Entanglement Entropy and Conformal Field Theory
Pasquale Calabrese
Dipartimento di Fisica Universit`adi Pisa
Les Houches Winter School 2010
Mainly joint work with John Cardy but also V. Alba, M. Campostrini, F. Essler, M. Fagotti, A. Lefevre, B. Nienhuis, L. Tagliacozzo, E. Tonni Review: PC & JC JPA 42, 504005 (2009)
Pasquale Calabrese Entanglement and CFT If you want to know more:
Extensive reviews by Amico et al., Eisert et al. [RMP] ⊕ Special issue of JPA
Pasquale Calabrese Entanglement and CFT − c (n−1/n) ` 6 Trρn = c +. . . A n a ?
And what about more intervals?
n SA =? TrρA =?
What is new?
c ` S = ln + c0 A 3 a 1
Pasquale Calabrese Entanglement and CFT +. . . ?
And what about more intervals?
n SA =? TrρA =?
What is new?
− c (n−1/n) ` 6 Trρn = c A n a
c ` S = ln + c0 A 3 a 1
Pasquale Calabrese Entanglement and CFT And what about more intervals?
n SA =? TrρA =?
What is new?
− c (n−1/n) ` 6 Trρn = c +. . . A n a ?
c ` S = ln + c0 A 3 a 1
Pasquale Calabrese Entanglement and CFT What is new?
− c (n−1/n) ` 6 Trρn = c +. . . A n a ?
c ` And what about more intervals? S = ln + c0 A 3 a 1
n SA =? TrρA =?
Pasquale Calabrese Entanglement and CFT Entanglement Entropy and path integral PC and John Cardy ’04
Lattice QFT in 1+1 dimensions: {φˆ(x)} a set of fundamental fields with eigenvalues {φ(x)} and eigenstates ⊗x |{φ(x)}i The density matrix at temperature β−1 is (Z = Tr e−βHˆ )
−1 −βHˆ ρ({φ1(x)}|{φ2(x)}) = Z h{φ2(x)}|e |{φ1(x)}i
Euclidean path integral: Z [dφ(x, τ)] Y Y −SE ρ = = δ(φ(x, 0)−φ2(x)) δ(φ(x, β)−φ1(x)) e Z x x R β SE = 0 LE dτ, with LE the Euclidean Lagrangian The trace sews together the edges along τ = 0 and τ = β to form a cylinder of circumference β. A = (u1, v1),..., (uN , vN ): ρA sewing together only those points x which are not in A, leaving open cuts for (uj , vj ) along the the line τ = 0.
Z ρA = = [dφ(x, 0)]δ(φ(x, β) − φ(x, 0))ρ x∈B
Pasquale Calabrese Entanglement and CFT Replica trick PC and John Cardy ’04
∂ n SA = −TrρA log ρA = − lim TrρA n→1 ∂n n TrρA (for integer n) is the partition function on n of the above cylinders attached to form an n−sheeted Riemann surface
ij jk kl li =“ρAρA ρA ρA”
n TrρA has a unique analytic continuation to Re n > 1 and that its first derivative at n = 1 gives the required entropy:
∂ Zn(A) SA = − lim n→1 ∂n Z n
Pasquale Calabrese Entanglement and CFT CFT: a remind
A physical systems at a quantum critical point is scale invariant 2∆φ −2∆φ hφ(r1)φ(r2)i = b hφ(br1)φ(br2)i hφ(r1)φ(r2)i = |r1 − r2| A Hamiltonian that is invariant under translations, rotations, and scaling transformations has usually the symmetry of the larger conformal group defined as the set of transformations that do not change the angles. YOUIn 2D the DON’T consequences are extraordinary: NEED THIS! all the analytic functions w(z) are conformal
