Entanglement Entropy in Non Unitary CFT

Entanglement Entropy in Non Unitary CFT
Francesco Ravanini
Collaboration with
D. Bianchini, O. Castro-Alvaredo, B. Doyon, E. Levi
arXiv:1405.2804, D.B. M.Sc. thesis & work in progress
Surrey Univ., 21 aug 2014
F. Ravanini
EE in non-unitary CFT
Outline
Introduction: Von Neumann and Renyi entropies as a “measure” of
Entanglement
Entanglement entropy in 1D spin chains and in 2D CFT
Non unitary CFT models and EE
Corner Transfer Matrix (CTM) method
Entanglement entropy in spin chains related to ABF and FB models
Numerical results
F. Ravanini
EE in non-unitary CFT
Introduction
Entanglement: fundamental quantum property
Different reasons for interest:
1
Quantum Information, Quantum computers
2
Telecommunication and Teleportation
3
Black holes, Information paradox & Quantum Gravity
4
Condensed matter physics −→ non-local correlations
5
Universality in Quantum Fluctuations and Phase Transitions
6
NON LOCALITY intrinsic in Quantum Mechanics?
EPR paradox (1935): uncompleteness of QM or non-locality?
Bell inequalities (1962)−→ local hidden variables exist only if a
certain correlation P < 2
Clauser Friedmann (1966) & Aspect (1980) experiments −→
P > 2 =⇒ possible non-locality of QM
F. Ravanini
EE in non-unitary CFT
Entanglement and density matrix
Consider a system divided in two complementary subsystems A and B
Define reduced density matrix for subsystem A
ρA = TrB |0ih0|
Quantum entropy (Von Neumann) of Entanglement (E-Entropy)
SA = −TrA (ρA log ρA ) = SB
[Bennett, Bernstein, Popescu, Schumacher (1996)]
For a separable state SA = 0, for a maximally entangled state it is
maximal =⇒ SA is a measure of Entanglement
Area law [Srednicki (1993)]
SA ∝ Area(∂A)
Rényi Entropy
Sn =
1
log TrA ρnA
1−n
F. Ravanini
=⇒
SA = S1 = lim Sn
EE in non-unitary CFT
n→1
Entanglement in a Spin Chain
Hamiltonian of a chain of length L
H=
L
X
Hk,k+1
k=1
Block of spins in the space interval [1, `] is subsystem A
The rest is subsystem B
=⇒ Entanglement of a block of spins in the space interval
[1, `] with the rest of the ground state as a function of `
F. Ravanini
EE in non-unitary CFT
Entanglement entropy in CFT
If the chain is critical, use CFT [Holzhey, Larsen, Wilczek 1994 Calabrese, Cardy 2004]
Partiton function of a theory with Lagrangean L on a Riemann
surface R= n sheets sewn by the segment [a, b]. It has zero
curvature but for points a, b with conical singularities
ˆ
ˆ
Z [L, R] = Dφ exp −
dxdy L[φ](x, y )
R
n copies of the theory
ˆ
´
Z [L, R] = Dφ1 ...Dφn e − C dxdy {L[φ1 ]+...+L[φn ]} = Z [L(n) , C] ≡ Zn
F. Ravanini
EE in non-unitary CFT
Twist fields (I)
Path integral with b.c.
