Exploring Kondo Physics through Quantum Information

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

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

References A. Bayat, H. Johannesson, S. Bose, P. Sodano To appear in Nature Communication. An order parameter for impurity systems at quantum criticality A. Bayat, S. Bose, P. Sodano, H. Johannesson, Phys. Rev. Lett. 109, 066403 (2012) Entanglement probe of two-impurity Kondo physics in a spin chain A. Bayat, S. Bose, P. Sodano, Phys. Rev. Lett. 105, 187204 (2010) Entanglement Routers Using Macroscopic Singlets Spin-Chain Kondo Model A. Bayat, P. Sodano, S. Bose, Phys. Rev. B 81, 064429 (2010) Negativity as the Entanglement Measure to Probe the Kondo Regime in the P. Sodano, A. Bayat, S. Bose Phys. Rev. B 81, 100412(R) (2010) Kondo Cloud Mediated Long Range Entanglement After Local Quench in a Spin Chain 3

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

Entanglement of Mixed States Separable states: i i = = A i B i , 0 1 p p p AB i i i i A i B i p Entangled states: AB i How to quantify entanglement for a general mixed state? There is not a unique entanglement measure 5

Negativity Separable: i i = A i B i = T T A i t B i ip 0 i ( ) p A A Valid density matrix = T ( 0) A Entangled: = 2 ( = T ) , N Negativity: A 0 6

Gapped Systems Excited states Ground state | | i j x i x j : e The intrinsic length scale of the system impose an exponential decay 7

Gapless Systems Continuum of excited states : 0 N Ground state x i x j | | i j There is no length scale in the system so correlations decay algebraically 8

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

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). 10

Kondo Spin Chain = i = 1 . + . + 1 . ++ 1 . ( ' J ) H J J J J + 1 2 2 1 3 2 2 i i i i 2 J = c . 0 2412 Kondo : (gapless) 2 J 2 J 1 J c : Dimer (gapfull) 2 J 2 J 1 E. S. Sorensen et al., J. Stat. Mech., P08003 (2007) 11

Entanglement as a Witness of the Cloud A B Impurity L / ' J ( ) ' J e K : 1 0 L E E K SA SB = = = : 1 0 L E E K SA SB = = : 1 0 L E E K SA SB 12

Entanglement versus Length Entanglement decays exponentially with length 13

Scaling B A Impurity L N-L-1 N L = ( , , ) ( , ) E L N E Kondo Regime: K N K L N ( , , ) ( , ) E L N E Dimer Regime: L K 14

Scaling of the Kondo Cloud e = / J ' K K N L = ( , , ) ( , ) E L N E Kondo Phase: N K N = 4 K K ( , , ) E L N Dimer Phase: 15

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

Simple Example ' 1. J ' 3. J 2 L 4 R m 2. J 3 = ) 0 ( = iHt ( ) ) 0 ( t e 1 3 cos( 4 ) J t = ' + ' J J J = ( ) max{ , 0 } m E t 14 m L R 4 17

Extended Singlet J' opt = N ( ' ) 1 Jopt K With tuning J we can generate a proper cloud which extends till the end of the chain 18

Quench Dynamics J J' J' m L R = ( ' ) 1 J N = ( ' ) 1 J N R R R L L L = ) 0 ( GS GS L R ( ) t ( ) EN t = iHLR t ( ) ) 0 ( t e 1 N 1 19

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 20

Distance Independence For simplicity take a symmetric composite: J ' J ' J m = N / ) 2 ( ) ' J ( 2 = N / ) 2 ( ) ' J ( 2 R L 2 t N t N = = = ( , , ) ' J ( , , ) ( , ) ( , ) E t N E t N E E 2 N N N 21

Optimal Quench J' J' J L R m = ' + ' J J J m L R J J' J' m L R = + 1 ( )( ' ' ) J N J J m L R 2 N N ( ) ( ) Log = ' / ' J e J 2 K 2 log 22

Optimal Parameter 23

Non-Kondo Singlets (Dimer Regime) J J' J' m L R Clouds are absent K: Kondo (J2=0) D: Dimer (J2=0.42) 24

Asymmetric Chains J J' J' m L R = ( ' ) 1 J N = ( ' ) 1 J N R R R L L L 25

Entanglement in Asymmetric Chains Symmetric geometry gives the best output 26

Entanglement Router 27

Two Impurity Kondo Model 28

Two Impurity Kondo Model 2 N L = N = . + . + 1 . + . L L L L L i L i L i L i ( ' J ' ) H J J J J + + 1 2 2 1 3 1 2 2 L L 2 i 2 R = i = . + . + 1 . + . R R R R R i R i R i R i ( ' J ' ) H J J J J + + 1 2 2 1 3 1 2 2 R R 2 = L 1. R 1 H J I I RKKY interaction 29

Impurities 3 1 p = , 0 k = + k k p T T GS 1 1 L R 1 = ( ) 0 p N 1 1 2 L R Entanglement 1 ( ) 0 p N 1 1 2 L R 30

Entanglement of Impurities ' J J ' J I c IJ Entanglement can be used as the order parameter for differentiating phases 31

Scaling at the Phase Transition 1 / ' c I J J T e K K The critical RKKY coupling scales just as Kondo temperature does 32

Entropy of Impurities = , 0 3 k 1 p = + k k p T T 1 1 L R = ( ) log( ) 1 ( ) log( 1 ) S p p p p 1 1 , L R = = 0 : ( ) log( ) 3 J S 0 : 0 J p Triplet 1 1 , L I I R = = = = 0 : 1/4 J p 0 : ( ) 2 J S Identity I 1 1 , L I R = 0 : 1 J p = 0 : ( ) 0 J S Singlet I 1 1 , L I R 33

Impurity-Block Entanglement 34

Block-Block Entanglement 35

2nd Order Phase Transition 36

Order Parameter for Two Impurity Kondo Model 37

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

Entanglement Spectrum Entanglement spectrum: 39

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

Schmidt Gap Schmidt gap: I J 41

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

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

Diverging Kondo Length I J 44

Finite Size Scaling = / / 1 c I (| | ) N f J J N S I = / / 1 c I (| | ) N f J J N S I | = | c I | J J S I = c I | J J I = 2 . 0 = 2 I 5 . 0 c I ( ) J J N 45

Schmidt Gap as an Observable 46

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

References A. Bayat, H. Johannesson, S. Bose, P. Sodano To appear in Nature Communication. An order parameter for impurity systems at quantum criticality A. Bayat, S. Bose, P. Sodano, H. Johannesson, Phys. Rev. Lett. 109, 066403 (2012) Entanglement probe of two-impurity Kondo physics in a spin chain A. Bayat, S. Bose, P. Sodano, Phys. Rev. Lett. 105, 187204 (2010) Entanglement Routers Using Macroscopic Singlets Spin-Chain Kondo Model A. Bayat, P. Sodano, S. Bose, Phys. Rev. B 81, 064429 (2010) Negativity as the Entanglement Measure to Probe the Kondo Regime in the P. Sodano, A. Bayat, S. Bose Phys. Rev. B 81, 100412(R) (2010) Kondo Cloud Mediated Long Range Entanglement After Local Quench in a Spin Chain 48