1. Kondo Physics from a Quantum Information Perspective
Pasquale Sodano
International Institute of Physics, Natal, Brazil

2. Sougato Bose
UCL (UK)
Abolfazl Bayat
UCL (UK)
Henrik Johannesson
Gothenburg (Sweden)

3. References
An order parameter for impurity systems at quantum criticality
A. Bayat, H. Johannesson, S. Bose, P. Sodano
To appear in Nature Communication.
Entanglement probe of two-impurity Kondo physics in a spin chain
A. Bayat, S. Bose, P. Sodano, H. Johannesson, Phys. Rev. Lett. 109, (2012)
Entanglement Routers Using Macroscopic Singlets
A. Bayat, S. Bose, P. Sodano, Phys. Rev. Lett. 105, (2010)
Negativity as the Entanglement Measure to Probe the Kondo Regime in the
Spin-Chain Kondo Model
A. Bayat, P. Sodano, S. Bose, Phys. Rev. B 81, (2010)
Kondo Cloud Mediated Long Range Entanglement After Local
Quench in a Spin Chain
P. Sodano, A. Bayat, S. Bose
Phys. Rev. B 81, (R) (2010)

4. Contents of the Talk Negativity as an Entanglement Measure
Single Kondo Impurity Model
Application: Quantum Router
Two Impurity Kondo model: Entanglement
Two Impurity Kondo model: Entanglement Spectrum

5. Entanglement of Mixed States
Separable states:
Entangled states:
How to quantify entanglement for a general mixed state?
There is not a unique entanglement measure

6. Negativity
Separable:
Valid density matrix
Entangled:
Negativity:

7. Gapped Systems
Excited states
Ground state
The intrinsic length scale of the system impose an exponential decay

8. Gapless Systems
Continuum of excited states
Ground state
There is no length scale in the system so correlations decay algebraically

9. Kondo Physics
Despite the gapless nature of the Kondo system, we have a length scale in the model

10. Realization of Kondo Effect
Semiconductor quantum dots
D. G. Gordon et al. Nature 391, 156 (1998).
S.M. Cronenwett, Science 281, 540 (1998).
Carbon nanotubes
J. Nygard, et al. Nature 408, 342 (2000).
M. Buitelaar, Phys. Rev. Lett. 88, (2002).
Individual molecules
J. Park, et al. Nature 417, 722 (2002).
W. Liang, et al, Nature 417, 725–729 (2002).

11. Kondo Spin Chain
E. S. Sorensen et al., J. Stat. Mech., P08003 (2007)

12. Entanglement as a Witness of the CloudB
Impurity L

13. Entanglement versus Length
Entanglement decays exponentially with length

14. Scaling
Impurity A B L
N-L-1
Kondo Regime:
Dimer Regime:

15. Scaling of the Kondo Cloud
Kondo Phase:
Dimer Phase:

16. Application: Quantum Router
Converting useless entanglement into useful
one through quantum quench

17. Simple Example

18. Extended Singlet With tuning J’ we can generate a proper cloud which
extends till the end of the chain

19. Quench Dynamics

20. Attainable Entanglement
1- Entanglement dynamics is very long lived and oscillatory
2- maximal entanglement attains a constant values for large chains
3- The optimal time which entanglement peaks is linear

21. Distance Independence
For simplicity take a symmetric composite:

22. Optimal Quench

23. Optimal Parameter

24. Non-Kondo Singlets (Dimer Regime)
Clouds are absent
K: Kondo (J2=0) D: Dimer (J2=0.42)

25. Asymmetric Chains

26. Entanglement in Asymmetric Chains
Symmetric geometry gives the best output

27. Entanglement Router

28. Two Impurity Kondo Model

29. Two Impurity Kondo Model
RKKY interaction

30. Impurities
Entanglement

31. Entanglement of Impurities
Entanglement can be used as the order parameter
for differentiating phases

32. Scaling at the Phase Transition
The critical RKKY coupling scales just as Kondo temperature does

33. Entropy of Impurities
Triplet
Identity
Singlet

34. Impurity-Block Entanglement

35. Block-Block Entanglement

36. 2nd Order Phase Transition

37. Order Parameter for Two Impurity Kondo Model

38. Order Parameter Order parameter is: 1- Observable
2- Is zero in one phase and non-zero in the other
3- Scales at criticality
Landau-Ginzburg paradigm:
4- Order parameter is local
5- Order parameter is associated with a symmetry breaking

39. Entanglement Spectrum

40. Entanglement Spectrum
NA=NB=400
J’=0.4
NA=600, NB=200
J’=0.4

41. Schmidt Gap
Schmidt gap:

42. Thermodynamic Behaviour
J’=0.4
J’=0.5
In the thermodynamic limit Schmidt gap takes
zero in the RKKY regime

43. In the thermodynamic limit the first derivative of
Diverging Derivative
In the thermodynamic limit the first derivative of
Schmidt gap diverges

44. Diverging Kondo Length

45. Finite Size Scaling

46. Schmidt Gap as an Observable

47. Summary Negativity is enough to determine the Kondo length and the
scaling of the Kondo impurity problems.
By tuning the Kondo cloud one can route distance independent
entanglement between multiple users via a single bond quench.
Negativity also captures the quantum phase transition in two
impurity Kondo model.
Schmidt gap, as an observable, shows scaling with the right
exponents at the critical point of the two Impurity Kondo model.

48. References
An order parameter for impurity systems at quantum criticality
A. Bayat, H. Johannesson, S. Bose, P. Sodano
To appear in Nature Communication.
Entanglement probe of two-impurity Kondo physics in a spin chain
A. Bayat, S. Bose, P. Sodano, H. Johannesson, Phys. Rev. Lett. 109, (2012)
Entanglement Routers Using Macroscopic Singlets
A. Bayat, S. Bose, P. Sodano, Phys. Rev. Lett. 105, (2010)
Negativity as the Entanglement Measure to Probe the Kondo Regime in the
Spin-Chain Kondo Model
A. Bayat, P. Sodano, S. Bose, Phys. Rev. B 81, (2010)
Kondo Cloud Mediated Long Range Entanglement After Local
Quench in a Spin Chain
P. Sodano, A. Bayat, S. Bose
Phys. Rev. B 81, (R) (2010)