Entanglement Entropy and Conformal Field Theory Pasquale Calabrese Dipartimento di Fisica Universit` a di 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
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Entanglement Entropy and Conformal Field Theory
Pasquale Calabrese
Dipartimento di FisicaUniversita di Pisa
Les Houches Winter School 2010
Mainly joint work with John Cardybut 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
What is new?
SA =c
3ln`
a+ c ′1
TrρnA = cn
(`
a
)− c6
(n−1/n)
+. . .?
And what about more intervals?
SA =? TrρnA =?
Pasquale Calabrese Entanglement and CFT
What is new?
SA =c
3ln`
a+ c ′1
TrρnA = cn
(`
a
)− c6
(n−1/n)
+. . .?
And what about more intervals?
SA =? TrρnA =?
Pasquale Calabrese Entanglement and CFT
What is new?
SA =c
3ln`
a+ c ′1
TrρnA = cn
(`
a
)− c6
(n−1/n)
+. . .?
And what about more intervals?
SA =? TrρnA =?
Pasquale Calabrese Entanglement and CFT
What is new?
SA =c
3ln`
a+ c ′1
TrρnA = cn
(`
a
)− c6
(n−1/n)
+. . .?
And what about more intervals?
SA =? TrρnA =?
Pasquale Calabrese Entanglement and CFT
Entanglement Entropy and path integral PC and John Cardy ’04
Lattice QFT in 1+1 dimensions: {φ(x)} a set of fundamental fields witheigenvalues {φ(x)} and eigenstates ⊗x |{φ(x)}〉The density matrix at temperature β−1 is (Z = Tr e−βH)
ρ({φ1(x)}|{φ2(x)}) = Z−1〈{φ2(x)}|e−βH |{φ1(x)}〉
Euclidean path integral:
ρ = =
Z[dφ(x , τ)]
Z
Yx
δ(φ(x , 0)−φ2(x))Yx
δ(φ(x , β)−φ1(x)) e−SE
SE =R β
0 LE dτ , with LE the Euclidean Lagrangian
The trace sews together the edges along τ = 0 and τ = β to form a cylinder ofcircumference β.A = (u1, v1), . . . , (uN , vN): ρA sewing together only those points x which arenot in A, leaving open cuts for (uj , vj) along the the line τ = 0.
ρA = =
Zx∈B
[dφ(x , 0)]δ(φ(x , β)− φ(x , 0))ρ
Pasquale Calabrese Entanglement and CFT
Replica trick PC and John Cardy ’04
SA = −TrρA log ρA = − limn→1
∂
∂nTrρn
A
TrρnA (for integer n) is the partition function on n of the above
cylinders attached to form an n−sheeted Riemann surface
=“ρijAρ
jkA ρ
klA ρ
liA”
TrρnA has a unique analytic continuation to Re n > 1 and that its
first derivative at n = 1 gives the required entropy:
SA = − limn→1
∂
∂n
Zn(A)
Zn
Pasquale Calabrese Entanglement and CFT
CFT: a remind
A physical systems at a quantum critical point is scale invariant
A Hamiltonian that is invariant under translations, rotations, and scalingtransformations has usually the symmetry of the larger conformal groupdefined as the set of transformations that do not change the angles.
But x > 2 . . . not only:At the conical singularities relevant operators can be generated locallyThe marginal case x = 2 introduces logarithms and has to do with c-theorem
Pasquale Calabrese Entanglement and CFT
Disjoint intervals: History
A = [u1, v1] ∪ [u2, v2]
In 2004 we obtained
TrρnA = c2
n
„|u1 − u2||v1 − v2|
|u1 − v1||u2 − v2||u1 − v2||u2 − v1|
« c6
(n−1/n)
Tested for free fermions in different ways Casini-Huerta, Florio et al.
For more complicated theories in 2009 Furukawa-Pasquier-Shiraishi andCaraglio-Gliozzi showed that it is wrong!
TrρnA = c2
n
„|u1 − u2||v1 − v2|
|u1 − v1||u2 − v2||u1 − v2||u2 − v1|
« c6
(n−1/n)
Fn(x)
x = (u1−v1)(u2−v2)(u1−u2)(v1−v2)
= 4− point ratio
Pasquale Calabrese Entanglement and CFT
Disjoint intervals: History
A = [u1, v1] ∪ [u2, v2]
In 2004 we obtained
TrρnA = c2
n
„|u1 − u2||v1 − v2|
|u1 − v1||u2 − v2||u1 − v2||u2 − v1|
« c6
(n−1/n)
Tested for free fermions in different ways Casini-Huerta, Florio et al.
For more complicated theories in 2009 Furukawa-Pasquier-Shiraishi andCaraglio-Gliozzi showed that it is wrong!