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Quantum impurity physics and the “NRG Ljubljana” code

Quantum impurity physics and the “NRG Ljubljana” code. Rok Žitko. J. Stefan Institute, Ljubljana, Slovenia. UIB, Palma de Mallorca, 1 2 . 12 . 2007. Quantum transport theory prof. Janez Bon ča 1,2 prof. Anton Ramšak 1,2 Tomaž Rejec 1,2 Jernej Mravlje 1.

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Quantum impurity physics and the “NRG Ljubljana” code

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  1. Quantum impurity physicsandthe “NRG Ljubljana” code Rok Žitko J. Stefan Institute, Ljubljana, Slovenia UIB, Palma de Mallorca, 12. 12. 2007

  2. Quantum transport theory prof. Janez Bonča1,2 prof. Anton Ramšak1,2 Tomaž Rejec1,2 Jernej Mravlje1 Experimental surface science and STM prof. Albert Prodan1 prof. Igor Muševič1,2 Erik Zupanič1 Herman van Midden1 Ivan Kvasić1 1 J. Stefan Institute, Ljubljana, Slovenia 2 Faculty of Mathematics and Physics, Uni. of Ljubljana, Ljubljana, Slovenia

  3. Outline • Impurity physics • Numerical renormalization group • SNEG– Mathematica package for performing symbolic calculations with second quantization operator expressions • NRG Ljubljana • project goals • features • some words about the implementation • Impurity clusters • N parallel quantum dots (N=1...5, one channel)

  4. Classical impurity

  5. Quantum impurity This is Kondo model!

  6. Nonperturbative behaviour The perturbation theory fails for arbitrarily small J !

  7. Screening of the magnetic moment Kondo effect!

  8. “Asymptotic freedom” ... T >> TK

  9. ... and “infrared slavery” T << TK Analogy: TK QCD

  10. Nonperturbative scattering

  11. Why are quantum impurity problems important? • Quantum systems in interaction with the environment (decoherence) • Magnetic impurities in metals (Kondo effect) • Electrons trapped in nanostructures (transport phenomena) • Effective models in dynamical mean-field theory (DMFT) of strongly-correlated materials

  12. Renormalization group ?

  13. Many energy scales are locally coupled (K. G. Wilson, 1975) Cascade effect

  14. L-n/2 Numerical renormalization group (NRG)

  15. Iterative diagonalization Recursion relation:

  16. Tools: SNEG and NRG Ljubljana Add-on package for the computer algebra system Mathematica for performing calculations involving non-commuting operators • Efficient general purpose numerical renormalization group code • flexible and adaptable • highly optimized (partially parallelized) • easy to use Both are freely available under the GPL licence: http://nrgljubljana.ijs.si/

  17. t e, U e, U Package SNEGhttp://nrgljubljana.ijs.si/sneg

  18. SNEG - features • fermionic (Majorana, Dirac) and bosonic operators, Grassman numbers • basis construction (well defined number and spin (Q,S), isospin and spin (I,S), etc.) • symbolic sums over dummy indexes (k, s) • Wick’s theorem (with either empty band or Fermi sea vacuum states) • Dirac’s bra and ket notation • Simplifications using Baker-Campbell-Hausdorff and Mendaš-Milutinović formula

  19. SNEG - applications • exact diagonalization of small clusters • perturbation theory to high order • high-temperature series expansion • evaluation of (anti-)commutators of complex expressions • NRG • derivation of coefficients required in the NRG iteration • problem setup

  20. “NRG Ljubljana” - goals • Flexibility (very few hard-coded limits, adaptability) • Implementation using modern high-level programming paradigms (functional programming in Mathematica, object oriented programming in C++)  short and maintainable code • Efficiency (LAPACK routines for diagonalization) • Free availability

  21. Package “NRG Ljubljana”http://nrgljubljana.ijs.si/ open source,GPL

  22. f0,L f0,R a b Definition of a quantum impurity problem in “NRG Ljubljana” t Himp = eps (number[a[]]+number[b[]])+U/2 (pow[number[a[]]-1,2]+pow[number[b[]]-1,2]) Hab = t hop[a[],b[]] Hc = Sqrt[Gamma] (hop[a[],f[L]] + hop[b[],f[R]]) + J spinspin[a[],b[]] + V chargecharge[a[],b[]]