0 0 2∆φ hφ(z1)φ(z2)i = |w (z1)w (z2)| hφ(w(z1))φ(w(z2))i
Under an arbitrary transformation x µ → x µ + µ Z 2 µν S → S + δS, with δS = d xT ∂µν
2 2 Under a conformal „ dz « c z 000z 0 − 3/2z 00 T (w) = T (z)+ transformation w → z dw 12 z 02
Pasquale Calabrese Entanglement and CFT c(1 − (1/n)2) (v − u)2 hT (z)i = 0 ⇒ hT (w)i = C Rn 24 (w − u)2(w − v)2
2 To be compared with the hT (w)Φn(u)Φ−n(v)iC ∆Φ(v − u) = 2 2 Conformal Ward identities hΦn(u)Φ−n(v)iC (w − u) (w − v)
n th Zn/Z transforms under conformal transformations as n power of the two point function of a (fake) primary field on the plane with scaling dimension
−(c/6)(n−1/n) c ` 1 ´ n Zn “ v − u ” ∆Φ = ∆Φ = 1 − 2 ⇒ Tr ρ = = c 24 n A Z n n a
c ` Finally with the replica trick (v − u = `) ⇒ S = ln + c0 A 3 a 1
Entropy and CFT PC and John Cardy ’04
n Single interval A = [u, v]. We need Zn/Z = h0|0iRn ⇒ compute hT (w)iRn 1/n w−u 1/n “ w−u ” w → ζ = w−v ; ζ → z = ζ ⇒ w → z = w−v
Pasquale Calabrese Entanglement and CFT c ` Finally with the replica trick (v − u = `) ⇒ S = ln + c0 A 3 a 1
Entropy and CFT PC and John Cardy ’04
n Single interval A = [u, v]. We need Zn/Z = h0|0iRn ⇒ compute hT (w)iRn 1/n w−u 1/n “ w−u ” w → ζ = w−v ; ζ → z = ζ ⇒ w → z = w−v
c(1 − (1/n)2) (v − u)2 hT (z)i = 0 ⇒ hT (w)i = C Rn 24 (w − u)2(w − v)2
2 To be compared with the hT (w)Φn(u)Φ−n(v)iC ∆Φ(v − u) = 2 2 Conformal Ward identities hΦn(u)Φ−n(v)iC (w − u) (w − v)
n th Zn/Z transforms under conformal transformations as n power of the two point function of a (fake) primary field on the plane with scaling dimension
−(c/6)(n−1/n) c ` 1 ´ n Zn “ v − u ” ∆Φ = ∆Φ = 1 − 2 ⇒ Tr ρ = = c 24 n A Z n n a
Pasquale Calabrese Entanglement and CFT Entropy and CFT PC and John Cardy ’04
n Single interval A = [u, v]. We need Zn/Z = h0|0iRn ⇒ compute hT (w)iRn 1/n w−u 1/n “ w−u ” w → ζ = w−v ; ζ → z = ζ ⇒ w → z = w−v
c(1 − (1/n)2) (v − u)2 hT (z)i = 0 ⇒ hT (w)i = C Rn 24 (w − u)2(w − v)2
2 To be compared with the hT (w)Φn(u)Φ−n(v)iC ∆Φ(v − u) = 2 2 Conformal Ward identities hΦn(u)Φ−n(v)iC (w − u) (w − v)
n th Zn/Z transforms under conformal transformations as n power of the two point function of a (fake) primary field on the plane with scaling dimension
−(c/6)(n−1/n) c ` 1 ´ n Zn “ v − u ” ∆Φ = ∆Φ = 1 − 2 ⇒ Tr ρ = = c 24 n A Z n n a
c ` Finally with the replica trick (v − u = `) ⇒ S = ln + c0 A 3 a 1
Pasquale Calabrese Entanglement and CFT Open systems (generally systems with boundaries)
c 1 n ` 2` ´ 12 (n− n ) c 2` 0 Tr ρA =c ˜n a ⇒ SA = 6 log a +c ˜1 c “ β 2π` ” 0 finite temperature SA(β) = 6 log πa sinh β +c ˜1
c ` 2L π` ´ 0 and finite size SA(L) = 6 log πa sin L +c ˜1
0 0 c˜1 − c1/2 = ln g boundary entropy [Affleck, Ludwig]
