Quantum impurity physics and the “NRG Ljubljana” code

Rok Žitko

UIB, Palma de Mallorca, 12. 12. 2007

J. Stefan Institute, Ljubljana, Slovenia

• 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

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 >> TK

... and “infrared slavery”

T << TK

S=0

Analogy: TK QCD

Nonperturbative scattering

S+ S- S- S+

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

1 k e V

1 0 0 m e v

1 e V

1 m e V?

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

Cascade effect

Numerical renormalization group (NRG)

-n/2

Iterative diagonalization

Recursion relation:

1

1/ 2 † †1 1, , , 1,( )

N N

N N N N N N N

H T H

H H f f f f

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/

Package SNEG

http://nrgljubljana.ijs.si/sneg

, U , 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”http://nrgljubljana.ijs.si/

open source,GPL

Definition of a quantum impurity problem in “NRG Ljubljana”

f0,L f0,R

a bt

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”

f0,L f0,R

a bt

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=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

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, 216601 (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, 036807 (2006).

M. Korkusinski, I. P. Gimenez, P. Hawrylak,L. Gaudreau, S. A. Studenikin, A. S. Sachrajda,Phys. Rev. B 75, 115301 (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, 045312 (2006)R. Žitko, J. Bonča: Quantum phase transitions in systems of parallel quantum dots, Phys. Rev. B 76, .. (2007).

Vk = eikL vkVk≡V (L0)

Conduction-band mediated inter-impurity exchange interaction

RKKY exchange Super-exchange

Effective single impurity S=N/2 Kondo model

The RKKY interaction is ferromagnetic, JRKKY>0:

S is the collective S=N/2 spin operator of the coupled impurities,

S=P(Si)P

Effective model (T<JRKKY):

JRKKY0.62 U(0JK)2 4th order perturbation in Vk

Free orbital regime

(FO)

Local moment regime

(LM)

Ferro-magnetically frozen (FF)

Strong-coupling

regime (SC)

o o

The spin-N/2 Kondo effect

Full line: NRG Symbols: Bethe Ansatz

Conductance as a function of the gate voltage

Kondo model Kondo 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 KondoS=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.

http://nrgljubljana.ijs.si/