Entanglement entropy and TT deformation

William Donnelly and Vasudev Shyam

Perimeter Institute for Theoretical Physics, 31 Caroline St. N, N2L 2Y5, Waterloo ON, Canada

E-mail: wdonnelly@perimeterinstitute.ca, vshyam@perimeterinstitute.ca

Abstract: Quantum gravity in a finite region of spacetime is conjectured to be dual to a

conformal field theory deformed by the irrelevant operator TT . We test this conjecture with

entanglement entropy, which is sensitive to ultraviolet physics on the boundary while also

probing the bulk geometry. We find that the entanglement entropy for an entangling surface

consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-

Takayanagi formula applied to a finite region consistent with the conjecture of McGough,

Mezei and Verlinde. We also consider a one-parameter family of conical entropies, which

are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local,

we conclude that the TT deformation acts as an ultraviolet cutoff on the entanglement

entropy. Our results support the conjecture that the TT -deformed CFT is the holographic

dual of a finite region of spacetime.

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Contents

1 Introduction 1

2 Sphere partition function and entanglement entropy 2

3 Conical entropy 5

4 Discussion 9

1 Introduction

The TT deformation of two dimensional conformal field theories (CFT) provides an ex-

actly solvable model of quantum field theory with an ultraviolet (UV) cutoff [1]. TT is a

composite operator satisfying the factorization property

〈TT 〉 =1

8

(〈T ab〉〈Tab〉 − 〈T aa 〉2

). (1.1)

Any CFT can be deformed by this operator, defining a one-parameter family of theories

labelled by a deformation parameter µ with dimensions of length squared. Here we take

µ to be positive. Eq. (1.1) is reminiscent of the factorization of multi-trace operators into

products of single-trace operators in large-N conformal field theories. The factorization

property determines many properties of the deformed theory, including the spectrum [1]

for which µ acts as a UV cutoff. We will work in the limit of large central charge, under the

assumption that 〈Tab〉 can be treated as a classical field, and Eq. (1.1) holds at all length

scales.

The factorization property (1.1) determines the partition function of the TT -deformed

theory on an arbitrary curved background. The response of the partition function to the

change of background metric is given by the stress tensor

δ logZ = −1

2

∫d2x√g 〈T ab〉δgab. (1.2)

The deformed theory is defined through the flow equation:

〈T aa 〉 = − c

24πR− µ

4

(〈T ab〉〈Tab〉 − 〈T aa 〉2

)(1.3)

which reduces to the CFT trace anomaly in the limit µ → 0. The flow equation together

with stress tensor conservation

∇a〈T ab〉 = 0 (1.4)

– 1 –

define a system of equations for 〈Tab〉 that are in some cases sufficient to determine it

completely. In what follows we drop the angular brackets, and simply use Tab to refer to

the expectation value.

Ref. [2] argued that for holographic CFTs, the TT deformation corresponds to in-

troducing a finite boundary in the bulk acting as an infrared cutoff. This correspon-

dence can be most easily seen by identifying the momentum conjugate to the metric

πab =√g(T ab− 2

µgab), under which the flow equation and stress tensor conservation become

the ADM (Arnowitt-Deser-Misner [3]) Hamiltonian constraint and momentum constraint,

respectively [2, 4]:

8πG√g

(πabπab − (πaa)2

)+

√g

8πG

(R+

2

`2

)= 0, (1.5)

∇aπab = 0. (1.6)

This fixes the identification between field theory and gravity quantities as:

c =3`

2G, µ = 16πG`. (1.7)

The former is the famous Brown-Henneaux central charge [5], while the latter is the iden-

tification of µ proposed in [2].1 This proposal passes a number of consistency checks [6].

In this letter we consider entanglement entropy of TT deformed field theory. Entan-

glement entropy is a natural probe of the field theory which is sensitive to the UV induced

by the TT deformation. Moreover, it has a well-known holographic dual given by the Ryu-

Takayanagi formula [7]. We will consider an entangling surface consisting of two antipodal

points on the sphere, which gives the entanglement entropy of the de Sitter vacuum state

across the horizon [8]. We find that this entanglement entropy is finite, and matches the

Ryu-Takayanagi formula with a finite cutoff surface consistent with the proposal of Ref. [2].

We further consider the conical entropy, a close relative of the Renyi entropy [9–11],

which carries more detailed information about the spectrum of the reduced density matrix.