  23. f0,L f0,R a b Definition of a quantum impurity problem in “NRG Ljubljana” t Himp = epsa number[a[]] + epsb number[b[]] +U/2 (pow[number[a[]]-1,2]+pow[number[b[]]-1,2]) Hab = t hop[a[],b[]] Hc = Sqrt[Gamma] (hop[a[],f[L]] + hop[b[],f[R]])

  24. Computable quantities • Finite-site excitation spectra (flow diagrams) • Thermodynamics: magnetic and charge susceptibility, entropy, heat capacity • Correlations: spin-spin correlations, charge fluctuations,...spinspin[a[],b[]] number[d[]]pow[number[d[]], 2] • Dynamics: spectral functions, dynamical magnetic and charge susceptibility, other response functions

  25. Spectral function Charge fluctuations Occupancy Sample input file [param]model=SIAMU=1.0Gamma=0.04 Lambda=3Nmax=40keepenergy=10.0keep=2000 ops=q_d q_d^2 A_d Model and parameters NRG iteration parameters Computed quantities

  26. Kondo effect in quantum dots Conduction as a function of gate voltage for decreasing temperature W. G. van der Wiel, S. de Franceschi, T. Fujisawa, J. M. Elzerman, S. Tarucha, L. P. Kouwenhoven, Science 289, 2105 (2000)

  27. Scattering theory “Landauer formula” See, for example, M. Pustilnik, L. I. Glazman, PRL 87, 216601 (2001).

  28. Keldysh approach One impurity: Y. Meir, N. S. Wingreen. PRL 68, 2512 (1992).

  29. Conductance of a quantum dot (SIAM) Computed using NRG.

  30. Systems of coupled quantum dots triple-dot device L. Gaudreau, S. A. Studenikin, A. S. Sachrajda, P. Zawadzki, A. Kam, J. Lapointe, M. Korkusinski, and P. Hawrylak,Phys. Rev. Lett. 97, 036807 (2006). M. Korkusinski, I. P. Gimenez, P. Hawrylak,L. Gaudreau, S. A. Studenikin, A. S. Sachrajda,Phys. Rev. B 75, 115301 (2007).

  31. Parallel quantum dots and the N-impurity Anderson model Vk = eikL vk Vk≡V (L0) R. Žitko, J. Bonča: Multi-impurity Anderson model for quantum dots coupled in parallel, Phys. Rev. B 74, 045312 (2006)R. Žitko, J. Bonča: Quantum phase transitions in systems of parallel quantum dots, Phys. Rev. B 76, .. (2007).

  32. RKKY exchange Super-exchange Conduction-band mediated inter-impurity exchange interaction

  33. Effective single impurity S=N/2 Kondo model The RKKY interaction is ferromagnetic, JRKKY>0: JRKKY0.62 U(r0JK)2 4th order perturbation in Vk Effective model (T<JRKKY): S is the collective S=N/2 spin operator of the coupled impurities, S=P(SSi)P

  34. Free orbital regime (FO) Ferro-magnetically frozen (FF) Local moment regime (LM) Strong-coupling regime (SC)

  35. The spin-N/2 Kondo effect Full line: NRG Symbols: Bethe Ansatz

  36. Conductance as a function of the gate voltage

  37. Kondo model Kondo model + potential scattering

  38. S=1 Kondo model + potential scattering S=1/2 Kondo model + strong potential scattering S=1 Kondo model

  39. Gate-voltage controlled spin filtering

  40. Spectral functions

  41. Kosterlitz-Thouless transition d1=+D, d2=-D S=1/2 Kondo S=1 Kondo

  42. Conclusions • Impurity clusters can be systematically studied with ease using flexible NRG codes • Very rich physics: various Kondo regimes, quantum phase transitions, etc. But to what extent can these effects be experimentally observed? • Towards more realistic models: better description of inter-dot interactions, role of QD shape and distances. http://nrgljubljana.ijs.si/

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