1. Quantum entanglement, Kondo effect, and electronic transport in
quantum dots system
Sahib Babaee Tooski
Institute of Molecular Physics, Polish Academy of Sciences,
Poznań, Poland
Collaboration Slovenia

2. Outline Introduction (Entanglement, Kondo effect, NRG)
Entanglement and Kondo effect in triple quantum dots
Friedel-Luttinger sum rule and Kondo effect in triple quantum dots
Effect of assisted-hopping on transport in a quantum dot
Summary

3. Outline Introduction (Entanglement, Kondo effect, NRG)
Entanglement and Kondo effect in triple quantum dots
Friedel-Luttinger sum rule and Kondo effect in triple quantum dots
Effect of assisted hopping on transport in a quantum dot
Summary

4. Concurrence to quantify quantum entanglementB
Concurrence to quantify quantum entanglement
Fully enanglement
Un-enanglement

5. Kondo effect in metal with magnetic impurities (uo kondo)
Kondo effect: 1) Scattering of conduction electron on a magnetic imputity via a spin-flip process. 2) Formation of singlet state entanglament

6. Quantum dot quantum dot Gates on top of GaAs/AlGaAs heterostructure
segregate a portion of 2D electron gas
quantum dot

7. S=1/2 Kondo effect in quantum dot
Signature of Kondo Physics, T<<TK :
Increase of the conductance for odd number of electrons
Abrikosov-Suhl peak in DOS
Unitary transmission with T=1
Goldhaber-Gordon, et al. Nature 391 (1998) 156

8. Landauer approach for electron transport using NRG Ljubljana
Conductance
Thermopower
In are the transport integrals
Transmission coefficient
Single-particle effects
Correlation effects
Wilson Numerical renormalization group (NRG)
1982 Nobel Prize

9. Introduction
Entanglement and Kondo effect in triple quantum dots
Underscreened Kondo effect in triple quantum dots
Effect of assisted hopping on transport in quantum dots
Summary

10. Experimental TQD system color is not the same bloch and states
Qubit in doublet subscpace
Pseudo-spin
L. Gaudreau, et al. Nature Phys. 8, 54 (2012);

11. Pairwise entanglement in triple quantum dots (increase thikness notation of t2 larger) one notation for states
(t1 <t2)
(t1 >t2) t1 t2 A C B t1 t2 A C B
Monogamy

12. One electron in each dot
Triple quantum dots
One electron in each dot

13. Triple QDs and Anderson model

14. quantum phase transition
Entanglement and Kondo effect
T=0
(t1 <t2)
(t1 >t2)
Screened S=1/2 Kondo effect
One stage Kondo effect
Underscreened S=1 Kondo effect
Two stages Kondo effect
Kondo singlet Γ t1 t2 A C B R L Γ t1 t2 A C B R L
Kondo singlet
quantum phase transition

15. Phase diagram between entangled and unentangled ground state(figure thiknes made scheme larger, larger fluctuation spin, plot be larger) Γ t1 t2 A C B R L
Kondo singlet
S=1/2 Kondo
Kondo singlet Γ t1 t2 A C B R L
S=1 Kondo
First-order quantum phase transition
Concurrencies independent from charge fluctuation

16. Introduction
Entanglement and Kondo effect in triple quantum dots
Friedel-Luttinger sum rule and Kondo effect in triple quantum dots
Effect of assisted hopping on transport in a quantum dot
Summary

17. Singular and Non-Fermi liquids in both Kondo peaks
Friedel sum rule
Singular and Non-Fermi liquids in both Kondo peaks
The Friedel sum rule relates the total charge displaced in the field of an impurity to the phase shift of a free electron at the Fermi momentum scattered at the impurity
where is the so-called Luttinger integral.
Screened S=1/2 Kondo effect
Normal Fermi Liquid
Underscreened S=1 Kondo effect
Singular Fermi Liquid
Kondo peaks in both Fermi liquids
L. I. Glazman and M. E. Raikh, JETP Lett. 47, 452 (1988); T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988);
D. E. Logan, A. P. Tucker, and Ma. R. Galpin, Phys. Rev. B 90, (2014).

18. Friedel sum rule (Conductance)
Linear conductance related with the phase shift and
this related with the dot occupancy
Screened S=1/2 Kondo effect
Normal Fermi-liquid
Underscreened S=1 Kondo effect
singular Fermi-liquid
Knowing the dot occupancy and conductance,
one can determine the phase of the system
D. E. Logan, A. P. Tucker, and Ma. R. Galpin, Phys. Rev. B 90, (2014).