These are calculated for index n < 1, and they verify a proposal due to Dong [12] relating

the conical entropy to the length of a brane anchored to the boundary. Again, the TT

deformation renders the entropy UV finite, and corresponds to a finite infrared cutoff

surface in the bulk. This gives further confirmation of the conjecture of Ref. [2].

2 Sphere partition function and entanglement entropy

We first calculate the sphere partition function in the TT -deformed CFT. We will see that

this is sufficient to calculate the entanglement entropy when the entangling surface is two

antipodal points on the sphere.

1In [2] the metric of the CFT was related to the induced metric of the bulk theory by a factor rc; herewe set rc = 1 which corresponds to measuring bulk and boundary distances in the same units.

– 2 –

To find the sphere partition function, we consider the metric ds2 = r2(dθ2+sin(θ)2dφ2)

and vary the radius r:d

drlogZ = −1

r

∫d2x√g T aa . (2.1)

By symmetry, the stress tensor takes the form Tab = αgab, where α is determined by the

flow equation (1.3):

α =2

µ

(1−

√1 +

cµ

24πr2

). (2.2)

In solving the quadratic equation we have chosen the branch that gives the CFT trace

anomaly in the limit µ → 0. This yields a differential equation for the partition function

as a function of radiusd logZ

dr=

16π

µ

(√r2 +

cµ

24π− r). (2.3)

This can be integrated with the help of the substitution r =√

cµ24π sinh(x), giving the

sphere partition function

logZ =c

3

(x− 1

2e−2x

)=c

3sinh−1

(√24π

cµr

)+

8π

µ

(r

√cµ

24π+ r2 − r2

)(2.4)

Note that we have chosen the boundary condition logZ = 0 at r = 0; this would not have

been possible in a CFT, where the partition function continues to change as a function of

scale at arbitrarily short distances. Here we see that the flow equation is consistent with a

trivial theory in the UV.

The Euclidean path integral on S2 corresponds to the de Sitter vacuum. We will

consider the entanglement entropy of this state across an entangling surface consisting

of two antipodal points. This entropy can be obtained directly from the sphere partition

function as follows.2 To calculate the entanglement entropy by the replica trick, we consider

the n-sheeted cover of the sphere:

ds2 = r2(dθ2 + n2 sin(θ)2dφ2). (2.5)

The entropy is then obtained from the partition function as:

S =

(1− n d

dn

)logZ

∣∣∣n=1

. (2.6)

In the absence of rotational symmetry, this formula requires analytic continuation in n,

but in the case of antipodal points we can continuously vary n.

Under a change of n, the partition function changes as

d logZ

dn

∣∣∣n=1

= −∫√g T φφ . (2.7)

Since the stress tensor on the sphere is isotropic, T φφ = 12T

aa . From (2.1) we conclude that

2We thank Aron Wall for teaching us this trick.

– 3 –

Log(2r/l)

Arcsinh(r/l)

0.5 1.0 1.5 2.0 2.5 3.0

r

l

-1.5

-1.0

-0.5

0.5

1.0

1.5

2.0f(r/l)

Figure 1. Entanglement entropy in the TT theory agrees with the CFT result for r � √µc, butis UV finite.

the entropy can be expressed in terms of the sphere partition function as

S =

(1− r

2

d

dr

)logZ =

c

3sinh−1

(√24π

cµr

). (2.8)

For r � √µc we see that this formula reproduces the well-known CFT result [13, 14] with

subleading corrections (see figure 1)

S =c

3log

(√96π

cµr

)+

c2µ

288πr2+O

(µ2). (2.9)

The corrections to the logarithmic term in the entanglement entropy are polynomial in

µ starting from order one. In the UV limit r → 0 the entanglement entropy vanishes,

indicating that the theory flows to a trivial theory.

We can also compare this result with the holographic proposal of [2]. The metric of

Euclidean AdS3 in global coordinates is

ds2 = `2(dρ2 + sinh(ρ)2(dθ2 + sin(θ)2dφ2)

). (2.10)

The sphere of radius r embeds into this geometry as a surface ρ = ρ0, where ρ0 is defined

by r = ` sinh(ρ0). According to the conjecture of [2], the TT deformed theory is dual to

quantum gravity in the region ρ < ρ0. According to the Ryu-Takayanagi formula [7], the

holographic entanglement entropy is given by L/(4G) where L is the length of a minimal

geodesic connecting the points of the entangling surface. In the case of antipodal points,

the geodesic passes through the center and has length L = 2`ρ0. Thus the Ryu-Takayanagi

formula yields:

S =L

4G=

`

2Gsinh−1

(r`

), (2.11)

which agrees precisely with (2.8) with the identifications (1.7).