Entanglement and CFT (generalizations)PC and John Cardy ’04
Finite temperature 8 πc ` “ ” <> , ` β classical extensive S = c log β sinh π` + c0 ' 3 β A 3 πa β 1 c ` :> log , ` β T = 0 non − extensive 3 a
Finite size c ` L π` ´ 0 SA = 3 log πa sin L + c1 Symmetric ` → L − `. Maximal for ` = L/2
Pasquale Calabrese Entanglement and CFT Entanglement and CFT (generalizations)PC and John Cardy ’04
Finite temperature 8 πc ` “ ” <> , ` β classical extensive S = c log β sinh π` + c0 ' 3 β A 3 πa β 1 c ` :> log , ` β T = 0 non − extensive 3 a
Finite size c ` L π` ´ 0 SA = 3 log πa sin L + c1 Symmetric ` → L − `. Maximal for ` = L/2
Open systems (generally systems with boundaries)
c 1 n ` 2` ´ 12 (n− n ) c 2` 0 Tr ρA =c ˜n a ⇒ SA = 6 log a +c ˜1 c “ β 2π` ” 0 finite temperature SA(β) = 6 log πa sinh β +c ˜1
c ` 2L π` ´ 0 and finite size SA(L) = 6 log πa sin L +c ˜1
0 0 c˜1 − c1/2 = ln g boundary entropy [Affleck, Ludwig]
Pasquale Calabrese Entanglement and CFT [Laflorencie et al ’06]
Correction to the scaling (Hystory)
1 c L π` S ≡ ln Trρα = (1 + α−1) ln sin + c0 α 1 − α A 6 π L α
Pasquale Calabrese Entanglement and CFT Correction to the scaling (Hystory)
1 c L π` S ≡ ln Trρα = (1 + α−1) ln sin + c0 α 1 − α A 6 π L α
[Laflorencie et al ’06]
Pasquale Calabrese Entanglement and CFT + `−1 ? Too naive : NOOO! + `−2(x−2) with x dim. Irr Op? You know FSS, but it is WRONG! + `−2x/α ? What? and x < 2 WHAT ? Sorry, this is the right answer
Numerical evidence for XX model (with x = 1):
CFT dα(`) ≡ Sα(`) − Sα (`)
Corrections to the scaling: Guess? c S = (1 + α−1) ln ` + c0 +? α 6 α
Pasquale Calabrese Entanglement and CFT + `−2x/α ? What? and x < 2 WHAT ? Sorry, this is the right answer
Numerical evidence for XX model (with x = 1):
CFT dα(`) ≡ Sα(`) − Sα (`)
Corrections to the scaling: Guess? c S = (1 + α−1) ln ` + c0 +? α 6 α + `−1 ? Too naive : NOOO! + `−2(x−2) with x dim. Irr Op? You know FSS, but it is WRONG!
Pasquale Calabrese Entanglement and CFT Numerical evidence for XX model (with x = 1):
CFT dα(`) ≡ Sα(`) − Sα (`)
Corrections to the scaling: Guess? c S = (1 + α−1) ln ` + c0 +? α 6 α + `−1 ? Too naive : NOOO! + `−2(x−2) with x dim. Irr Op? You know FSS, but it is WRONG! + `−2x/α ? What? and x < 2 WHAT ? Sorry, this is the right answer
Pasquale Calabrese Entanglement and CFT Corrections to the scaling: Guess? c S = (1 + α−1) ln ` + c0 +? α 6 α + `−1 ? Too naive : NOOO! + `−2(x−2) with x dim. Irr Op? You know FSS, but it is WRONG! + `−2x/α ? What? and x < 2 WHAT ? Sorry, this is the right answer
Numerical evidence for XX model (with x = 1):
CFT dα(`) ≡ Sα(`) − Sα (`)
Pasquale Calabrese Entanglement and CFT Exact Solution XX model PC, F Essler
CFT ` −2/α Sα(`) − Sα (`) = (−) ` fα + ...
