Entanglement Entropy in Three-Dimensional Higher Spin Theories · formulation of the higher spin...

Entanglement Entropy in Three-Dimensional

Higher Spin Theories

Juan Jottar

Based on:

arXiv:1302.0816,1306.4347with Jan de Boer

+

to appearwith G. Compère and W. Song

Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 1

Outline

1 Motivation: higher spin theories in AdS3

2 Entanglement entropy proposal

3 Testing the proposal

4 Outlook


Motivation: higher spin theories in AdS3

Outline




4 Outlook



Higher spin theories

Broad definition: interacting theories of gravity coupled to a finite (orinfinite) number of massless fields of spin s > 2 .

Motivation from holography: explore AdS/CFT in a regime where thebulk theory is not just classical (super-)gravity, and the dual theory isnot necessarily strongly-coupled:

◮ Critical O(N) vector models in 3d in the large-N limit dual to higherspin theories of Fradkin-Vasiliev type in AdS4. (Klebanov, Polyakov2002; Giombi, Yin 2010-12; Maldacena, Zhiboedov 2011-12)

◮ Two-dimensional minimal model CFTs with extended (W -)symmetriesin the large-N limit dual to higher spin theories in AdS3. (Gaberdiel,Gopakumar 2010)

Motivation from GR: curvature, causal structure, etc are not invariantunder the higher spin gauge symmetries ⇒ challenge traditionalgeometric notions.



AdS3 gravity as a Chern-Simons theory

Standard AdS3 gravity can be written as an SL(2,R)× SL(2,R)Chern-Simons theory. (Achucarro, Townsend 1986; Witten 1988)

Take 3d gravity with a negative cosmological constant Λ = −1/ℓ2 .Combine dreibein ea and (dual) spin connection ωa = (1/2!)ǫabcωbc

into SL(2,R) connections

A = AaJa = ω +e

ℓ, A = AaJa = ω −

e

ℓ

where the Ja satisfy the so(2, 1) ≃ sl(2,R) algebra [Ja, Jb] = ǫ cab Jc .

Defining CS(A) = A ∧ dA+ 2

3A ∧ A ∧ A one finds

ICS ≡k

4π

∫

M

Tr[

CS(A)− CS(A)]

=1

16πG3

[∫

M

d3x√

|g |

(

R+2

ℓ2

)

−

∫

∂Mωa ∧ ea

]

k = ℓ4G3



Higher spin theories in AdS3

The higher spin theories in 3d are constructed by generalizing thegauge algebra.

The pure higher spin sector of the 3d Vasiliev theory is hs[λ]⊕ hs [λ]Chern-Simons theory.

The dual CFT has an infinite tower of conserved currents and W∞[λ]symmetry. (Henneaux, Rey 2010; Gaberdiel, Hartman 2011)

For λ = N ∈ Z it is possible to truncate the tower of higher spin fieldsto s ≤ N . The bulk theory reduces to sl(N,R)⊕ sl(N,R)Chern-Simons. Different sl(2) embeddings are possible.

The asymptotic symmetry algebra is of WN type. (Campoleoni,

Fredenhagen, Pfenninger, Theisen 2010)



Extending the AdS3/CFT2 dictionary

The BTZ black hole entropy (via Bekenstein-Hawking) andholographic entanglement entropy (via Ryu-Takayanagi) matchuniversal CFT results:

Cardy entropy formula

Sthermal = 2π

√

c

6L0 + 2π

√

c

6L0

(Single interval) Entanglement entropy at finite temperature T = β−1

SA =c

3log

(

β

πasinh

(

π∆x

β

))

where c is the central charge and a the UV cutoff.

Question: How do we compute these entropies in higher spin theories?


Entanglement entropy proposal

Outline




4 Outlook



Entanglement entropy

Consider a quantum system in a pure (or mixed) state, with densityoperator ρ = |Ψ〉〈Ψ| (or ρ = e−βH).

Partition the Hilbert space as H = HA ⊗HB (B = Ac), the reduceddensity matrix for subsystem A is defined as ρA = TrBρ.

The entanglement entropy SA associated with A is then given by theVon Neumann entropy of ρA: SA = −TrA ρA log ρA.

Universal results in 2d CFTs. (Holzhey, Larsen, Wilczek 1994; Calabrese,

Cardy, 2004) If A is a single interval of length ∆x in an 1d system:◮ System on the ∞ line at zero temperature: SA ≃ (c/3) log (∆x/a)

◮ Periodic system of total length L: SA ≃ (c/3) log(

Lπa

sin (π∆x/L))

◮ System on the ∞ line in a thermal (mixed) state:

SA ≃ (c/3) log(

βπa

sinh (π∆x/β))



Holographic entanglement entropy

Generically, entanglement entropy is notoriously difficult to compute infield theory, even for free theories.