– 4 –

3 Conical entropy

The entanglement entropy is just one measure of entanglement. More information about

the spectrum of the reduced density matrix is encoded in the conical entropy 3:

Sn =

(1− n d

dn

)logZn, (3.2)

which reduces to the entanglement entropy when n = 1. In the case of two antipodal points

on the sphere, Sn is the de Sitter entropy at a temperature ∼ 1/n. By varying n we probe

the density of states at different energy scales.

To calculate Sn via the replica trick, we consider the partition function of the theory

on the conical sphere (2.5). We will assume rotational symmetry in φ, so that we can

parametrize the stress tensor in terms of two nonzero components Tθθ(θ) and Tφφ(θ). The

problem simplifies if we consider the variables u = µ2T

θθ − 1 and v = µ

2Tφφ − 1, in terms of

these variables stress tensor conservation and the flow equation are

du

dθ= cot(θ)(v − u), uv = 1 +

cµ

24πr2. (3.3)

This has the solution

u2 = 1 +cµ

24πr2+

cnsin(θ)2

, (3.4)

where cn is independent of θ. The value of cn is determined by the coupling of the theory

to the conical singularity at θ = 0, π.

Boundary conditions We will define the theory on a surface with a conical singularity

via a limiting procedure similar to Ref. [15]. We first define a family of smoothed replica

geometries in which a small neighborhood of the conical singularity is replaced by a smooth

“cap”. We will take this cap to be a portion of maximally symmetric space; for n < 1 this

is a sphere, and for n > 1 a hyperbolic space.

Near the conical singularity, the geometry is approximately a flat cone ds2 = dτ2 +

τ2n2dφ2, from which we cut out the region τ < ε. For n > 1 we attach this to a cap which

is the region σ < σc of the hyperbolic space with metric ds2 = `2c(dσ2+sinh(σ)2dφ2), which

has constant negative curvature R = −2/`2c . Matching the intrinsic and extrinsic geometry

of the circle determines

`c =εn√n2 − 1

, σc = cosh−1(n). (3.5)

3The conical entropy Sn is related to the Renyi entropy Sn as

Sn = n2∂n

(n− 1

nSn

). (3.1)

– 5 –

The nonsingular solution on the cap takes the form Tab = 2µ(u+ 1) where

u2 = 1 +cµ

24πε2

(1

n2− 1

). (3.6)

However, (3.6) has no real solutions for small ε which prevents us from taking the limit

ε → 0.4 In terms of bulk variables, there are no solutions when `c < `: this corresponds

to the fact that we cannot embed a hypersurface with more negative curvature than the

ambient space without breaking rotational symmetry.

In the spherically symmetric case, we can instead analytically continue from n < 1; a

similar strategy has been employed for calculating entanglement in string theory [16–19].

For n < 1, we can replace a neighborhood of the conical singularity with a spherical cap

consisting of the region θ < θc of the sphere with radius rc:

ds2 = r2c (dθ2 + sin(θ)2dφ2). (3.7)

Matching the length and extrinsic curvature determines

rc =εn√

1− n2, θc = cos−1(n). (3.8)

The equation for the stress tensor is given by (3.6) just as in the hyperbolic case, except

that for n < 1 it always has real solutions.

We can now use this solution to determine the constant in (3.4). Stress tensor conser-

vation implies that u should be continuous, which fixes the singular part of u:

u2 = 1 +cµ

24πr2+

cµ

24πr2 sin(θ)2

(1

n2− 1

). (3.9)

This equation determines the stress tensor

T θθ =2

µ

(1−

√1 +

cµ

24πr2+

cµ

24πr2

(1

n2− 1

)1

sin(θ)2

), (3.10)

T φφ =2

µ

1−1 + cµ

24πr2√1 + cµ

24πr2+ cµ

24πr2

(1n2 − 1

)1

sin(θ)2

(3.11)

The sign of u in (3.9) has been chosen so that the CFT limit µ→ 0 is finite. In this limit,

the stress tensor agrees with the result obtained from the Schwarzian transformation law

of the stress tensor [13].