φi (x, 0+ ) = φi+1 (x, 0− )
,
x ∈ [a, b]
Twist fields
φi (y )T (x) = Θ(x1 − y1 )T (x)φi+1 (y ) + Θ(y1 − x1 )T (x)φi (y )
φi (y )T˜ (x) = Θ(x1 − y1 )T˜ (x)φi−1 (y ) + Θ(y1 − x1 )T˜ (x)φi (y )
Orbifold construction [Knizhnik (1987)]
Zn
∝
hO(x)iL,R
=
hT (a, 0)T˜ (b, 0)iL(n) ,C
hO(x)T (a, 0)T˜ (b, 0)iL(n) ,C
F. Ravanini
hT (a, 0)T˜ (b, 0)iL(n) ,C
EE in non-unitary CFT
Twist fields (II)
Conformal transformation
w ∈ R 7→ z ∈ C
:
z=
w −a
w −b
n1
Stress-energy tensor T (n) of replica theory
T (n) (z) =
n
X
Tj (z) transforms as T (w ) =
j=1
dz
dw
2
T (z)+
c
{z, w }
12
1-pt function hT (w )iL,R
hT (x)T (a, 0)T˜ (b, 0)iL(n) ,C
1
(a − b)2
c
1− 2
=
24
n
(w − a)2 (w − b)2
hT (a, 0)T˜ (b, 0)iL(n) ,C
Comparing, get the conformal dimensions of the twist fields
c
1
˜
n−
∆ n = ∆n =
12
n
General definition off-criticality [Cardy, Castro-Alvaredo, Doyon (2008)]
F. Ravanini
EE in non-unitary CFT
Density matrix
Density matrix of the vacuum (ground state |Ωi, not to be confused
with conformal vacumm |0i)
ρ = |ΩihΩ|
Reduced density matrix
ρA = TrB ρ
Traces
TrA ρnA ∝ Zn
normalized
TrA ρˆnA =
Zn
Z1n
Renyi (Sn ) & Von Neumann (S1 ) entropies
Sn =
1
Zn
1
log TrA ρˆnA =
log n
1−n
1−n
Z1
F. Ravanini
,
EE in non-unitary CFT
S1 = lim Sn
n→1
EE in CFT - Results
Thermodynamic limit L → ∞
S(`) ∼
`→∞
c
log ` + O(1)
3
c = central charge of CFT, O(1) = non-universal
Obtain results for L finite through conformal map plane → strip
L
`π
c
S(`, L) = log
sin
+ O(1)
3
π
L
O(1) does not depend on `/L.
Off-criticality S is finite for `, L → ∞ and computable exactly in
integrable spin chains through CTM approach.
S(ξ) ∼
ξ→∞
F. Ravanini
c
log ξ + O(1)
3
EE in non-unitary CFT
Non-unitary models
Free energy (β = 1/kT ) [Affleck; Blote, Cardy, Nightingale (1986)]
F (β) = f Lβ + f˜ β
−
bulk boundary
πc
+ ...
6β
Casimir
For non-unitary models [Itzykson, Saleur, Zuber (1986)]
c 7→ ceff = c − 24∆min
Is it true also for EE?
F. Ravanini
EE in non-unitary CFT
EE in non-unitary CFT
TrA (ρnA ) in the vacuum in a critical chain with boundary.
Zn =orbifold on the half-plane
b−state
A
time
A
time
B
time
l
A
B
B
a−state
Exchange role of time and space, then transform to the cylinder
z 7→ w = i log
`
Zn = ha|e − log ε ·Horb |bi
F. Ravanini
,
`−z
`+z
`
Z1n = ha|e − log ε ·Hrep |bi
EE in non-unitary CFT
Stress-energy tensor
T (z) =
n
X Lk
X
=
T (j) (z)
z k+2
T (j) (x + 2π)
T
(x + 2π)
= T
n
X
(j)
(x)
(j)
Lk ,
(j)
Lk ∈ Virnc
c
12
orbifold (cyclic)
= T (j+1) (x)
replica: n commuting Virc :
Lrep
k =
¯0 −
H = L0 + L
j=1
k∈Z
(j)
=⇒
replica (periodic)
k ∈Z
¯rep
Hrep = Lrep
0 + L0 −
=⇒
j=1
orbifold: Torb (x) = Tb x c (x mod 2π)
2π
X Lk
Torb (x) =
with
z k+2
nc
12
x ∈ [0, 2πn[
Lk ∈ Virc
k ∈Z
k∈Z
T (x) =
n
X
Torb (x + 2πj)
has modes
j=1
F. Ravanini
EE in non-unitary CFT
Lnk , k ∈ Z
Orbifold / (Replica)n
Define: [Doyon, Hoogeveen, Bernard (2013)]
Lorb
k =
Lnk
+ ∆T δ0,k ∈ Virnc
n
=⇒
¯orb nc
Horb = Lorb
0 + L0 −
12
Insert a complete set of states
X
`
nc
`
Zn = ha|e − log ε ·Horb
|sihs|bi ∝ e −2 log ε (∆:T φ: − 12 )
s
Z1n
=
ha|e
− log
`
ε ·Hrep
X
nc
|sihs|bi ∝ e −2 log ε (∆min − 12 )
`
s
ε
Zn
n
=
Tr
ρ
=
A
A
Z1n
`
c
eff
12
(n− n1 )+...
