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

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

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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) 3

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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 4

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

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Negativity Separable: Valid density matrix Entangled: Negativity: 6

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

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

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

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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). 10

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

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Entanglement as a Witness of the Cloud A B Impurity L 12

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Entanglement versus Length Entanglement decays exponentially with length 13

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

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

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

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Simple Example 17

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

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Quench Dynamics 19

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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 Attainable Entanglement 20

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For simplicity take a symmetric composite: Distance Independence 21

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Optimal Quench 22

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Optimal Parameter 23

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

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Asymmetric Chains 25

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Symmetric geometry gives the best output Entanglement in Asymmetric Chains 26

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Entanglement Router 27

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Two Impurity Kondo Model 28

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Two Impurity Kondo Model RKKY interaction 29

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Impurities Entanglement 30

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

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

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Entropy of Impurities Triplet Identity Singlet 33

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Impurity-Block Entanglement 34

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Block-Block Entanglement 35

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2 nd Order Phase Transition 36

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Order Parameter for Two Impurity Kondo Model 37

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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 38

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Entanglement Spectrum Entanglement spectrum: 39

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Entanglement Spectrum N A =N B =400 J’=0.4 N A =600, N B =200 J’=0.4 40

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Schmidt Gap Schmidt gap: 41

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

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

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Diverging Kondo Length 44

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Finite Size Scaling 45

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Schmidt Gap as an Observable 46

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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. 47

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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) 48

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