2 Γ2((1 + 1/α)/2) f = α−→→1 0 α 1 − α Γ2((1 − 1/α)/2)
π2 1 ` odd 12 ln b` CFT S∞(`) − S∞ (`) = π2 1 − ` even 24 ln b` In a magnetic field:
CFT −2/α Sα(`) − Sα (`) = fα cos(2kF `)|2` sin kF | + ...
PS: For the Ising model it is the same pα = 2/α
Pasquale Calabrese Entanglement and CFT Odd L Even L
pα and Fα(x) are universal Similar plots for any ∆, odd and even L, any α
pα = 2K/α
Fα depends on the parity of L
Many interesting details in the paper
XXZ Heisenberg chain PC, M Campostrini, F Essler, B Nienhuis
L π`−pα S (`, L) = SCFT(`, L) + (−)`f sin F (`/L) α α α π L α XX in Finite Size All ` for several odd L from 17 to 4623 [∼ 105 points]:
F2(x) = cos πx ? Pasquale Calabrese Entanglement and CFT XX in Finite Size All ` for several odd L from 17 to 4623 [∼ 105 points]:
F2(x) = cos πx ?
XXZ Heisenberg chain PC, M Campostrini, F Essler, B Nienhuis
L π`−pα S (`, L) = SCFT(`, L) + (−)`f sin F (`/L) α α α π L α
Odd L Even L
pα and Fα(x) are universal Similar plots for any ∆, odd and even L, any α
pα = 2K/α
Fα depends on the parity of L
Pasquale Calabrese Entanglement and CFT Many interesting details in the paper Corrections to Scaling in CFT J Cardy, PC
Z 2 Add an irrelevant operator Φ of dimension x: S = SCFT + λ Φ(z)d z ,
∞ N Z Z X (−λ) 2 2 Change in free energy: −δFn = ··· hΦ(z1) ... Φ(zN )iRn d z1 ... d zN , N=1 N! Rn Rn 1/n ζ = (z/(z − `)) maps Rn on the plane
Z Z (2−x)(n−1) (2) 1 2 4−2x |ζ1ζ2| 2 2 δFn = − g (n`/) 4−2x 4−2x 2x d ζ1d ζ2 . 2 0 0 n n C C |ζ1 − 1| |ζ2 − 1| |ζ1 − ζ2| For n − 1 small the integral is convergent! ⇒ corrections `−2(x−2) For larger values of n divergences as ζj → 0 or ∞. Regulator in z plane! Any of them gives a multiplicative factor ∝ 2−x+(x/n) Summing up: `−2(x−2)−(2−x+(x/n))−(2−x+(x/n)) = `−2x/n But x > 2 . . . not only: At the conical singularities relevant operators can be generated locally The marginal case x = 2 introduces logarithms and has to do with c-theorem
Pasquale Calabrese Entanglement and CFT For more complicated theories in 2009 Furukawa-Pasquier-Shiraishi and Caraglio-Gliozzi showed that it is wrong!
c „ « 6 (n−1/n) n 2 |u1 − u2||v1 − v2| TrρA = cn Fn(x) |u1 − v1||u2 − v2||u1 − v2||u2 − v1|
x = (u1−v1)(u2−v2) = 4 − point ratio (u1−u2)(v1−v2)
Disjoint intervals: History
A = [u1, v1] ∪ [u2, v2]
In 2004 we obtained
c „ « 6 (n−1/n) n 2 |u1 − u2||v1 − v2| TrρA = cn |u1 − v1||u2 − v2||u1 − v2||u2 − v1|
Tested for free fermions in different ways Casini-Huerta, Florio et al.