S=
=⇒
Sn =
ceff
`
log + ...
6
ε
F. Ravanini
EE in non-unitary CFT
ceff (n + 1)
`
log +...
12n
ε
New twist field
We have introduced a new field that acts as a twist [Castro-Alvaredo, Doyon,
Levi (2012)]
1
: T φ : (x) = lim ε2(1− n )∆min T (x + ε)φ(x)
ε→0
allowing to express the trace of powers of ρ in a natural way in
non-unitary models where the vacuum is not the conformally invariant
state |0i, but
|φi = φ(0)|0i
where φ(z) is the field with lower (negative) conformal dimesion ∆min
 h:T φ:(`)i
on the half-plane

 hφ(`)in
n
TrA ρA ∝

 h:T φ:(`):T φ:(0)i
on the plane
hφ(`)φ(0)in
The same approach could be used also for negativity
F. Ravanini
EE in non-unitary CFT
Corner Transfer Matrix
CTM is a very useful tool [Baxter (1972)]
A¯s ,¯s 0 =
XY
wi
•
and analogously B, C , D with 90° rotations.
F. Ravanini
EE in non-unitary CFT
Partition function and CTM
Now we can build up the whole lattice by using the 4 CTM’s
Partition function
X
Z=
Aσ¯ σ¯ 0 Bσ¯ 0 σ¯ 00 Cσ¯ 00 σ¯ 000 Dσ¯ 000 σ¯ = Tr(ABCD)
σ
¯ ,¯
σ 0 ,¯
σ 00 ,¯
σ 000
F. Ravanini
EE in non-unitary CFT
Reduced density matrix and CTM
Now suppose to divide the spins in two subsystems A:
σ
¯A = (σ1 , ..., σp ) and B: σ
¯B = (σp+1 , ..., σL ), i.e. σ
¯ = (¯
σA , σ
¯B )
Reduced density matrix of subsystem A
X
ρA (¯
σA , σ
¯A0 ) =
h¯
σA , σ
¯B |0ih0|¯
σA0 , σ
¯B i = TrB h¯
σA |0ih0|¯
σA0 i
σ
¯B
ρA = (ABCD)σ¯ ,¯σ0
=⇒
F. Ravanini
Sn =
1
log TrA ρnA
1−n
EE in non-unitary CFT
EE in FB models
Continuum limit of ABF models on square lattice (RSOSm ). CTM
diagonalization is given and the calculation of ρA has been done
[Franchini, De Luca (2012)]
Can be generalized to FB non-unitary RSOSm,m0 models
c
a = 1, ..., m0 − 1 , d = 1, ..., m − 1 and t = T −T
Tc
0
m
−1
X
Zn =
a=1
4π 2
E (x a , y )n F (a, d ; x 2n ) ,
y = e log t ,
x =y
m 0 −m
m0
,
0
mod 2
b dm
m c
X
k(k−1)
E (x, y ) =
(−1)k y 2 x k
a=
F (a, d ; q) = q
(a−d )(a−d −1)
4
n∈Z
Renyi entropy
Sn =
1
1
n
log TrA ρnA =
log Zn −
log Z1
1−n
1−n
1−n
expanding for t → 0 with ξ ∼ t −ν , with ν =
Sn =
m0
4(m0 −m)
(n + 1)ceff
log ξ + ...
12n
F. Ravanini
EE in non-unitary CFT
c
q 24 −∆da χda (q)
Numerical results
Spin chain [von Gehlen (1994)]
L
H(λ, h) =
1X z
x
(σi + λσix σi+1
+ ihσix )
2
i=1
2
has a critical line in the (λ, h)-plane with c = − 22
5 (ceff = 5 ): Lee-Yang
universality class.
S=
ceff
3
log
L
π
sin
`π
L
+α
Numerically ceff = 0.4056 and α = 0.3952
F. Ravanini
EE in non-unitary CFT
Quantum critical hamiltonian
“The answer is yes, but... what was the question?” [W. Allen]: We
know the 2D classical lattice model, we can compute formally Sn ,
but what is the quantum Hamiltonian we are dealing with?