19. quantum phase transition
t1 < t2 Γ t1 t2 A C B R L
ntot=5
ntot=2 t1 t2 A C B
n=3
n=5
quantum phase transition
Screened S=1/2
Kondo effect
Normal Fermi liquid t1 t2 A C B
n=4
n=6
no Kondo effect
ntot=4
Underscreened S=1
Kondo effect
Singular Fermi liquid
Nagaoka ferromagnetic
n=2 t1 t2 A C B

20. quantum phase transitionΓ t1 t2 A C B R L
t1 > t2
ntot=5
ntot=2 t1 t2 A C B
n=5
Screened S=1/2
Kondo effect
Normal Fermi liquid t1 t2 A C B
n=4
n=6
no Kondo effect t1 t2 A C B
n=3
n=2
ntot=4
Underscreened S=1
Kondo effect
Singular Fermi liquid
quantum phase transition

21. Introduction
Entanglement and Kondo effect in triple quantum dots
Friedel-Luttinger sum rule and Kondo effect in triple quantum dots
Effect of assisted hopping on transport in a quantum dot
Summary

22. Anderson model with assisted-hopping
describes the Coulomb-interaction-mediated transfer of electron from the state |k> in the lead to the QD, when the QD is already occupied by an electron with the opposite spin

23. Perturbative analysis (i)
(i) Different level renormalisations
Haldane scaling theory
f is the Fermi-Dirac distribution.

24. Perturbative analysis (ii)
(ii) modified Kondo exchange coupling constant
Schrieffer-Wolff transformation
Effective single impurity S=1/2 Kondo model
Exchange constant
Kondo temperature
Exponential reduction of the Kondo temperature for x 1

25. Static properties
S.B.Tooski, A.Ramšak, B.R.Bułka,R.Žitko, New J. Phys. 16 (2014)

26. reduced Kondo coupling JK
Static properties
consequence of
reduced Kondo coupling JK
it makes the local spin
more decoupled from the leads.
At x=1 the upper level becomes decoupled

27. Gate-voltage dependence
thermopower S and conductance G at high temperature T>TK
Increasing x, upper atomic peak starts to be decouple from the leads
For x=1, we have a single resonant level

28. Gate-voltage dependence
thermopower S and conductance G at high temperature T>TK
hopping transport is dominated
A quantum dot with x has large Kondo induced S
S increases 5 times
Reasons for enhanced S:
large charge fluctuation
Increasing x, upper atomic peak starts to be decouple from the leads
For x=1, we have a single resonant level

29. Gate-voltage dependence
thermopower S and conductance G at low temperature T<TK
x induces deviations in G from universality
x kills Kondo

30. Gate-voltage dependence
thermopower S and conductance G at low temperature T<TK
Large x results in:
large S
Asymmetry in S
x induces deviations in G from universality
Sharp peaks in transport

31. Introduction
Entanglement and Kondo effect in triple quantum dots
Friedel-Luttinger sum rule and Kondo effect in triple quantum dots
Effect of assisted hopping on transport in a quantum dot
Summary

32. Summary
Kondo singlet formation in TQD can lead to switching between entangled and unentangled states.
TQD system is sensitive to symmetry breaking which can induce a quantum phase transition between the S=1/2 and S=1 ground states.
Friedel sum rule can be used to describe transport for regular as well as singular Fermi liquids.
Assisted hopping exponentially reduces Kondo temperature, significantly modifies electronic transport.

33. Publication list Journals:
S. B. Tooski, R. Zitko, B. R. Bułka, A. Ramšak
Friedel sum rule for the Anderson impurity model
In preparation.
3. S. B. Tooski, A. Ramšak, B. R. Bułka, R. Zitko
Effect of assisted hopping on thermopower in an interacting quantum dot
New Journal of Physics 16, (2014).
4. S. B. Tooski, A. Ramšak, R. Zitko, B. R. Bułka
Entanglement switching via the Kondo effect in triple quantum dots
The European Physical Journal B 87, 145 (2014)
Selected Editorial Board as Highlight of the Year 2014.
5. M. Urbaniak, S. B. Tooski, A. Ramšak, B. R. Bułka
Thermal entanglement in a triple quantum dot system
The European Physical Journal B 86, 1 (2013)
6. S. B. Tooski, B. R . Bułka
Dark States and Transport through Quantum Dots
Acta Physica Polonica A 121, 1231 (2012)

34. Thank you for your attention