Quantum entanglement and local excitations
Paweł Caputa

Quantum Matter, Spacetime and Information YKIS 2016, YITP 15/06/2016

Based on“work in progress” with Marek M. Rams (Jagiellonian U.) …’16
1602.06542 [hep-th] with Tokiro Numasawa (YITP) and Alvaro Veliz-Osorio (WITS) ’16

previous works:[PC, A. Veliz-Osorio’15][PC, J.Simon, A.Stikonas, T.Takayanagi, K.Watanabe ’15]
[PC, J.Simon, A.Stikonas, T.Takayanagi ’14]
[PC, M. Nozaki, T. Takayanagi ’14]

Outline•Introduction (already in Nozaki’s talk)•Evolution of entanglement after local operator excitation in 2d CFTs

•

RCFTs

•

Numerics: Ising model•

Large c limit•

SU(N) k WZW•Conclusions

Introduction•

Quantum Information gives a new perspective on quantum field theories•

New Tools in QFT: measures of entanglement, distance between states,complexity, OTO correlators etc.

•2d CFT is a perfect playground for “defining”/exploring these tools•

Advantage: Symmetry (computation), numerics (critical points), RT (HRT)•We can scan and classify 2d CFTs by the properties of these tools in variouscorners of parameter space. (Ideally holographic vs non-holographic CFTs).

Universality in CFT 1+1[Holzhey,Larsen,Wilczek’94][Calabrese, Cardy’04][Calabrese, Cardy’16]

Ground statec|A|
SA = log+ c̃n3
aQuenchesct
SA ⇠ log
3✏

SA ⇠ c t

SA ⇠ c ⇥ const

Useful to extract the central charge (numerics)!
RT (HRT) results confirm the large c behaviour!

Can we be less universal but still under control?
Local Operator Excitations!

CFT in 1+1 d[Nozaki,Numasawa,Takayanagi’13]ohl“Local quench” setupA⇢(t) = e iHt e ✏H O( l)|0ih0|O† ( l)e ✏H eiHt
How does this change (Renyi) entanglement entropies of A?

⇢A (t) = T rAc [⇢(t)](n)
SA (t)?

(n)
SA =

1
1

hO(w1 , w̄1 )O† (w2 , w̄2 )...O(w2n 1 , w̄2n 1 )O† (w2n , w̄2n )i⌃nlogn†
n(hO(w1 , w̄1 )O (w2 , w̄2 )i⌃1 )•Q1: How much data about a CFT can we extract numerically ?

•Q2: Large c vs “holographic” behaviour?

This talk: Some modest progress in these directions

of operator and CFT as we will study below.

RCFTs

where Sab
we conside[He,Numasawa,Takayanagi,Watanabe’13]
result for(n)!SSAA!2"
1.0
0.8
0.6log dO

0.4
0.2

0.0

o0.5

1.0

1.5

2.0

2.5

3.0

t

when l <For exa
imal modeerator [8],the expres
for l < t <h(2)FIG. 2. The time evolution of ∆SA
1, L = 1.

A

Quasi-particle(2)
for O2 . We setl =∆S =AIn general CFTs, the function G(z, z̄) can be expressed[P.C, Veliz-Osorio’15],[Chen,Guo,He,Wu’15]
for any using
membera conformalfamily!
theofconformal
blocks[8]:
2+1 d:[Fradkin,Dong,Leigh,Nowling'08]REGa (z, z̄) =!

b 2(Caa ) Fa (b|z)F̄a (b|z̄),(16)

Can we see this numerically at the critical point?
bThe n-tIsing Model

H=X⇥

J

critical for⇤zx x
i i+1 + h i

i

1, ",

CFT operators (families)(n)
SA = log dO

g = h/J = ±1

d1 = d" = 1,d =p2

lattice operators(i) ⇠xi"(i) ⇠x x
i i+1ziJordan-Wigner map to free fermions and we can compute the reduced
density matrix for our locally excited states (+ checks with MPS)
[Verstraete….Vidal…]

Second Renyix[PC,M.M. Rams’16…..]lmatlab, log2

g=0.9

1

0.95
1S (2)

