Kondo Physics from a Quantum Information Perspective

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

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

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, 066403 (2012) • Entanglement Routers Using Macroscopic Singlets • A. Bayat, S. Bose, P. Sodano, Phys. Rev. Lett. 105, 187204 (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, 064429 (2010) • Kondo Cloud Mediated Long Range Entanglement After Local • Quench in a Spin Chain • P. Sodano, A. Bayat, S. Bose • Phys. Rev. B 81, 100412(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, 156801 (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 Cloud A B Impurity L

13. Entanglement versus Length Entanglement decays exponentially with length

14. Scaling B A Impurity 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 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. 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, 066403 (2012) • Entanglement Routers Using Macroscopic Singlets • A. Bayat, S. Bose, P. Sodano, Phys. Rev. Lett. 105, 187204 (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, 064429 (2010) • Kondo Cloud Mediated Long Range Entanglement After Local • Quench in a Spin Chain • P. Sodano, A. Bayat, S. Bose • Phys. Rev. B 81, 100412(R) (2010)