Quantum impurity physics
and the “NRG Ljubljana” code
Rok Žitko
J. Stefan Institute,
Ljubljana, Slovenia
UIB, Palma de Mallorca, 12. 12. 2007
• 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”
S= 0
T << TK
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
1keV
1 eV
?
1 meV
100 mev
Many energy scales are locally coupled
(K. G. Wilson, 1975)
Cascade effect
Numerical renormalization group (NRG)
-n/2
Iterative diagonalization
Recursion relation:
H N 1  T  H N 
H N 1  
1/ 2
H N  N ( f
†
N 1,
f N ,  f
†
N ,
f N 1, )
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/
t
e, U
e, U
Package SNEG
http://nrgljubljana.ijs.si/sneg
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-CampbellHausdorff 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
t
a
b
Himp = eps (number[a[]]+number[b[]])+
U/2 (pow[number[a[]]-1,2]+pow[number[b[]]-1,2])
Hab = t hop[a[],b[]] + V
J chargecharge[a[],b[]]
spinspin[a[],b[]]
Hc = Sqrt[Gamma] (hop[a[],f[L]] + hop[b[],f[R]])
Definition of a quantum impurity
problem in “NRG Ljubljana”
f0,L
f0,R
t
a
b
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
Model and parameters
Lambda=3
Nmax=40
keepenergy=10.0
keep=2000
ops=q_d q_d^2 A_d
NRG iteration parameters
Computed quantities
Spectral function
Charge fluctuations
Occupancy
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)
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
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).
Parallel quantum dots and
the N-impurity Anderson model
ikL v
=
e
VVk≡V
(L0)
k
k
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).
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:
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
Free orbital
regime
(FO)
Local
moment
regime
(LM)
o
o
Ferromagnetically
frozen (FF)
Strongcoupling
regime (SC)
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
d1=+D, d2=-D
S=1/2 Kondo
S=1 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/