Kondo Physics from a Quantum
Information Perspective
Pasquale Sodano
International Institute of Physics, Natal,
Brazil
1
Abolfazl Bayat
UCL (UK)
Sougato Bose
UCL (UK)
Henrik Johannesson
Gothenburg (Sweden)
2
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)
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:
 AB   pi  iA   iB
pi  0,
p
i
Entangled states:
i
1
i
 AB   pi   
A
i
B
i
i
How to quantify entanglement for a general mixed state?
There is not a unique entanglement measure
5
Negativity
Separable:
   pi  iA   iB
i
 T   pi (  iA )t   iB
A

TA
0
i
Valid density matrix
Entangled:
Negativity:

TA
  
N (  )  2  ,
(  0)
T    
A
 0
6
Gapped Systems
Excited states

Ground state
 :
 
x
i
x
j
e

|i  j |

The intrinsic length scale of the system impose an exponential decay
7
Gapless Systems
Continuum of excited states
N :
0
 
x
i
Ground state
x
j
| i  j |

There is no length scale in the system so correlations decay algebraically
8
Kondo Physics
K
 K  TK
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
H  J ' ( J11. 2  J 21. 3 )   J1 i . i 1  J 2 i . i 2
i 2
J2
 J 2c  0.2412 : Kondo (gapless)
J1
J2
 J 2c :
J1
Dimer (gapfull)
E. S. Sorensen et al., J. Stat. Mech., P08003 (2007)
11
Entanglement as a Witness of the Cloud
A
Impurity
B
L
 K ( J ' )  e /
J'
L   K : ESA  1  ESB  0
L   K : ESA  1  ESB  0
L   K : ESA  1  ESB  0
12
Entanglement versus Length
Entanglement decays exponentially with length
13
Scaling
A
Impurity
L
B
N-L-1
N
L
Kondo Regime: E ( L,  K , N )  E (
, )
K N
L
N
Dimer Regime: E ( L,  , N )  E (
, )
K L
14
Scaling of the Kondo Cloud
K
N
4
K
K  e /
J'
Kondo Phase: E ( L,  K , N )  E (
Dimer Phase: E ( L,  , N )
N
K
,
L
)
N
15
Application: Quantum Router
Converting useless entanglement into useful
one through quantum quench
16
Simple Example
 
J 'L  1. 2
 
J 'R  3. 4
 
J m 2 . 3
 ( 0)      
 (t )  eiHt  (0)
J m  J 'L  J 'R
E14 (t )  max{0,
1  3 cos(4 J mt )
}
4
17
Extended Singlet
J'opt
 K ( J 'opt )  N 1
With tuning J’ we can generate a proper cloud which
extends till the end of the chain
18
Quench Dynamics
Jm
J 'L
 L ( J 'L )  N L 1
J 'R
 R ( J 'R )  N R  1
 (0)  GSL  GSR
 (t )  eiH t  (0)
LR
1N (t )
E1N (t )
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'
Jm
 L ( J ' )  ( N  2) / 2
J'
 R ( J ' )  ( N  2) / 2
t N
t 2N
E (t , N , J ' )  E (t , N ,  )  E ( , )  E ( ,
)
N 
N N 2
21
Optimal Quench
J 'L
Jm
J 'R
J m  J 'L  J 'R
Jm
J 'L
J m  ( N )(J 'L  J 'R )
K  e
/ J'
1
 J ' 2
log 
J 'R
N
( N )  Log ( )
2
2
22
Optimal Parameter
23
Non-Kondo Singlets (Dimer Regime)
Jm
J 'L
J 'R
Clouds are absent
K: Kondo (J2=0)
D: Dimer (J2=0.42)
24
Asymmetric Chains
Jm
J 'L
 L ( J 'L )  N L 1
J 'R
 R ( J 'R )  N R  1
25
Entanglement in Asymmetric Chains
Symmetric geometry gives the best output
26
Entanglement Router
27
Two Impurity Kondo Model
28
Two Impurity Kondo Model
H L  J ' ( J ' L  1L . 2L  J 2 1L . 3L ) 
H R  J ' ( J ' R  1R . 2R  J 2 1R . 3R ) 
N L 2
L
L
L
L
J

.


J

.

 1 i i 1 2 i i  2
i 2
N R 2
R
R
R
R
J

.


J

.

 1 i i 1 2 i i  2
i 2
H I  J I  1L . 1R
RKKY interaction
29
Impurities
GS
Entanglement
1 1  p 
L R
1
p
2
1
p
2



1 p
k
k

T
T

3 k 0, 
N ( 1L1R )  0
N ( 1L1R )  0
30
Entanglement of Impurities
J'
JI
J'
J Ic
Entanglement can be used as the order parameter
for differentiating phases
31
Scaling at the Phase Transition
J  TK 
c
I
1
K
 e  / J '
The critical RKKY coupling scales just as Kondo temperature does
32
Entropy of Impurities
1 1  p 
L R



1 p
k
k

T
T

3 k 0, 
S ( 1L ,1R )   p log( p )  (1  p ) log(1  p )
J I  0 : S ( 1L ,1R )  log(3)
J I  0 : p  0
Triplet
J I  0 : S ( 1L ,1R )  2
J I  0 : p  1/4
Identity
J I  0 : S ( 1L ,1R )  0
J I  0 : p  1
Singlet
33
Impurity-Block Entanglement
34
Block-Block Entanglement
35
nd
2
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
JI
NA=NB=400
J’=0.4
JI
NA=600, NB=200
J’=0.4
40
Schmidt Gap
Schmidt gap:
JI
41
Thermodynamic Behaviour
J’=0.4
J’=0.5
JI
JI
In the thermodynamic limit Schmidt gap takes
zero in the RKKY regime
42
Diverging Derivative
JI
JI
In the thermodynamic limit the first derivative of
Schmidt gap diverges
43
Diverging Kondo Length
JI
44
Finite Size Scaling
S  N
  /
f (| J I  J | N
c
I
1 /
)
 S N  /  f (| J I  J Ic | N 1/ )
 S  | J I  J Ic |
  | JI  J |
c
I
( J I  J Ic ) N 0.5

  0. 2
 2
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
• 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)
48