Quantum impurity physics and the “NRG Ljubljana” code Rok Žitko UIB, Palma de Mallorca, J. Stefan Institute, Ljubljana, Slovenia

Quantum transport theory –prof. Janez Bonča 1,2 –prof. Anton Ramšak 1,2 –Tomaž Rejec 1,2 –Jernej Mravlje 1 Experimental surface science and STM –prof. Albert Prodan 1 –prof. Igor Muševič 1,2 –Erik Zupanič 1 –Herman van Midden 1 –Ivan Kvasić 1 1 J. Stefan Institute, Ljubljana, Slovenia 2 Faculty of Mathematics and Physics, Uni. of Ljubljana, Ljubljana, Slovenia

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)

Classical impurity

Quantum impurity This is Kondo model!

Nonperturbative behaviour The perturbation theory fails for arbitrarily small J !

Screening of the magnetic moment Kondo effect!

“Asymptotic freedom”... T >> T K

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

Nonperturbative scattering

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

Renormalization group ?

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

Numerical renormalization group (NRG)  -n/2

Iterative diagonalization Recursion relation:

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:

Package SNEG , U t

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,  ) 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

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

“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

Package “NRG Ljubljana” open source,GPL

Definition of a quantum impurity problem in “NRG Ljubljana” f 0,L f 0,R ab 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[]]

Definition of a quantum impurity problem in “NRG Ljubljana” f 0,L f 0,R ab 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]])

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

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

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

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

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

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

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

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

Conduction-band mediated inter-impurity exchange interaction RKKY exchangeSuper-exchange

Effective single impurity S=N/2 Kondo model The RKKY interaction is ferromagnetic, J RKKY >0: S is the collective S=N/2 spin operator of the coupled impurities, S=P(  S i )P Effective model (T<J RKKY ): J RKKY  0.62 U(  0 J K ) 2 4 th order perturbation in V k

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

The spin-N/2 Kondo effect Full line: NRGSymbols: Bethe Ansatz

Conductance as a function of the gate voltage

Kondo modelKondo model + potential scattering

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

Gate-voltage controlled spin filtering

Spectral functions

Kosterlitz-Thouless transition  1 =+ ,  2 =-  S=1 Kondo S=1/2 Kondo

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.