1.05
1.1

0.5

020

l

40

60

80

100timealso for " and@

n=1
Hamiltoniana
X is diagonalized1†
H=✏k k k, ✓X
2 condwhere we assume periodic
boundarykH=✏k~ N +1 . The model is critical for h = ±1k andwhere k are fermionic annihilation operators
otherwise,in ground
this article
wefocus
on h= 1.
hilates thestateofH
andthedispers
where k are fermionic annihil
Themodel canhilates
be mapped
intostate
the offreereads
thegroundH⇥ log 2⇥ log 2temand oreadsqtransformation(c)(b) by Jordan-Wigner✏k = 2 in(hthecosk)2 +qsin2 k. sefree-fermion
1 results are performed
12✏k =After
2 (h thecotails,
wereferto
theAppendix.
The dispersion relation is non-linear due toHamiltonianis diagonalizedasnatureofthesystem,
which
results
in velociThe
dispersion
relation
is non1
124✓ system,particles dependingon
momentumk. whichIn◆ordenatureoftherX
1†particles
on momeguish the resultsof that, depending
simultaneouslyH=✏we, ckk k
2 wenasguish
theresultsofthat,modelwithlinearized
dispersion
relation,k00
model withdispersi✓◆
X linearized1† operatorlin
linannihilation
1 where k are fermionicH =✏k 1 k k X ,2
lin the disper
2 ✏linhilates the ground state of
and=k HHk

CFT data:⇥ log 2(a)

S (2)

1
2

140S (1)

12

reads

140

140

128

256
t · vF

384klinwhere ✏k = vF k, vF = 2 is the1velocity of qulin2
where✏=vk,v = 2 is thFwith k ! 0 at the criticalpoint,
k2 Fand 2k arq✏k =with2 (hsin k.p
k ! cos0 atk)
the+criticalannihilation operators as in the Ising model.
annihilation
operators
as inthThe lattice relation
operators
thathaveand✏fie
0 The dispersion
0is non-lineardue
512 00.5
11.5 lattice
2 operators
00.5 tohThethatleading contributions in the continuum are knature of thewhich
results inveloct · vsystem,
F /N leading
contributionsin thec(n) momentum= nx, k. In xord
particles depending on(n)z = nx x
guish the results of "(n)that,=wen simultaneouslyn+1n.
"(n) = nx nx
modelWe
withlinearized dispersion relation, n
excite the ground state of critical Ising
We excite the✓ ground state◆S (2)

1| i = Oi ...Ok |0i(n)SA =kX
i=1log dOS (2)General excitations (quasi-particlephenomenology)
0.5
03(b)
2.5
21.5
10.500512[CFT 2 operators: Numasawa ’to appear]

1,024
t · vF

1,536

1t · vF /N

1.52,048⇥ log 2i=1xi |0ilarge c: [Anous, Hartman, Rovai, Sonner ‘16]

128

S (2)

| Gi =NY

2L log d64

000.5

2

ath. Theor. 42 (2009) 504005

P Calabrese and J Cardy

Second Renyi entropy (Purity)

ath. Theor. 42 (2009) 504005(n=2 and finite interval A,primary)
P Calabrese and J Cardyuv

Az(w) =uv

Arww

uv

Figure 2. A representation of the Riemann surface R3,1 . Reprinted with permission from [12].

gth L behaves as(2)SA (z, z̄) =⇥log |z(1z)|4h G(z, z̄)⇤Figure 2. A representationof the−2−1Riemann surface R3,1 . Reprinted with permission from [12].