Conical Entropy Having found an appropriate boundary condition for the stress tensor,

we can now proceed to calculate the entropy Sn for n < 1. To find the partition function

4In the CFT limit this issue does not arise, because we take µ→ 0 prior to taking ε→ 0.

– 6 –

at fixed n, we vary the radius to obtain:

d logZ

dr= −2πnr

∫dθ sin(θ)

(T θθ + T φφ

)(3.12)

=4πn

µ

∫dθ sin(θ)

[2r2 + cµ

24π + α2

√r2 + α2

− r]

(3.13)

where we have defined

α2 =cµ

24π

(1 +

(1

n2− 1

)1

sin(θ)2

). (3.14)

This can be integrated in r to obtain

logZ =n

4G`

∫dθ sin(θ)

[`2 sinh−1

( rα

)+ r√α2 + r2 − r2

]. (3.15)

For n = 1, this reduces to (2.4).

From (3.15) we obtain the entropy

Sn = −n2 ddn

(logZ

n

)(3.16)

=rc

6n

∫dθ

1

sin(θ)

(1− cµ

24πα2√α2 + r2

)(3.17)

=c

3

(1− n2)√cµ

24πr2+ n2

Π

(n2∣∣∣ r2 + cµ

24π

r2 + cµ24πn2

). (3.18)

Π is the complete elliptic integral of the third kind:

Π(η|m) :=

∫ π/2

0

dθ

(1− η sin(θ)2)√

1−m sin(θ)2. (3.19)

This function has a branch cut for n ≥ 1, corresponding to a pole in the integral (3.19).

However, we can take the principal value of this integral, which is real for all n > 0. This

function is displayed in figure 2.

The limit n→ 0 gives the logarithm of the rank of the reduced density matrix:

S0 =

√2πc

3µπr (3.20)

which scales with the length, πr, of the boundary. This is suggestive of a lattice theory in

which an interval of length L has a Hilbert space of dimension exp(√

2πc3µ L

).

Comparison with holography We now compare our result (3.18) with the prediction

from holography. According to the proposal of Ref. [12], Sn is given by L/4G, where L is

the length of a cosmic string whose tension induces an angle 2πn in the bulk. By rescaling

φ, this is equivalent to finding the length of a geodesic in a smooth geometry with induced

– 7 –

n=0.5

n=0.7

n=1,arcsinh(r/l)

n=2

n=3

0.5 1.0 1.5 2.0 2.5 3.0

r

l

0.5

1.0

1.5

G S˜n

0.5 1.0 1.5 2.0 2.5 3.0n

0.5

1.0

1.5

2.0

2.5

3.0

G S˜n

r/l=1

r/l=2

r/l=3

r/l=4

Figure 2. The entropy Sn is a smoothly increasing function of r. At fixed r, Sn is a decreasingfunction of n, with a kink at n = 1.

boundary metric given by (2.5). Finding this solution is equivalent to finding an embedding

of the metric (2.5) into Euclidean AdS3.

This embedding problem is most easily solved in the coordinates:

ds2 = `2(dϕ2 + e−2ϕ(dρ2 + ρ2dφ2)). (3.21)

We will assume the embedding preserves rotational symmetry, so that the φ coordinate on

the boundary is the same as the φ coordinate of the bulk. The embedding is then defined

by two functions ϕ(θ) and ρ(θ). Demanding that the embedding be isometric leads to the

equations

ρ = eϕr

`n sin(θ),

r2

`2=

(dϕ

dθ

)2

+ e−2ϕ

(dρ

dθ

)2

. (3.22)

This can be rewritten as a single differential equation for ϕ,

r2

`2=

(dϕ

dθ

)2

+n2r2

`2

[(dϕ

dθ

)sin(θ) + cos(θ)

]2, (3.23)

which can be solved to give

dϕ

dθ=r√`2(1− n2) + n2(r2 + `2) sin(θ)2 − n2r2 cos(θ) sin(θ)

`2 + n2r2 sin(θ)2. (3.24)

The resulting embedding in global coordinates is shown in figure 3.

