entanglement entropy in the xy modelffranchi/presentations... · →entanglement of a block of...
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Entanglement Entropy in the XY Model
Coauthors: V. E. Korepin, A. R. ItsB.-Q. Jin
Thanks: A.G. Abanov, B.M. McCoy, L.A. Takhtajan
Fabio Franchini
- JPA 40, 8467 (2007) - JPA 41, 2530 (2008)- work in progress…
&
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• Introduction: Von Neumann and Renyi Entropy as a measure of Entanglement
• Quantum Entropy of the XY model
• Ellipses of constant Entropy and theEssential Critical Point
• Modular properties of the entropyand of the partition function -Not enough time-
• Conclusions
Outline
Entanglement Entropy in the XY Model n. 2 Fabio Franchini
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• Consider a unique (pure) ground state
• Divide the system into two Subsystems: A & B
• If system wave-function is:
→ No Entanglement
(Measurements on B does not affect A state)
Understanding Entanglement
Entanglement Entropy in the XY Model n. 3 Fabio Franchini
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• If the system wave-function is:
(with d > 1, & linearly independent):
→ Entangled (Measurements on B affect A state):
i.e.
Understanding Entanglement (cont.)
Entanglement Entropy in the XY Model n. 4 Fabio Franchini
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• Compute Density Matrix of subsystem:
• Entanglement for pure state as Quantum Entropy (Bennett, Bernstein, Popescu, Schumacher 1996):
Von Neumann Entropy
How to measure Entanglement?
Entanglement Entropy in the XY Model n. 5 Fabio Franchini
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• Von Neumann Entropy:
• Renyi Entropy:(equal to Von Neumann for α → 1)
• Tsallis Entropy
• Concurrence (Two-Tangle)
• ...
More Entanglement Estimators
Entanglement Entropy in the XY Model n. 6 Fabio Franchini
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• NB: SAB = 0 < SA + SB (Unlike
thermodynamic entropy)
Entropy of a subsystem
Entanglement Entropy in the XY Model n. 7 Fabio Franchini
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• Assume Bell State as unity of Entanglement:
Entropy as a measure of entanglement
|ΨAi
• Von Neumann Entropy measures how many Bell-Pairs
are contained in a given state (i.e. closeness
of state to maximally entangled one)
Entanglement Entropy in the XY Model n. 8 Fabio Franchini
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• Consider the Ground state of a Hamiltonian:
Entanglement in a Spin Chain
• Block of spins in the space interval [1, n] is subsystem A
• The rest of the ground state is subsystem B.
→ Entanglement of a block of spins on a space interval [1, n]
with the rest of the ground state as a function of n
Entanglement Entropy in the XY Model n. 9 Fabio Franchini
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• We study the bi-partite entropy of the ground state of
a system:
Entropy as a Correlation Function
• Highly non-trivial correlation function: new insights?