log Z = f1 Aa+ f2 La+ ···(10)z12 z34
z=z13 z24nd f 2 are the non-universal bulk and boundary-free energies. Note, however, thatG(z,z̄)z̄)O(0)ing terms
cancelin⌘thehO(1)O(1)O(z,ratio of partition functionsin (8).
gth L behaves as
onformal field theory, as was argued by Cardy and Peschel [14], there are also
Z = f1 Aato−2log+ fa.2 LThesea −1 +arise· · · from the points of the non-zero curvature(10)
ermslog
proportionalofld
In our case,are conical singularities
the branchpoints.
ndandf 2 its
areboundary.
the non-universal
bulktheseand boundary-freeenergies. atNote,however,that
constantcontribution:
sngprecisely
these inlogarithmic
terms
whichfunctions
give riseinto(8).
the non-trivial dependence ofterms cancel
the ratio ofpartition/letAusare(z, z̄) ! (0, 0)
sult for thefield
entropyon as
thewasshort-distancethe moment
simplyonformaltheory,argued by cut-off
Cardy a.
andForPeschel[14],t 2
there
also
n orderto achievetoalog
finitelimit asa →thepointsright-handof (8)curvature
should beerms
proportionala. These
arise
from0,theof the side
non-zerooft2
A points.
(z, z̄) ! (1, 0)
Z (A,byandsomeld
its renormalization
boundary. In ourconstantcase, thesearen).
conical singularities at thebranch
precisely these logarithmic terms which give rise to the non-trivial dependence ofsult for theworld-sheet
entropy on tothereplicatedshort-distancecut-off a.twist
Forfields
the moment let us simply
replicatedtarget-space:✏!0Rational CFT (diagonal)
In rational CFTs

G(z, z̄) =Htot =

Xa

F(a|z)F̄(a|z̄)

aHa ⌦ H̄afinite

bd [O] are constants, which are called fusion matrices [27].

Thus by1 z̄ !to0 the contribution
to extract
constant when(z,inz̄)order! (1,0), thetheconformal
block zis!reducedF(a|1z) =X

Fab [O]F(b|z) !

G(z, z̄) ' F00 [O]
b · (1z)

2Oz̄

20fact Finallythat COO= 1.
e following expression from (33):(2)
SA =log F00 [O] = log dO ,O,G(z, z̄) ' F00 [O](1z) 2h z̄ 2h(69)quantum dimension

S0adim (Ha )da ==(70)S00
dim (H0 )called the quantum dimension [27] and is related to the S-matrix ofInformation about the modular S-matrix of a CFT!
tion byLarge c[PC,Nozaki,Takayanagi’14]

General arguments:•conformal blocks exponentiate (factorisation)(2)
SA ' 4[Zamolodchikov]

2t
O · log✏•

HRT in back-reacted geometry from a massive particle•

Heavy operator at large c CFT[Asplund,Bernamonti,Galli,Hartman’14]

ct(1)
SA ⇠ log
6•

Quasi-particle picture breaks-down✏[Asplund,Bernamonti,Galli,Hartman’15]

Origin of log(t) ?

nstant contribution
atedSU(N)to thek WZW
singularLet us compute the purity first. In order to exlate time value, we apply [P.C,Numasawa,Veliz-Osorio’16]
the fusion transforma
mixes conformal blocks
X rep.
XState excited by the operator in the fundamental(m)(n)2h¯G(1 z, z̄) =Ii Ij
Xnn cnm F3 i (z)Fj (z̄)(13)↵2n,mN
1h = h̄ =2N (k + N )i,jg (au-l) |0i! (1, 0). The
where the relevant coefficient israphic CFTs, where4-point correlator from K-Z equations
valid, such singu(N/) ( N/)
11c=
N
=
[N]
=d,
211gX
Xg of entanglement”. ¯(n)(n)(1/)(1/)G(z, z̄) =Ii Ij
Xnn Fi (z)Fj (z̄)
affine blocks
ance of a the (nonni,jwithdg being the quantum dimension for the fu
t late times is retal representation, where the quantum number
rowthoftheRényi
Now take: ✏ ! 0(z, z̄) ! (1, 0)
fined asis behavior emerges
x/2
x/2
WZW models.
2⇡iqq(2)N +k .
[x] = 1/2q=e
SA =canlog[N ]log dO increaseqq 1/2
tanglement entropy
onic excitation [22].
Taking the limit of the conformal blocks for
N <-> k! leaves us with the log of thetheExcitations
OTO arerespect
ratherlevel-rank(1, 0)duality(see App.A)