The length of the geodesic connecting the point θ = 0 and θ = π in the metric (3.21)

is given by

L = `(ϕ(π)− ϕ(0)) = `

∫ π

0dθdϕ

dθ. (3.25)

To find the total entropy, one simply has to integrate (3.24),

L

4G=

r

2G

1√1 + r2n2

[1 + r2

r2K(m)− 1

r2Π(η|m)

], (3.26)

– 8 –

n = 1

n = 0.8

n = 0.6

n = 0.4

n = 0.2

n = 0

Figure 3. Embedded surfaces with r = ` = 1 in Euclidean AdS3. For this graph we have

used Poincare disk coordinates in which the metric is ds2 = 4d~x2

(1−~x2)2 with the third coordinate

suppressed. The AdS boundary is the black circle, and finite boundaries are shown in color. Atn = 1, the embedding is a circle. As n is decreased the embeddings have an increasingly elongated“football” shape. At n = 0, the embedding degenerates to a line.

where K(m) = Π(0|m) is the complete elliptic integral of the first kind and

m =n2(1 + r2)

1 + n2r2, η =

r2n2

1 + r2n2. (3.27)

Using an identity of elliptic integrals5, this agrees with our expression (3.18) for the entropy,

with the identification of µ and c given in (1.7).

4 Discussion

We have carried out a calculation of entanglement entropy for an entangling surface con-

sisting of two antipodal points on the sphere. Our calculation verifies the conjecture of

Ref. [2] that the theory is dual to gravity in a finite region. The entanglement entropy is

UV finite and is given by the Ryu-Takayanagi formula. Since UV divergences are local,

this indicates that the entanglement entropy is finite for arbitrary entangling surfaces.

So far we have only considered the simplest case of two antipodal points, but to further

test the correspondence one should consider non-antipodal points, as well as multiple inter-

vals. Such generalizations do not have rotational symmetry about the entangling surface,

and so the method of continuation from n < 1 does not apply. Instead one would have

to find the appropriate solutions to equations (1.3), (1.4) for boundary conditions with

n > 1. Our analysis shows that these solutions must either be complex or break the replica

symmetry.

The conical entropy Sn encodes the full spectrum of the reduced density matrix. The

entanglement spectrum is given by the inverse Laplace transform of the partition function

Z, which can be found by integrating (3.2). In CFT, the n dependence of Z is universal,

leading to a universal entanglement spectrum [21]. Our result (3.18) has a highly nontrivial

5(m− η)Π(η|m) + (m− η′)Π(η′|m) = mK(m) where (1− η)(1− η′) = 1−m [20, Eq. 19.7.9].

– 9 –

n dependence compared to the CFT result Sn ∼ 1n , suggesting a modification of the

entanglement spectrum. In particular, we would like to understand the significance of

the branch cut in Sn, and whether the entanglement spectrum, like the spectrum on the

cylinder, has imaginary eigenvalues.

The fact that the entropy vanishes as r → 0 indicates that the theory is trivial in the

UV. This flow from a trivial theory in the UV to a CFT in the infrared is reminiscent

of the continuous Multiscale Entanglement Renormalization Ansatz (cMERA) [22]. It

would be interesting to see if the flow induced by TT can be viewed as locally building up

entanglement as in cMERA.

The relation between the flow equation and the Hamiltonian constraint suggests that

the natural generalization of TT to higher dimensions is 18(T abTab−(T aa )2). This generaliza-

tion has been considered in the case of a flat boundary [23], but a further generalization to

curved boundaries would be required to study the entanglement entropy using the methods

described here.

The TT deformation can be viewed as a result of dynamical gravity in two dimensions

[24]. In particular, Dubovsky et. al. in [25, 26], have shown that this deformation can

be defined through coupling the CFT to Jackiw-Teitelboim gravity. In a similar vein,

Cardy [27] defines this deformation perturbatively through a path integral prescription

involving integrating over Gaussian metric fluctuations. This is also consistent with the

idea that putting the CFT in the bulk leads to a finite induced gravitational constant on

the boundary [28–30], which leads to a finite entanglement entropy. Similar ideas in which

the entropy is regulated using an irrelevant deformation appear in Refs. [31, 32]. To study

the entanglement entropy from this perspective, one must face all the difficulties of defining

entanglement entropy in a theory with dynamical gravity. We hope that our results can

shed some light on how gravity acts as a UV cutoff for the entanglement entropy.

Acknowledgements

We thank Rob Myers for his comments on an earlier draft. We also thank Ed Witten

and Gabriel Wong for discussions relating to this work. We would also like to thank

Tomonori Ugajin for early conversations pointing us toward this interesting problem. This

research was supported in part by Perimeter Institute for Theoretical Physics. Research at

Perimeter Institute is supported by the Government of Canada through the Department

of Innovation, Science and Economic Development Canada and by the Province of Ontario

through the Ministry of Research, Innovation and Science.

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