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Entanglement Entropy and Conformal Field Theory

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Entanglement Entropy and Conformal Field Theory 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: fφ^(x)g a set of fundamental fields with eigenvalues fφ(x)g and eigenstates ⊗x jfφ(x)gi The density matrix at temperature β−1 is (Z = Tr e−βH^ ) −1 −βH^ ρ(fφ1(x)gjfφ2(x)g) = Z hfφ2(x)gje jfφ1(x)gi 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 x2B 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 = jr1 − r2j 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 = jw (z1)w (z2)j 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 = h0j0iRn ) 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 = h0j0iRn ) 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 = h0j0iRn ) 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) 8π2 1 > ` odd >12 ln b` CFT < S1(`) − S1 (`) = > π2 1 >− ` even : 24 ln b` In a magnetic field: CFT −2/α Sα(`) − Sα (`) = fα cos(2kF `)j2` sin kF j + ::: 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 ; 1 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 jζ1ζ2j 2 2 δFn = − g (n`/) 4−2x 4−2x 2x d ζ1d ζ2 : 2 0 0 n n C C jζ1 − 1j jζ2 − 1j jζ1 − ζ2j For n − 1 small the integral is convergent! ) corrections `−2(x−2) For larger values of n divergences as ζj ! 0 or 1.
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