Transcript Spin-1 Chains Critical Properties and CFT

Measures of Entanglement at
Quantum Phase Transitions
M. Roncaglia
Condensed Matter Theory Group in Bologna
G. Morandi
F. Ortolani
E. Ercolessi
C. Degli Esposti Boschi
L. Campos Venuti
S. Pasini
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• Entanglement is a resource for:
teleportation
dense coding
quantum cryptography
quantum computation
QUBITS
Spin chains are natural candidates
as quantum devices
• Strong quantum fluctuations in low-dimensional
quantum systems at T=0
• The Entanglement can give another perspective for
understanding Quantum Phase Transitions
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• Entanglement is a property of a state, not of an Hamiltonian.
But the GS of strongly correlated quantum systems are
generally entangled.
• Direct product states
A
  A  B
 Ai B j    Ai   Bj   0
• Nonzero correlations at T=0 reveal entanglement
• 2-qubit states
00 , 11
Product states


1
 01  10 

2
1
 00  11 
2
Maximally entangled
(Bell states)
 
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B
Block entropy
A
B
• Reduced density matrix for the subsystem A
 A  TrB  
• Von Neumann entropy
S A  T rA  A log  A
• For a 1+1 D critical system
CFT with central charge c
c
S A  log l
3
l= block size
Off-critical
c
S A  A log 
6
[ See P.Calabrese and J.Cardy, JSTAT P06002 (2004).]
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Renormalization Group (RG)
• c-theorem:
RG flow
cUV  cIR
UV
(Zamolodchikov, 1986)
IR
fixed point
fixed point
• Massive theory (off critical)
Block entropy saturation
RG flow
UV
fixed point
Irreversibility of RG trajectories
Loss of entanglement
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• Local Entropy: when the subsystem A is a single site.
• Applied to the extended
Hubbard model
• The local entropy
depends only on the
average double occupancy
• The entropy is maximal at
the phase transition lines
(equipartition)
[ S.Gu, S.Deng, Y.Li, H.Lin, PRL 93, 86402 (2004).]
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• Bond-charge Hubbard model
(half-filling, x=1)
• Critical points: U=-4, U=0
• Negativity


N (  AB )   TABA  1 / 2
1
• Mutual information
I  S (  A )  S (  B )  S (  AB )
• Some indicators show
singularities at transition
points, while others don’t.
[ A.Anfossi et al., PRL 95, 056402 (2005).]
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Ising model in
transverse field
• Critical point: l=1
• The concurrence
measures the
entanglement
between two sites
after having traced
out the remaining
sites.
• The transition is signaled by the first derivative of the
concurrence, which diverges logarithmically (specific heat).
[ A.Osterloh, et al., Nature 416, 608 (2002).]
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Concurrence
For a 2-qubit pure state the concurrence is (Wootters, 1998)
Cij (  )   *  iy   jy 
if
  a 00  b 01  c 10  d 11
C    2 ad  bc
• Is maximal for the Bell states and zero for product states
For a 2-qubit mixed state in a spin ½ system


Cij = max 0, Cij , Cij ;

1 x x
C =  σ i σ j  σ iy σ yj  1  σ iz σ zj
2

ij
  σ
2
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z
i
 σ
z
j
 
2

Ising model in transverse field
H   [ ix ix1  h iz ]
i
h  1
     
h  1
  
h 1
Critical point

     
P z  1
2D classical Ising model
CFT with central charge c=1/2
Jordan-Wigner transformation
Exactly solvable fermion model
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Near the transition (h=1):
σz 
2


1

h  1 lnh  1
S1 has the same
singularity as
Local (single site) entropy:
1
ρ1 = I + σ z σ z S1 = Tr  ρ1lnρ1 
2

σz

Local measures of entanglement based on the 2-site density
matrix depend on 2-point functions


σi σ j
Nearest-neighbour concurrence
inherits logarithmic singularity
 h  1 lnh  1
Accidental cancellation of the leading singularity may occur,
as for the concurrence at distance 2 sites
Ci ,i  2 