N=,Large c in WZW k(24)[P.C,Numasawa,Veliz-Osorio’16]

an define a ’t Hooft limit of large central charge with
2 four-point corre- N
Parameters
coupling
fixed (weak or strong).
Thek(N1)=c = in this limit by [25] andr has been analyzed in detailk+Nk
apply their analysis in our context. For c ! 1, theint correlator(Note that here, unlikeCorrelator becomes
becomes(27)(h~1/2)3], all our operators are light: h/c ! 0ras c✓! 1).
◆
1can see that
111ng this correlator,
oneforalargecenG(z, z̄) ' 2 +++2|1 z| get “scrambled”,
c z(1 z̄) (1 z)z̄charge the important |z|
singularities
ch leads to a logarithmic growth of the purityr
✓ ◆2t
c(2)
(2)
SA (t) ' 2h log
log(2).
(25)
S' logA✏

behavior comes from discarding terms proportional
Time-scaleif atthesuch
log ofquantum dimension
p1 . However,
wewhichincludecorrections,then the is reachedctime answer becomes the logarithm of the1/4quantumcension in the large-c limit. It tis'illustrativel + 1/4to✏ verify
in the strong coupling regime, where 2h = 1/2 andcan be computed by approximating the operators
Pk1↵, z̄ ) '(z ) ¯ (z̄ ) with complex fermions.

N=,Large c in WZW k(24)[P.C,Numasawa,Veliz-Osorio’16]

an define a ’t Hooft limit of large central charge with2 four-point corre- NParameterscoupling
fixed (weak or strong).
Thek(N1)=c = in this limit by [25] andr has been analyzed in detailk+Nk
apply their analysis in our context. For c ! 1, theint correlator(Note that here, unlike
Correlator becomes
becomes(27)(h~1/2)3], all our operators are light: h/c ! 0ras c✓! 1).
◆1can see that1
1
1ng this correlator,
oneforalargecenG(z, z̄) ' 2 +++2|1 z| get “scrambled”,
c z(1 z̄) (1 z)z̄charge the important |z|
singularitiesch leads to a logarithmic growth of the purityr✓ ◆2tc(2)(2)
SA (t) ' 2h log
log(2).
(25)
S' logA✏

behavior comes from discarding terms proportional
Time-scaleif atthesuchlog ofquantum dimension
p1 . However,
wewhichincludecorrections,then the is reachedctime answer becomes the logarithm of the1/4quantumcension in the large-c limit. It tis'illustrativel + 1/4to✏ verify
in the strong coupling regime, where 2h = 1/2 andcan be computed by approximating the operators
but no chaos!
Pk↵, z̄ ) ' 1(z ) ¯ (z̄ ) with complex fermions.

Conclusions/Future:•

Entanglement measures in field theory are very useful forcharacterising local operator excitations!

•Using QI tools we can extract more CFT data from the critical points(full S and modular T matrix?)•

How to see “holography” with QI tools (log(t)<->? )•Numerics for other Hamiltonians (Potts?)•General excitations at large c ?

•Measures of chaos and relative entropy ?

!

