Entanglement entropy in two-dimensional conformal field theory - - PowerPoint PPT Presentation

Entanglement entropy in two-dimensional conformal field theory

Paweł Grabiński

University of Wrocław

22 July 2016

Entanglement in 2dCFT Paweł Grabiński 1/12

Entanglement

◮ Density operator:

ˆ ρ = |ψ ⊗ ψ|

◮ Reduced density operator:

ˆ ρA = trAˆ ρ

◮ Separable subsystems:

|ψAB = |φA ⊗ |χB

◮ Entangled subsystems:

|ψAB =

• i

√pi |iA ⊗ |iB

◮ Entropy of entanglement(von Neuman Entropy):

SA = −tr (ˆ ρA log ˆ ρA) = −

• i

pi log pi

Entanglement in 2dCFT Paweł Grabiński 2/12

Density operator in QFT

Density operator in Quantum Field Theory can be defined as: ρ(φ1, φ2) = Z −1

• F(φ1,φ2)

Dφe−SE [φ] (1) τ x β [0, β] × R φ(β, x) φ(0, x) φ(0, x) = φ(β, x) x S1 × R τ

Entanglement in 2dCFT Paweł Grabiński 3/12

Reduced density operator in QFT

Reduced density operator can be defined as: ρ(φ1, φ2) = Z −1

• S\A

Dφe−SE [φ]

x τ φ(0, x) = φ(β, x) B B A φ(β, x)=φ(0, x) . .

Entanglement in 2dCFT Paweł Grabiński 4/12

2d Conformal Field Theory

◮ Conformal Transformation:

g′

µν(x′) = ∂xα

∂x

′µ

∂xβ ∂x

′ν gµν(x) = Ω(x)gµν(x),

µ, ν = 1, 2

◮ Primary Fields Transformation in CFT(w = f (z)):

φ′(w, ¯ w) = dw dz −h d ¯ w d ¯ z −¯

h

φ(z, ¯ z)

◮ Global Ward’s Identities(i = −1, 0, 1) from SL(2, C): n

• j=1

zi

j

• zj∂j + hj(i + 1)
• φ(z1, ¯

z1) . . . φ(zj, ¯ zj) . . . φ(zn, ¯ zn) = 0

◮ 2-point correlation function from GWI:

φ(z1, ¯ z1)φ(z1, ¯ z1) = A (z1 − z2)−2h ( ¯ z1 − ¯ z2)−2¯

h

Entanglement in 2dCFT Paweł Grabiński 5/12

Entropy and Replica trick

◮ Renyi Entropy:

S(n)

A

= − 1 n − 1 log trˆ ρn

A ◮ Entanglement Entropy:

SA = lim

n→1 S(n) a

= − d dn trˆ ρn

A|n=1 ◮ Replica trick (T = 0 case):

Entanglement in 2dCFT Paweł Grabiński 6/12

Entanglement entropy at T = 0

◮ From non-trivial geometry to Twist fields: n

• i=1

T(w, i)L,Rn = nT(w)L,Rn = Tn(z)T (a, a) ¯ T (b, b)Ln,C T (a, a) ¯ T (b, b)Ln,C

◮ Our conformal transformation:

f : Rn → C, C ∋ z = f (w) = w − a w − b 1

n

, w ∈ Rn

◮ From transformation rule of energy-momentum tensor:

T(w) = dz dw 2 T(z) + c 12{z, w} T(z)Ln,C = 0 ⇒ T(w)L,Rn = c(n2 − 1) 24n2 (a − b)2 (w − a)2(w − b)2

Entanglement in 2dCFT Paweł Grabiński 7/12

Entropy for interval A = [a, b] at T = 0

◮ Local Ward’s Identities:

T(z)φ(z1, ¯ z1) . . . φ(zn, ¯ zn) =

n

• i=1
• hi

(z − zi)2 + 1 z − zi ∂i

• φ(z1, ¯

z1) . . . φ(zn, ¯ zn)

◮ From Local Ward’s Identities we get: h = ¯

h = c(n2 − 1) 24n

◮ Trace of the reduced density operator takes form(α(n) =

A Z n ):

tr{ρn

A} = T (a, a) ¯

T (b, b)Ln,C Z n = α(n)(a − b)− c(n2−1)

6n

◮ Entropy of interval A = [a, b] at T = 0:

SA = cα(1) 3 ln(a − b) − dα dn |n=1 , α(1) = 1

Entanglement in 2dCFT Paweł Grabiński 8/12

Results for interval in T > 0

◮ Field transformation rule:

T (u, u) ¯ T (v, v)L,cyl =

• dw

dz

• 2h

a

• dw

dz

• 2h

b

T (a, a) ¯ T (b, b)Ln,C

◮ Mapping g : C → R × S

z = g(w) = 2π β log (w)

◮ Form of correlation function:

T (u, u) ¯ T (v, v)L,cyl = A β π sh π β (u − v) −4h

◮ Entropy of interval A = [a, b] in T > 0:

SA = c 3 ln β π sh π β (u − v)

• − dα

dn |n=1

Entanglement in 2dCFT Paweł Grabiński 9/12

Interval on a circle at T = 0

Performing rotation of cylinder(β = iL): SA = c 3 ln L π sin π L(u − v)

• − dα

dn |n=1

0.2 0.4 0.6 0.8 1.0 l L 0.3 0.2 0.1 0.1 0.2 Entropy of entanglment Entanglement in 2dCFT Paweł Grabiński 10/12

Statistical models correspondance

2-dimensional Conformal Field Theory

• Statistical Models at Critical Point

0.2 0.4 0.6 0.8 1

n/L

• 0.4
• 0.2

0.2 0.4 0.6 0.8

Sn(L)-log(L/π)/3

n=1 n=2 n=3 n=4 n=5 n=6

Example: Quantum 1d Ising model with transverse field with hamiltonian: H = −

N−1

• i=1
• J Sx

i Sx i+1 + hSz i

• Entanglement in 2dCFT

Paweł Grabiński 11/12

Literature

Thank You!

• P. Di Francesco, P. Mathieu, D. Senechai, Conformal Field Theory,

Springer-Verlag, New York 1997

• P. Calabrese, J. Cardy, Entanglement Entropy and Quantum Field

Theory

• F. Leja, Analytic and harmonic functions, PTW, Warsaw 1952

Entanglement in 2dCFT Paweł Grabiński 12/12