1
2
σ ix σ ix 2  σ iy σ iy 2  σ iz σ iz 2  1  h  1 ln h  1
2
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Seeking for QPT point
Alternative: FSS of magnetization
N = 20,30,100,
Standard route: PRG
~
~
N h , N = M h , M
 
 
h,N  = E1 N   E0 N 
First excited state needed
C. Hamer, M. Barber, J. Phys. A: Math. Gen. (1981) 247.
Exact scaling function in the
critical region N < ξ 
2 h  1
lnN + ln8 / π + γC  1+ π 12
σz = +
π
π
12 N
Crossing points:
~
π2 1
hN = 1+
6 N2
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Shift
term
Quantum phase transitions (QPT’s)
Let
H ( g )  H 0  gV
• First order: discontinuity in
 e
• Second order:
g n
n
e
g
(level crossing)
g  gc
diverges for some n  2
g  gc
• At criticality the correlation length diverges
• GS energy:
g
gc
e( g )  e(r ) g  e(s)  ( g )
  g  gc

scaling hypothesis
• Differentiating w.r.t. g
V
( s)
 Og(s)  sgn( g  gc ) g  gc

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  (d  1)  1
• The singular term Og(s) appears in every reduced density
matrix containing the sites connected by V .
• Local algebra hypothesis: every local quantity can be expanded
in terms of the scaling fields permitted by the symmetries.
• Any local measure of entanglement contains the singularity
of the most relevant term.
• Warning: accidental cancellations may occur depending on
the specific functional form
next to leading singularity
• The best suited operator for detecting and classifying QPT’s
is V , that naturally contains O(s). Moreover, FSS at criticality ( L   )
g
V
( s)
L Og(s) L sgn( g  gc )L  / L /  
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0 0
Spin 1 lD model
HJ
N
 [S
i 1
l = Ising-like
x
i
 S ix1  S iy  S iy1  l S iz  S iz1  D ( S iz ) 2 ]
D = single ion
Phase
Diagram
• Symmetries: U(1)xZ2
In this case
e
= Siz
D
 
2
Around the c=1 line:
H


1
 x  2   2   cos
2
  D  Dc 

4K 
Critical
exponents

(sine-Gordon)
  K /( 2  K )
  1 /( 2  K )
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S 
z 2
λ = 2.59
Derivative
~ sgn D  Dc D  Dc 

Dc = 2.294
 = 0.64
 = 0.82
The same for
S iz S iz1 ~ sgn l  lc l  lc 

Crossing effect
• What about local measures
of entanglement?
Using symmetries:
S1 =  z lnz / 21  z ln1  z 
 
z = Siz
2
Single-site entropy
( 2)
• Two-sites density matrix ij contains the same leading singularity
[ L.Campos Venuti, et. al., PRA 73, 010303(R) (2006).]
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[ F.Verstraete, M.Popp, J.I.Cirac, PRL 92, 27901 (2004).]
Localizable Entanglement
• LE is the maximum amount of entanglement that can
be localized on two q-bits by local measurements.
j
i
Lij  max ps E(| s )
{ s}
N+2 particle state 
s
• Maximum over all local measurement basis
s
 s1 ,  , s N
ps = probability of getting s
E| s

is a measure of entanglement (concurrence)
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
[ L. Campos Venuti, M. Roncaglia, PRL 94, 207207 (2005).]
Calculating the LE requires finding an optimal
basis, which is a formidable task in general
However, using symmetries some maximal (optimal) basis
are easily found and the LE takes a manageable form
Spin 1/2


Spin 1
Ising model
Quantum XXZ chain


LE = max of correlation
λD
MPS (AKLT)
LE = string correlations



L1N   exp iStot
  1
Lij  max   i  j 

•  E  C :
The lower bound is attained
• The LE shows that spin 1 are
perfect quantum channels but is
insensitive to phase transitions.
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A spin-1 model: AKLT
 
1   2
H  J  [ Si  S j  ( Si  S j ) ]
3
i 1
N
=Bell state
Optimal basis:
0 ,   1  1 / 2
• Infinite entanglement length but finite correlation length
• Actually in S=1 case LE is related to string correlation
O
String
  lim
| k  j | 

S j expi



Typical configurations
k 1
  Sl
l  j 1



 Sk


0
   0  00  0    0  0   
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Conclusions
•
Low-dimensional systems are good candidates for Quantum Information devices.
•
Several local measures of entanglement have been proposed recently for the
detection and classification of QPT. (nonsystematic approach)
•
Apart from accidental cancellations all the scaling properties of local
entanglement come from the most relevant (RG) scaling operator.
•
The most natural local quantity is e / g, where g is the driving parameter
across the QPT.
• it shows a crossing effect
Advantages:
• it is unique and generally applicable
•
Localizable Entanglement  It is related to some already known correlation
functions. It promotes S=1 chains as perfect quantum channels.
•
Open problem: Hard to define entanglement for multipartite systems,
separating genuine quantum correlations and classical ones.
References:
L.Campos Venuti, C.Degli Esposti Boschi, M.Roncaglia, A.Scaramucci, PRA 73, 010303(R) (2006).
L.Campos Venuti and M. Roncaglia, PRL 94, 207207 (2005).
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The End
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