Entanglement Entropy in the XY Modelffranchi/presentations... · →Entanglement of a block of spins on a space interval [1, n] with the rest of the ground state as a function of

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…

&

• 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

• 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


• If the system wave-function is:

(with d > 1, & linearly independent):

→ Entangled (Measurements on B affect A state):

i.e.

Understanding Entanglement (cont.)


• Compute Density Matrix of subsystem:

• Entanglement for pure state as Quantum Entropy (Bennett, Bernstein, Popescu, Schumacher 1996):

Von Neumann Entropy

How to measure Entanglement?


• Von Neumann Entropy:

• Renyi Entropy:(equal to Von Neumann for α → 1)

• Tsallis Entropy

• Concurrence (Two-Tangle)

• ...

More Entanglement Estimators


• NB: SAB = 0 < SA + SB (Unlike

thermodynamic entropy)

Entropy of a subsystem


• 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)


• 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


• 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


Von Neumann

Renyi

• 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)


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⎣ ⎦∑


• 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

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


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:


• For h > 2 :

• For h < 2 :

Asymptotic Entropy


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


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


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 ��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 )


3-D plot of the EntropyVon NeumannEntropy( α = 1 )


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 )


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)


Entropy around the ECP

(Von Neumann Entropy)

α = 1


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


• For h > 2 :

• For h < 2 :

The entropy depends just on one parameter (k)

Recalling the formulae


• 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


• 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


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


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


Conjecture:

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)


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

Von Neumann Entropy of the XY model


Documents

Entanglement Entropy in the XY Modelffranchi/presentations... · →Entanglement of a block of spins on a space interval [1, n] with the rest of the ground state as a function of