Recently, Kitaev proposed that chaoticingbehaviorformation
the theoryis thf
this monodromy
frompabout
the explicit
form ofntum systems can be diagnosed by computing
theblockFor
we first
fillbutterfly
the existi
conformal[17]that,
one cansee thee↵tation value of the square of commutators
local
late
time
OTO
c
Atofearly
times,
sincezvalue⇡ 0 ofandz̄ ⇡0, th[P.C,Numasawa,Veliz-Osorio’16]
CFT.
In RCFTs
themonodromy
ofthe
conformal
in essentially
RCFT amounts to calculating[Gu,Xi’16]
tors OTO[2]. ThisRCFT.
a ntionfromidentity
channel(pconsider= 0) domi
givenbythe
athefinite
matrixNext,
and wewehave
PRL
115,131603(20-time order (OTO) thermal correlator f (z, z̄) ⇡ 1. AtCFT,theSU(N)
WZW
modlatetimes,0X oncekagain z ⇡ii
iiput in the nknown
results
canbeF(p|z)!
MFDE
jj goes by, thepqcross-ratiojj (q|z) .
However,
astimezate ordering of epsilons✏(seethefigures)aswei ††
scrambling
qand measures of quO(t)OO(t)Oijij✏4e t the cross-ratio z = (z12 z34 )/(z13 z24 ) encircles
non-trivial ✏contour
com
4
O around z = 1 in theE D plane
E andconsider a large-c ’t Hooftlim
G(z, z̄)(1) ✏2 O weCij (t)⌘1 Din the complex. =comes
ise the pointz=✏(0)
2(thisis notcase
for aroundz̄). Asz =shownin††Becausecrosstheratioz goes
1 and fina
OOOOturesO with holographic CFTs
anj role of thei iwith z̄).j Theo 0 (this doesn’t happen✏✏3
3
time
evolutionbackz = 0, the fromonly therelevant
component
ing
thistomonodromy
explicitform oofpurityandOTO
correlators
ature in this specific behavior of z is not crucial✏1 O (0)✏1Therefore,block
we obtain
conformal[17]
onecansee
thebutterfly
serve that, in the large-c limit,
isonly used
to extract
the universal
quantity
vanishes
exponentiallyatpredictions
late times thenCFT.
In RCFTs
themonodromyof conformxquantumsystemchaos. Moreprecisely,
inchaotic
CFTs,ii
ii a cons
while
theOTO
approachuantumis X
chaotic.
Anumberof
universal
iiiilimG(z,z̄)=MF(0|z)F̄00 jj
jj (0|z̄).
G(z,z̄)=F(p|z)F̄(p|z̄)
givenby In
a t!1
finitematrixofand
we have
orrelators
areobject
expected
damp
afterjjthe
jjobtained
absencequantumchaos.
rtiesof thiscantobe
forso-called
2dCFTs.
plingtime[3].
In
contrast,
forchaotic
RCFTs,behavior
which aremightXThis
letteris organized
as folinFIG.
2(coloronline).
Tular, its been argued that
FIG.
The
orbitofchiralpartof ii
operators
Moreover,for1. RCFTs
thismonodromy
matrixii
ble systems,
one expectsCij (t) to reachconstantF(p|z)!
MF(q|z).
pqjjjjpute
thelatetimevalueofOTO
ellingcharacteristicof holographicCFTs[3–6].
can be expressed thein termsof the modular
S-mcontinuations
corresp
Indeed, as we shall see, they are given byt I
q theory (TQFT).
quantum field
the other hand, one of the characteristic [18]:features ofhWWVVi,andhWVVW⇤lationship between purity and eat large central charge
1 isSija so-called scrambling of⇤Because
cross ratiozgoesaroundz = 1pointand fij(3)Cij (t) !
SaroundthebranchSSij 00 00
In Sec.
4,illustrate
bothquantiO
glement [12]. One particulardi dj S00 incarnation of scramM=.
00only relevant compone
back to z = 0,
theSTQFT
model. Finally,
Sec. 5 studs the evolution⇤ of Rényi entanglement entropies af0iinSpicture0j
O (t)S003D
times, where Sij is the complex conjugateTherefore,of theO
weobtainquantities
in the ’t Hooft limit.
al operator excitation and here we will focus on theO(t)ar S-matrix. The argument proceeds as follows,
fz; z̄ithis
g. inlate
Theinteres
place
twoappendices.
idetails
de writeRényi entropy or simply the purity. Various
stud-we also
Below,
derivevalue
iitimeii ofG(z,z̄) = M00 Fjj(0|z)F̄jj (0|z̄“Monodromy scalar”lim
anyon
interferometry
experiments
DE purity2.
Late
timeof OTO
insy
Rt!1[19].
owed
that,
for
rational
CFTs(RCFTs),satuing
3dTQFTAs
timepasses,theoperatofðz;z̄Þ.
Bycrossing††
Shtengel,section
Slingerland’06]
O(z,z̄)O(z,z̄)O(z3 , logarithmz̄3 )Oj (z4 , z̄4of) the quantum [Bonderson,11i
22In the present
we compu
o aiconstant equal tojtheontheEuclidean
secti
Moreover,for
RCFTs
thismonodromymat
4h4htheOTO
correlators(1)
withinijsion of the local operators
conformal
family [7–9].
FIG. 2. 3d TQFT counterpart. Here, 0 denoji†j†ij†ji†i