However, in theories with a gravity dual it can be computed using theRyu-Takayanagi prescription:

t = const.

A

r

bulk

x

y

SA =

Area(γA)4GN



Entanglement entropy for higher spin theories

What replaces the R-T prescription in a 3d higher spin theory?. Writesomething that:

◮ Involves the natural objects in the h.s. theory: Wilson lines.

◮ Is invariant under the diagonal gauge group (rotations of the localframe), and under gauge transformations that do not change the statein the dual theory.

◮ Encodes the geodesic length in the pure gravity case.

One is lead to consider

W (P ,Q) ≡ TrR

[

P exp

(∫ P

Q

A

)

P exp

(∫ Q

P

A

)]



Entanglement entropy for higher spin theories

For AAdS3 solutions of pure gravity, placing P and Q on theboundary, and at equal times, we find

W2d (P ,Q)|ρP=ρQ=ρ0−−−−→ρ0→∞

exp d(P ,Q)

where d(P ,Q) is the geodesic distance.

Since ent ent in AdS3 is related to the geodesic distance via theRyu-Takayanagi prescription, we propose

Sent = kcs log TrR

[

P exp

(∫ P

Q

A

)

P exp

(∫ Q

P

A

)]∣

∣

∣

∣

ρP=ρQ=ρ0→∞

This is a well-defined object in the higher spin theory as well. In theabsence of field-theoretical results to compare against, we will performsome plausibility tests.


Testing the proposal

Outline




4 Outlook



Test 1: Recover AdS3 results

For solutions of pure gravity we obtain

Poincaré-patch: SPAdS3=

c

3log

[

∆x

a

]

global: SAdS3=

c

3log

[

ℓ

asin

(

∆ϕ

2

)]

black hole: SBTZ =c

6log

[

β+β−

π2a2sinh

(

π∆x

β+

)

sinh

(

π∆x

β−

)]

in agreement with CFT results and holographic calculations.(Ryu,Takayanagi, 2006; Hubeny, Rangamani, Takayanagi, 2007)

The higher spin theory with diagonally-embedded sl(2) contains atruncation to Einstein gravity plus Abelian Chern-Simons fields. Ourprescription also gives the right result for black holes carrying U(1)charges, as recently confirmed by an independent CFT calculation.(Caputa, Mandal, Sinha 2013)



Test 2: Thermal entropy

The entanglement entropy should approach the thermal entropy in thelimit ∆x ≫ β (i.e. as subsystem A approaches the full system). E.g.

SA =c

3log

(

β

πasinh

(

π∆x

β

))

−−−−→∆x≫β

πc

3β∆x = sthermal ∆x

Since the causal structure is not invariant under the higher spin gaugesymmetries, we cannot calculate the entropy with the usual methods.

Proceed indirectly by demanding existence of a well-defined partitionfunction. E.g. N = 3:

Z (τ, α) = Tr[

e2πi(τ L+αW )]

= eS+2πi(τL+αW)

with the integrability condition

∂L

∂α=

∂W

∂τand τ =

i

2π

∂S

∂L, α =

i

2π

∂S

∂W



Gutperle and Kraus showed that integrability follows from demandingsmoothness (connection has trivial holonomy around contractible cycleof the Euclidean torus) and obtained:

S = 4π

√

c

6L

√

1 −3

4C

with C = C (W/L3/2) .

Using the canonical formalism for charges in GR it was found instead(Perez, Tempo, Troncoso 2013)

Scan = 4π

√

c

6L

(

1 −3

2C

)−1√

1 −3

4C

which agrees with a perturbative result obtained using the metricformulation of the higher spin theory and Wald’s entropy formula.(Campoleoni et. al. 2012)



Thermodynamics from the on-shell action

From the point of view of the action, the two formulations follow byadding different boundary terms, and in general one finds (de Boer,

J.I.J. 2013)

Sholo = − 2πikcs Tr[

Az (τAz + τAz)− barred]

Scan = − 2πikcsTr[

(Az + Az) (τAz + τAz)− barred]

The holonomy conditions imply (τAz + τAz) = u−1(iL0)u . Since forconstant connections Az and Az commute by the e.o.m.:

Sholo = 2πkcs TrN

[

λzL0 − barred]

Scan = 2πkcsTrN

[

λϕL0 − barred]

where λj is a diagonal matrix containing the eigenvalues of Aj .The entanglement functional involves the spectrum of Aϕ, so it favorsthe canonical expression.