At criticality Uq (sl (2)) invariant XXZ model [Alcaraz, Barber, Batchelor
(1988) - Pasquier, Saleur (1990)]
#
"N−1
X
q − q −1 z
q + q −1 z z
z
x x
y y
σn σn+1 ) +
(σ1 − σN )
H = −J
(σn σn+1 + σn σn+1 +
2
2
n=1
Can be rewritten in terms of Temperley-Lieb operators
H = −J
N−1
X
en
n=1
en2 = −(q+q −1 )en
,
en en±1 en = en
F. Ravanini
,
en em = em en if |n−m| > 1
EE in non-unitary CFT
Quantum critical hamiltonian
“The answer is yes, but... what was the question?” [W. Allen]: We
know the 2D classical lattice model, we can compute formally Sn ,
but what is the quantum Hamiltonian we are dealing with?
At criticality Uq (sl (2)) invariant XXZ model [Alcaraz, Barber, Batchelor
(1988) - Pasquier, Saleur (1990)]
#
"N−1
X
q − q −1 z
q + q −1 z z
z
x x
y y
σn σn+1 ) +
(σ1 − σN )
H = −J
(σn σn+1 + σn σn+1 +
2
2
n=1
Can be rewritten in terms of Temperley-Lieb operators
H = −J
N−1
X
en
n=1
en2 = −(q+q −1 )en
,
en en±1 en = en
F. Ravanini
,
en em = em en if |n−m| > 1
EE in non-unitary CFT
Quantum off-critical hamiltonian
What happens off-criticality? Introduce the tile operators
(a = (a1 , a2 , ..., aN ))
Y
a0
δai ,ai0
1|a =
i
a0
ej |a
a0
gj |a
=
=



Y
s(aj0 λ)
 δa ,a0  δaj −1 ,aj +1 
,
s(u) = ϑ1 (u, t)
i i
s(aj+1 λ)
i6=j
"
!
#
Y
0
s 0 (a +1 λ)
s 0 (λ)
s 0 (0) s(a λ)
δai ,ai0 δaj −1 ,aj +1 (aj0 −aj +1 ) s(ajj+1
+ s(λ) − s(λ) s(a j λ)
λ)
j +1
i
Hamiltonian
H=−
N−1
X s 0 (0)
d
s 0 (λ)
log T(u)|u=0 = −J
ej −
1 + gj
du
s(λ)
s(λ)
j=1
Limit t → 0: gj → 0 while ej →TL-algebra
In general, algbera with two parameters (=⇒ elliptic algebras?)
F. Ravanini
EE in non-unitary CFT
Summary
Von Neumann and Rényi E-Entropies are crucial tools to study
entanglement in quantum systems. In integrable models, they can
be calculated using integrable techniques.
Corner Transfer Matrix technique allows the exact calculation of
bipartite E-Entropy in spin chains. Having the exact formula at
hand, one can test some of the open issues about entanglement in
these models.
In the case of non-unitary theories, the coefficient of the logarithmic
divergence near criticality gives ceff instead of c. Although this
result is not surprising, it sheds more light on the general way to
compute finite interval density matrices in generic CFT’s.
An integrable way to compute finite size E-Entropy is to be
developed. It would complement the present knowledge by new
precious information.
F. Ravanini
EE in non-unitary CFT
Conclusions
Entanglement entropy is a new way to approach interesting problems
in theoretical physics and it should be better understood in
(integrable) QFT, as it seems crucial in the solution of challenging
paradoxes, like the information loss in black holes.
It also stimulates progresses in mathematics, in the best tradition of
the integrability approach.
Thank you!!!
F. Ravanini
EE in non-unitary CFT
Conclusions
Entanglement entropy is a new way to approach interesting problems
in theoretical physics and it should be better understood in
(integrable) QFT, as it seems crucial in the solution of challenging
paradoxes, like the information loss in black holes.
It also stimulates progresses in mathematics, in the best tradition of
the integrability approach.
Thank you!!!
F. Ravanini
EE in non-unitary CFT

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