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Kondo Physics from a Quantum Information Perspective

Pasquale Sodano

International Institute of Physics, Natal, Brazil

2

Abolfazl BayatUCL (UK)

Sougato BoseUCL (UK)

Henrik JohannessonGothenburg (Sweden)

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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, 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)

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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

Entanglement of Mixed States

Entangled states: i

Bi

AiiAB p

How to quantify entanglement for a general mixed state?

There is not a unique entanglement measure 5

Separable states:

i

iii

Bi

AiiAB ppp 1 ,0

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Negativity

Separable:

i

Bi

Aiip

i

Bi

tAii

T pA )(

Valid density matrix

0AT

Entangled: 0)( AT

Negativity:

ATN ,2)(0

7

Gapped Systems

Ground state

Excited states

||

:ji

xj

xi e

The intrinsic length scale of the system impose an exponential decay

8

Gapless Systems

0 : N Ground state

Continuum of excited states

|| jixj

xi

There is no length scale in the system so correlations decay algebraically

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Kondo Physics

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

K

KK T

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Realization of Kondo EffectSemiconductor 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).

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Kondo Spin Chain

2

2211312211 ..)..('i

iiii JJJJJH

(gapfull)Dimer :

(gapless) Kondo :2412.0

21

2

21

2

c

c

JJ

J

JJ

J

E. S. Sorensen et al., J. Stat. Mech., P08003 (2007)

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Entanglement as a Witness of the Cloud

A BImpurity

L

01 : SBSAK EEL

01 : SBSAK EEL

01 : SBSAK EEL

'/)'( JK eJ

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Entanglement versus Length

Entanglement decays exponentially with length

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Scaling

),(),,(N

LNENLE

KK

),(),,(L

NLENLE

K

Kondo Regime:

Dimer Regime:

A BImpurity

L N-L-1

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Scaling of the Kondo Cloud

'/ JK e

4K

N

),(),,(N

LNENLE

KK

),,( NLE

Kondo Phase:

Dimer Phase:

K

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Application: Quantum Router

Converting useless entanglement into useful one through quantum quench

Simple Example

43.' RJ21.'

LJ

32. mJ

)0(

)0()( iHtet

RLm JJJ '' }4

)4cos(31,0max{)(14

tJtE m

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18

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

1)'( NJ optK

optJ '

Extended Singlet

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1)'( LLL NJ

LJ ' RJ '

1)'( RRR NJ

mJ

RL GSGS )0(

)0()( tiHLRet )(1 tN )(1 tE N

Quench Dynamics

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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

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For simplicity take a symmetric composite:

2/)2()'( NJL

'J 'J

2/)2()'( NJR

mJ

)2

2,(),(),,()',,(

N

N

N

tE

N

N

tENtEJNtE

Distance Independence

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RJ 'LJ 'mJ

RLm JJJ ''

LJ ' RJ 'mJ

)'')(( RLm JJNJ )

2()( 2 NLogN

Optimal Quench

2'/

log

1' Je J

K

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Optimal Parameter

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LJ ' RJ 'mJ

K: Kondo (J2=0) D: Dimer (J2=0.42)

Clouds are absent

Non-Kondo Singlets (Dimer Regime)

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1)'( LLL NJ

LJ ' RJ '

1)'( RRR NJ

mJ

Asymmetric Chains

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Symmetric geometry gives the best output

Entanglement in Asymmetric Chains

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Entanglement Router

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Two Impurity Kondo Model

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Two Impurity Kondo Model

2

2221131221

2

2221131221

..)..'('

..)..'('

R

L

N

i

Ri

Ri

Ri

Ri

RRRRRR

N

i

Li

Li

Li

Li

LLLLLL

JJJJJH

JJJJJH

RKKY interaction

RLII JH 11 .

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Impurities

GS

,011 3

1

k

kk TTp

pRL

0)( 2

1

0)( 2

1

11

11

RL

RL

Np

Np

Entanglement

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Entanglement of Impurities

Entanglement can be used as the order parameter for differentiating phases

' ' JJJ I

cIJ

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Scaling at the Phase Transition

'/1 J

KK

cI eTJ

The critical RKKY coupling scales just as Kondo temperature does

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Entropy of Impurities

)1log()1()log()( 1,1 ppppSRL

,011 3

1

k

kk TTp

pRL

0)( :0

2)( :0

)3log()( :0

1,1

1,1

1,1

RL

RL

RL

SJ

SJ

SJ

I

I

I

1 :0

1/4 :0

0 :0

pJ

pJ

pJ

I

I

I Triplet

Identity

Singlet

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Impurity-Block Entanglement

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Block-Block Entanglement

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2nd Order Phase Transition

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Order Parameter for Two Impurity Kondo Model

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Order ParameterOrder parameter is:

1- Observable2- Is zero in one phase and non-zero in the other3- Scales at criticality

Landau-Ginzburg paradigm:

4- Order parameter is local5- Order parameter is associated with a symmetry breaking

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Entanglement Spectrum

Entanglement spectrum:

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Entanglement Spectrum

NA=NB=400J’=0.4

NA=600, NB=200J’=0.4

IJ IJ

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Schmidt GapSchmidt gap:

IJ

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Thermodynamic Behaviour

IJ IJ

J’=0.4 J’=0.5

In the thermodynamic limit Schmidt gap takes zero in the RKKY regime

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Diverging Derivative

IJ

In the thermodynamic limit the first derivative of Schmidt gap diverges

IJ

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Diverging Kondo Length

IJ

Finite Size Scaling

2

2.0

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)|(|

)|(|/1/

/1/

NJJfN

NJJfNcIIS

cIIS

5.0)( NJJ cII

||

|| cII

cIIS

JJ

JJ

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Schmidt Gap as an Observable

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SummaryNegativity 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 twoimpurity Kondo model.

Schmidt gap, as an observable, shows scaling with the right exponents at the critical point of the two Impurity Kondo model.

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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, 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)