Pasquale Calabrese Entanglement and CFT Disjoint intervals: History
A = [u1, v1] ∪ [u2, v2]
In 2004 we obtained
c „ « 6 (n−1/n) n 2 |u1 − u2||v1 − v2| TrρA = cn |u1 − v1||u2 − v2||u1 − v2||u2 − v1|
Tested for free fermions in different ways Casini-Huerta, Florio et al. For more complicated theories in 2009 Furukawa-Pasquier-Shiraishi and Caraglio-Gliozzi showed that it is wrong!
c „ « 6 (n−1/n) n 2 |u1 − u2||v1 − v2| TrρA = cn Fn(x) |u1 − v1||u2 − v2||u1 − v2||u2 − v1|
x = (u1−v1)(u2−v2) = 4 − point ratio (u1−u2)(v1−v2)
Pasquale Calabrese Entanglement and CFT Compactified boson (Luttinger) Furukawa Pasquier Shiraishi
4 θ3(ητ)θ3(τ/η) θ2(τ) 1 2 F2(x) = 2 , x = η = ∝ R [θ3(τ)] θ3(τ) 2K Compared again exact diagonalization in XXZ chain
Pasquale Calabrese Entanglement and CFT Compactified boson PC Cardy Tonni
Using old results of CFT Θ 0|ηΓ Θ 0|Γ/η Fn(x) = on orbifolds Dixon et al 86 [Θ 0|Γ]2 Γ is an (n − 1) × (n − 1) matrix
n−1 „ « » – 2i X k k 2F1(y, 1 − y; 1; 1 − x) Γrs = sin π βk/n cos 2π (r − s) , βy = n n n 2F1(y, 1 − y; 1; x) k = 1 X Riemann theta function Θ(z|Γ) ≡ exp ˆ iπ m · Γ · m + 2πim · z ˜ m ∈ Zn−1 •F n(x) invariant under x → 1 − x and η → 1/η • We are unable to analytic continue to real n for general x and η • Only for η 1 and for x 1
Pasquale Calabrese Entanglement and CFT x 1 n−1 “ x ”α X l/n F (x) = 1+2n P +. . . α = min(η, 1/η) P = n 4n2 n n 2α l=1 [sin (πl/n)]
0 1−2α α 0 −F1(x) = 2 x P1 + ...
Compactified boson II PC Cardy Tonni
η 1
0 0 Z i∞ 0 1 D1(x) + D1(1 − x) 0 dz πz −F1(x) = ln η− with D1(x) = − 2 log Hz (x) 2 2 −i∞ i sin πz
Pasquale Calabrese Entanglement and CFT Compactified boson II PC Cardy Tonni
η 1
0 0 Z i∞ 0 1 D1(x) + D1(1 − x) 0 dz πz −F1(x) = ln η− with D1(x) = − 2 log Hz (x) 2 2 −i∞ i sin πz
x 1 n−1 “ x ”α X l/n F (x) = 1+2n P +. . . α = min(η, 1/η) P = n 4n2 n n 2α l=1 [sin (πl/n)]
0 1−2α α 0 −F1(x) = 2 x P1 + ...
Pasquale Calabrese Entanglement and CFT The XX model Fagotti PC
The RDM of two intervals is not trivial because of JW string
CFT OK and δα = 2/α
Pasquale Calabrese Entanglement and CFT Von Neumann
The Ising model Alba Tagliacozzo PC
Monte Carlo for 2D and TTN for 1D 2 TrρA MC
2 TrρA TTN
Large monotonic correction to the scaling! FSS analysis confirms:
" √ √ 1/2 #1/2 „ (1 + x)(1 + 1 − x) « (x 1/4 + ((1 − x)x)1/4 + (1 − x)1/4) F (x) = + 2 2 2
Pasquale Calabrese Entanglement and CFT Monte Carlo for 2D and TTN for 1D 2 TrρA MC
2 TrρA TTN
Large monotonic correction to the scaling! FSS analysis confirms:
" √ √ 1/2 #1/2 „ (1 + x)(1 + 1 − x) « (x 1/4 + ((1 − x)x)1/4 + (1 − x)1/4) F (x) = + 2 2 2
The Ising model Alba Tagliacozzo PC
Von Neumann
Pasquale Calabrese Entanglement and CFT The Ising model Fagotti PC
δα = 1/α because of Ising fermion!
Pasquale Calabrese Entanglement and CFT Take home message
Entanglement entropy naturally encodes universal features of quantum systems
Pasquale Calabrese Entanglement and CFT