Kondo Physics from a Quantum Information Perspective
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Kondo Physics from a Quantum Information Perspective
Pasquale SodanoInternational Institute of Physics, Natal,
Brazil
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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
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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
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Gapped Systems
Ground state
Excited states
||
:ji
xj
xi e
The intrinsic length scale of the system impose an exponential decay
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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
JJJ
JJJ
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
),(),,(NLNENLE
KK
),(),,(LNLENLE
K
Kondo Regime:
Dimer Regime:
A BImpurity
L N-L-1
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Scaling of the Kondo Cloud
'/ JK e
4K
N
),(),,(NLNENLE
KK
),,( NLE
Kondo Phase:
Dimer Phase:
K
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Application: Quantum Router
Converting useless entanglement into useful one through quantum quench
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Simple Example
43.'
RJ21.'
LJ
32.
mJ )0(
)0()( iHtet
RLm JJJ '' }4
)4cos(31,0max{)(14tJtE m
17
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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,(),(),,()',,(
NN
NtEN
NtENtEJNtE
Distance Independence
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RJ 'LJ 'mJ
RLm JJJ ''
LJ ' RJ 'mJ
)'')(( RLm JJNJ )
2()( 2 NLogN
Optimal Quench
2'/
log1' 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
1k
kk TTppRL
0)( 21
0)( 21
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
1k
kk TTppRL
0)( :0
2)( :0
)3log()( :0
1,1
1,1
1,1
RL
RL
RL
SJ
SJ
SJ
I
I
I
1 :01/4 :00 :0
pJpJpJ
I
I
I TripletIdentitySinglet
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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
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Finite Size Scaling
22.0
45
)|(|
)|(|/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)