• Multi-Point correlation function with contributions
from all two-point correlators
Entanglement Entropy in the XY Model n. 10 Fabio Franchini
Von Neumann
Renyi
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• We study the behavior for block size n → ∞
(Double scaling limit: 0 << n << N )
General Behavior
• For gapped phases: (Vidal, Latorre, Rico, Kitaev 2003)
• For critical phases: (Calabrese, Cardy, 2004)
Entanglement Entropy in the XY Model n. 11 Fabio Franchini
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The Anisotropic XY Model( ) ( )x x y y z
i i 1 i i 1 ii
H 1 1 h+ +⎡ ⎤= − + γ σ σ + − γ σ σ + σ⎣ ⎦∑
Entanglement Entropy in the XY Model n. 12 Fabio Franchini
• Jordan-Wigner followed by Bogoliubov transformation to diagonalize the Hamiltonian
• The XY Model is essentially Free Fermions
• Correlators for physical quantities involve inverting the
transformation to FF: complications
( )†q q q
qH 1/ 2= ε χ χ −∑ 2 2 2
q (h / 2 cosq) sin qε = − + γ
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2 2 2q (h / 2 cosq) sin qε = − + γ
Phase Diagram:
• 3 non-critical regions (2,1a,1b)
• 2 critical phases:Ω0: Isotropic XYΩ+: Critical magnetic field
Phase Diagram of the XY Model(only γ>0 shown)
h Γ
Ω
Ω
1
1
o
I
2
(2)
(1a)
(1b)
γ
Ω+
( ) ( )x x y y zi i 1 i i 1 i
iH 1 1 h+ +⎡ ⎤= − + γ σ σ + − γ σ σ + σ⎣ ⎦∑
The Phase Diagram of the XY Model
Entanglement Entropy in the XY Model n. 13 Fabio Franchini
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h Γ
Ω
Ω
1
1
o
I
2
(2)
(1a)
(1b)
γ
Ω+
Entropy on the gapped phases for |GS>
• We define an Elliptic Parameter:
Entanglement Entropy in the XY Model n. 14 Fabio Franchini
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• For h > 2 :
• For h < 2 :
Asymptotic Entropy
Entanglement Entropy in the XY Model n. 15 Fabio Franchini
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h Γ
Ω
Ω
1
1
o
I
2
(2)
(1a)
(1b)
γ
Ω+
• We have a completely analytical expression for the asymptotic entropy
• Let’s extract some physics out of it!!
Asymptotic Entropy of the XY model
Entanglement Entropy in the XY Model n. 16 Fabio Franchini
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Fabio Franchini
h Γ
Ω
Ω
1
1
o
I
2
(2)
(1a)
(1b)
γ
Ω+
• Absolute minimum at h → ∞ or γ → 0 (h > 2) : S∞ → 0
as the ground state becomes ferromagnetic ( )
• Local minimum S∞ = ln 2at the boundary between cases 1a and 1b ( )
• The ground states is factorized:(each state is factorized and has no entropy)
Minima of the Entropy
Entanglement Entropy in the XY Model n. 17 Fabio Franchini
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1
2 ���������������1 � Γ2
2 3 4 5 6h
0.2
0.4
0.6Ln20.8
1
S
1 A
1 B 2
k�0k�1
k�0
k�����������������1� Γ2Von NeumannEntropy
Entropy at fixed γ
Entanglement Entropy in the XY Model n. 18 Fabio Franchini
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1
2 ���������������1 � Γ2
2 3 4 5 6h
0.2
0.4
0.6Ln20.8
1
S
1 A
1 B 2
k�0k�1
k�0
k�����������������1� Γ2Von NeumannEntropy
Entropy at fixed γ
1 2 3 4h
0.2
0.4
0.6Ln20.81
1.2
1.4
SR�0.5�
k�0
k�0
k�1
Renyi Entropyα = 1/2
1 2 3 4h
0.2
0.4
0.6Ln20.8
1
1.2
1.4
SR�2�
k�0
k�0
k�1Renyi Entropyα = 2
( α = 1 )
Entanglement Entropy in the XY Model n. 19 Fabio Franchini
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3-D plot of the EntropyVon NeumannEntropy( α = 1 )
Entanglement Entropy in the XY Model n. 20 Fabio Franchini
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3-D plot of the Entropy0
0.5
1
1.5
Γ
01
23
4h 00.5
1
1.5
2
S
�Γ,h���0,2�Essential
Critical Point
00.5
1
1
00.5
1
1.5
Γ
0
1
2
3
h
0
0.5
1
1.5
SR�2�
�Γ,h���0,2
Essential
Critical Poin
0
0.5
1
1.5
S
Renyi Entropyα = 1/2 α=2
Von NeumannEntropy( α = 1 )
Entanglement Entropy in the XY Model n. 21 Fabio Franchini
