECTRONS (not quasiparticles) Theory seminar, Grenoble

15. Inelastic scattering for Kondo model: T=0 Consider impurity in ground state + electron wave packet far away Elastic scattering Inelastic Scattering electron leaves behind excitations Theory seminar, Grenoble

16. Definition of the S-matrix in the interaction representation: Many-body operator ! T-matrix: Note: describes the scattering ofELECTRONS, not quasiparticles! Theory seminar, Grenoble

17. Scattering of single electron states single electron states: i.e. eigenstates of H0, but as or, in terms of wave packets: Electron is still far away when the interaction is on Electrons far away when the interaction is off Note: we send in ELECTRONS and watch outgoing electrons, not quasiparticles! Theory seminar, Grenoble

18. Connection to scattering cross sections Many-body operator ! T-matrix: Total cross section (optical theorem): forward scattering of single particles total cross section Elastic cross section: elastic cross section is also related to : Sum over all final states with precisely one outgoing electron Inelastic scattering cross section: Theory seminar, Grenoble

19. How to compute Reduction formulas relate the time-ordered Green’s function with Follow, e.g., Itzikson & Zuber to obtain: Full time-ordered Green’s function Theory seminar, Grenoble

20. Application to Kondo problem Theory seminar, Grenoble

21. Case of Kondo model Impurity spin Conduction electron [Costi PRL2000] Theory seminar, Grenoble

22. Method: Wilson’s numerical renormalization group [Wilson75] • one defines a sequence of discretized Hamiltonians • diagonalize iteratively • at N-th iterationone 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

23. Results obtained by numerical renormalization group: G.Z., László Borda, Jan von Delft, Natan Andrei, Phys. Rev. Lett. 93, (2004) • roughly linear for • Similar behavior is expected as a function of temperature [ as indeed found by T. Micklitz et al, PRL 2005] Theory seminar, Grenoble

24. Results obtained by numerical renormalization group: [C. Bauerle, F. Mallet, F. D. Mailly, G. Eska, and L. Saminadayar, PRL 95, 266805 (2005)] Theory seminar, Grenoble

25. High energy scattering rate: At very high frequencies all the scattering is inelastic ! Inelastic scattering Elastic scattering See also M. Garst, P. Wölfle, L. Borda, J. von Delft, L. I. Glazman, cond-mat/0507431 Theory seminar, Grenoble

26. Magnetic field dependence already a very small field results in a strong spin- dependence Theory seminar, Grenoble

27. 2CK model ~ anisotropy Theory seminar, Grenoble

28. Conclusions: • can be computed by exploiting reduction formulas and using NRG • the quadratically vanishing inelastic rate appears only well below • even a very small results in a strong spin asymmetry of the inelastic rate • For we obtain • Our formalism carries over to other quantum impurity models • The finite version of our formula describes the dephasing from magnetic impurities in weak localization experiments [T. Micklitz et al, cond-mat/0509583] [confirmed in M. Garst, et al., cond-mat/0507431] Theory seminar, Grenoble