Quantum%20impurity%20physics%20and%20the%20

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Quantum impurity physics and the NRG Ljubljana
code

• Rok Žitko

J. Stefan Institute, Ljubljana, Slovenia
UIB, Palma de Mallorca, 12. 12. 2007
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• Experimental surface science and STM
• prof. Albert Prodan1
• prof. Igor Muševic1,2
• Erik Zupanic1
• Herman van Midden1
• Ivan Kvasic1

• Quantum transport theory
• prof. Janez Bonca1,2
• prof. Anton Ramšak1,2
• Tomaž Rejec1,2
• Jernej Mravlje1

1 J. Stefan Institute, Ljubljana, Slovenia 2
Faculty of Mathematics and Physics, Uni. of
Ljubljana, Ljubljana, Slovenia
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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 (N1...5, one channel)

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Classical impurity
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Quantum impurity
This is Kondo model!
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Nonperturbative behaviour
The perturbation theory fails for arbitrarily
small J !
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Screening of the magnetic moment
Kondo effect!
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Asymptotic freedom ...
T gtgt TK
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... and infrared slavery
T ltlt TK
Analogy TK ? ?QCD
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Nonperturbative scattering
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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

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Renormalization group
?
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Many energy scales are locally coupled (K. G.
Wilson, 1975)
Cascade effect
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Numerical renormalization group (NRG)
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Iterative diagonalization
Recursion relation
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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/
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Package SNEG http//nrgljubljana.ijs.si/sneg
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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)
• Wicks theorem (with either empty band or Fermi
  sea vacuum states)
• Diracs bra and ket notation
• Simplifications using Baker-Campbell-Hausdorff
  and Mendaš-Milutinovic formula

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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

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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

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Package NRG Ljubljanahttp//nrgljubljana.ijs.si
/ open source,GPL
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Definition of a quantum impurity problem in NRG
Ljubljana
t
Himp eps (numberanumberb)U/2
(pownumbera-1,2pownumberb-1,2) Hab
t hopa,b Hc SqrtGamma (hopa,fL
hopb,fR)
J spinspina,b
V chargechargea,b
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Definition of a quantum impurity problem in NRG
Ljubljana
t
Himp epsa numbera epsb numberb U/2
(pownumbera-1,2pownumberb-1,2) Hab
t hopa,b Hc SqrtGamma (hopa,fL
hopb,fR)
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Computable quantities

• Finite-site excitation spectra (flow diagrams)
• Thermodynamics magnetic and charge
  susceptibility, entropy, heat capacity
• Correlations spin-spin correlations, charge
  fluctuations,...spinspina,b
• numberdpownumberd, 2
• Dynamics spectral functions, dynamical magnetic
  and charge susceptibility, other response
  functions

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Sample input file
parammodelSIAMU1.0Gamma0.04 Lambda3Nmax
40keepenergy10.0keep2000 opsq_d q_d2 A_d
Model and parameters
NRG iteration parameters
Computed quantities
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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)
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Scattering theory
Landauer formula
See, for example, M. Pustilnik, L. I. Glazman,
PRL 87, 216601 (2001).
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Keldysh approach
One impurity
Y. Meir, N. S. Wingreen. PRL 68, 2512 (1992).
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Conductance of a quantum dot (SIAM)
Computed using NRG.
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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).
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Parallel quantum dots and the N-impurity
Anderson model
Vk eikL vk
VkV (L?0)
R. Žitko, J. Bonca Multi-impurity Anderson model
for quantum dots coupled in parallel, Phys. Rev.
B 74, 045312 (2006)R. Žitko, J. Bonca Quantum
phase transitions in systems of parallel quantum
dots, Phys. Rev. B 76, .. (2007).
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Conduction-band mediated inter-impurity exchange
interaction
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Effective single impurity SN/2 Kondo model
The RKKY interaction is ferromagnetic, JRKKYgt0
JRKKY??0.62 U(r0JK)2
4th order perturbation in Vk
Effective model (TltJRKKY)
S is the collective SN/2 spin operator of the
coupled impurities, SP(SSi)P
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Free orbital regime (FO)
Ferro-magnetically frozen (FF)
Local moment regime (LM)
Strong-coupling regime (SC)
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The spin-N/2 Kondo effect
Full line NRG
Symbols Bethe Ansatz
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Conductance as a function of the gate voltage
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Kondo model
Kondo model potential scattering
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S1 Kondo model potential scattering
S1/2 Kondo model strong potential scattering
S1 Kondo model
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Gate-voltage controlled spin filtering
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Spectral functions
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Kosterlitz-Thouless transition
d1D, d2-D
S1/2 Kondo
S1 Kondo
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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/