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h Γ
Ω
Ω
1
1
o
I
2
(2)
(1a)
(1b)
γ
Ω+
• Point (γ,h)=(0,2) is special:
• Theory is critical, but not CFT (quadratic spectrum)
• We can study the entropy close to this point:
–Approaching it along h=2, γ>0: S∞ = ∞
–Approaching it along γ=0, h>2: S∞ = 0
–Approaching it along γ=0, h<2: S∞ = ∞
–Approaching it along : S∞ = ln 2
The Essential Critical Point (ECP)
Entanglement Entropy in the XY Model n. 22 Fabio Franchini
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Entropy around the ECP
(Von Neumann Entropy)
α = 1
Entanglement Entropy in the XY Model n. 23 Fabio Franchini
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Entropy around the ECP
2 2.05 2.1 2.15 2.2 2.25h
1.75 1.8 1.85 1.9 1.95 2h
0
0.1
0.2
0.3
0.4
0.5
Γ
2 2.05 2.1 2.15 2.2 2.25h
1.75 1.8 1.85 1.9 1.95 2h
0
0.1
0.2
0.3
0.4
0.5
Γ
(Von Neumann Entropy)
α = 2
α = 1/2
α = 1
Entanglement Entropy in the XY Model n. 24 Fabio Franchini
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• For h > 2 :
• For h < 2 :
The entropy depends just on one parameter (k)
Recalling the formulae
Entanglement Entropy in the XY Model n. 25 Fabio Franchini
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• Curves of constant Entropy
are curves of constant k
• These curves are
Hyperbolae and Ellipses:
• All these curves pass through the Essential Critical Point!
h Γ
Ω
Ω
1
1
o
I
2(2)
(1a)(1b)
γ
Ω+
Curves of constant Entropy
Entanglement Entropy in the XY Model n. 26 Fabio Franchini
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• From any point in the phase diagram one reaches the ECP following a curve of constant Entropy
• The range of the Entropy in the phase diagram is the positive real axis
Near the ECP the Entropy reachesevery positive value!
• Small variations in the parameters change the Entropy dramatically!
• ECP important forQuantum Control
⇒
Importance of the Essential Critical Point
Entanglement Entropy in the XY Model n. 27 Fabio Franchini
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Critical Magnetic Field:
Isotropic XY model (XX Model):
h Γ
Ω
Ω
1
1
o
I
2
(2)
(1a)
(1b)
γ
Ω+
(Jin, Korepin 2003)
(Calabrese, Cardy, 2004)
Entropy on the critical phases
Entanglement Entropy in the XY Model n. 28 Fabio Franchini
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Critical Magnetic Field:
Isotropic XY model (XX Model):
h Γ
Ω
Ω
1
1
o
I
2
(2)
(1a)
(1b)
γ
Ω+
Entropy on the critical phases
Entanglement Entropy in the XY Model n. 29 Fabio Franchini
Conjecture:
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Entropy as a function of α0.5 1 1.5 2
Α
0.5
1
1.5
Γ
1
1.5
2
2.5
SR�Α�
1
1
2
0.5
1
1.5
Α1
2
3
h
00.511.5
2
SR�Α�
0.5
1Α
0011
• Diverges for α → 0
• Except at the factorizing field ( ):SR= ln 2
• Limit α → ∞ gives largest eigenvalue of density matrix (Single copy entanglement)
Entanglement Entropy in the XY Model n. 30 Fabio Franchini
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Conclusions• We studied analytically the entropy (Von Neumann and Renyi) as a measure of
bipartite entanglement in double scaling limit of the XY model
• Entropy diverges for critical phases, approaches a constant in gapped phases
• We achieved detail knowledge of the behavior of the entropy (also in α)
• Near Essential Critical Point, entropy reaches every positive value
• We can access the spectrum of the density matrix (we have the largest eigenvalue, we are working on the others)
• Entropy is sensitive to previously unnoticed modular properties of the model
Thank you!Entanglement Entropy in the XY Model n. 31 Fabio Franchini
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Von Neumann Entropy of the XY model
Entanglement Entropy in the XY Model n. 32 Fabio Franchini