Holomorphic vs. canonical

The canonical expression for the entropy is also obtained byconsidering a generalized version of the conical singularity method.(Ammon, Castro, Iqbal 2013)

The holomorphic formalism has been well understood from the pointof view of the dual CFT: the black hole partition function correspondsto the theory deformed perturbatively by

∫

d2x µW . (Ammon et. al.

2011, Kraus, Perlmutter 2011; Gaberdiel, Hartman, Jin 2012)

On the other hand, the CFT interpretation of the canonical formalismhas proven elusive.

The key to remedy this is to analyze the different notions of conservedcharges at play.



Conserved charges at finite chemical potential

In the holomorphic formulation the charges (expectation values) areobtained from traces of Az , while the chemical potentials (sources) areincorporated by turning on the Az component. The energy (mass) ofthe higher spin black hole is still L ∼ Tr

[

A2z

]

, as for BTZ.

On the other hand, the canonical charges in the presence of thedeformation are (Bañados, Canto, Theisen 2012; Compère, Song 2013)

L = L+ 3µW +16

3kµ

2L

2∼ Tr

[

A2ϕ

]

W = W +32L2µ

3k+

16LWµ2

k+ . . . ∼ Tr

[

A3ϕ

]

One can show that the canonical entropy, as a function of the tildedcharges, has the same form as the holomorphic entropy. The same istrue about the partition function as a function of the tilded sources τ ,α which are conjugate to the canonical charges. (Compère, J.I.J, Song,

to appear)



Selecting the representation

In the principal embedding we can rewrite our thermal entropy formulaas

sthermal = kcsTrN

[

(

λx − λx

)

L0

]

= kcs〈~λx −~λx , ~ρ〉

where ~ρ is the Weyl vector of sl(N) .

In the principal embedding, the representation R that gives the rightthermal limit ∆x ≫ β of the entanglement functional is then the onewith highest weight Λ = ~ρ . The dimension of this rep isdim(R) = 2N(N−1)/2 .

With this criterion one can determine the representation in otherembeddings as well.



Test 3: Strong subadditivity

An important property of entanglement entropy is that it is stronglysubadditive

S(A) + S(B) ≥ S(A ∪ B) + S(A ∩ B)

S(A) + S(B) ≥ S(A ∩ Bc) + S(B ∩ Ac)

In one spatial dimension, these inequalities imply that thesingle-interval entanglement entropy is a concave and non-decreasingfunction of the interval length.

We have verified these properties numerically in different examples,including the spin-3 black hole. Caveat: Short distance singularities atthe scale ∆x ≃ µ suggest a redefinition of the cutoff is needed due tothe effects of the irrelevant perturbation.



Example: entanglement at finite β and finite spin-3 charge

Result depends on the dimensionless ratios (∆x/β) and (µ/β):

SA =c

3log

[

β

π aF

(

∆x

β,µ

β

)]

When the higher spin deformation is switched off (µ = W = 0)

SA =c

3log

[

β

π asinh

(

π∆x

β

)]

1 2 3 4 5 60

2

4

6

8

Π Dx

Β

LogHFL


Outlook

Outline




4 Outlook


Outlook

Summary

We provided a holographic "guess" for the entanglement entropy in2d CFTs with higher spin gravity duals, in particular in the presence ofdeformations by higher spin currents (irrelevant operators).

The proposal passes several checks, but seems to breakdown fordistances on the scale of the higher spin chemical potentials.

We expect to clarify these issues via a constructive proof based on thereplica trick and Rényi entropies:

SA = limn→1

S(n)A = − lim

n→1

1

n − 1ln Tr [ρnA]

The Ryu-Takayanagi prescription in AdS3/CFT2 has been recentlyproven in this way (Faulkner 2013; Hartman 2013), and extended beyondthe semiclassical limit. (Barrella et. al. 2013)


Outlook

Proving the higher spin entanglement proposal (in progress)

One way to attack the problem is by computing

Tr [ρnA] =Zn

Z1

where Zn is the partition function on an n-sheeted cover of the originalmanifold obtained by cutting along subsystem A and cyclically gluingn copies (replicas). In the present case this amounts to evaluating theChern-Simons action for bulk solutions with the topology of thebranched cover at the boundary.

Alternatively, perform a direct CFT calculation using twist fields:

Tr [ρnA] =⟨

σn(u1)σn(v1) . . . σn(uN)σn(vN)e∫d2x µW

⟩

C or T 2


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