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JHEP09(2015)091

Published for SISSA by Springer

Received: August 6, 2015

Accepted: August 26, 2015

Published: September 15, 2015

Universal corner entanglement from twist operators

Pablo Bueno,

Robert C. Myers

and William Witczak-Krempa

Instituto de F´ısica Te´orica UAM/CSIC,

C/Nicol´as Cabrera, 13–15, C.U. Cantoblanco, 28049 Madrid, Spain

Perimeter Institute for Theoretical Physics,

31 Caroline Street North, ON N2L 2Y5, Canada

E-mail: p.bueno@csic.es, wkrempa@physics.harvard.edu,

rmyers@perimeterinstitute.ca

Abstract: The entanglement entropy in three-dimensional conformal ﬁeld theories (CFTs)

receives a logarithmic contribution characterized by a regulator-independent function a(θ)

when the entangling surface contains a sharp corner with opening angle θ. In the limit

of a smooth surface (θ → π), this corner contribution vanishes as a(θ) = σ (θ − π)

. In

arXiv:1505.04804, we provided evidence for the conjecture that for any d = 3 CFT, this

corner coeﬃcient σ is determined by C

, the coeﬃcient appearing in the two-point function

of the stress tensor. Here, we argue that this is an instance of a much more general relation

connecting the analogous corner coeﬃcient σ

appearing in the nth R´enyi entropy and the

scaling dimension h

of the corresponding twist operator. In particular, we ﬁnd the simple

relation h

/σ

= (n − 1)π. We show how it reduces to our previous result as n → 1,

and explicitly check its validity for free scalars and fermions. With this new relation, we

show that as n → 0, σ

yields the coeﬃcient of the thermal entropy, c

. We also reveal a

surprising duality relating the corner coeﬃcients of the scalar and the fermion. Further,

we use our result to predict σ

for holographic CFTs dual to four-dimensional Einstein

gravity. Our ﬁndings generalize to other dimensions, and we emphasize the connection to

the interval R´enyi entropies of d = 2 CFTs.

Keywords: Field Theories in Lower Dimensions, Conformal and W Symmetry, Hologra-

phy and condensed matter physics (AdS/CMT)

ArXiv ePrint: 1507.06997

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP09(2015)091

JHEP09(2015)091

1 Introduction & main results

Understanding the structure of quantum entanglement in complex systems has become an

active area of study in a variety of areas of physics, including condensed matter physics,

e.g., [1–5]; quantum ﬁeld theory, e.g., [6–10]; and quantum gravity, e.g., [11–21]. For these

investigations, the entanglement entropy S

and R´enyi entropies S

[22, 23] have proven

to be two particularly useful measures of the relevant degrees of freedom. In the context

of quantum ﬁeld theory, these are deﬁned for a spatial region V with:

(V ) =

1 − n

log Tr ρ

, S

(V ) = lim

n→1

(V ) = −Tr (ρ

log ρ

) , (1.1)

where ρ

is the reduced density matrix computed by integrating out the degrees of freedom

in the complementary region V .

In the present paper, we will focus on the R´enyi entropy in three-dimensional conformal

ﬁeld theories (CFTs), which takes the form

= B

− a

(θ) log( `/δ ) + c

+ O



δ/`



, (1.2)

where δ is a UV cut-oﬀ and ` is a length scale characteristic of the size of the region V —

see ﬁgure 1a. The result is dominated by the ﬁrst contribution, the celebrated ‘area law’

term, but the corresponding coeﬃcient B

depends on the details of the UV regulator. The

subleading logarithmic contribution appears when the entangling surface (i.e., the boundary

of V ) contains a sharp corner of opening angle θ, as in ﬁgure 1a. The corresponding corner

function a

(θ) is regulator independent and hence it is a useful quantity to characterize the

underlying CFT. For instance, several groups have numerically studied the corner function

using lattice Hamiltonians [24–33], and obtained results independent of the lattice details

which probe the low energy degrees of freedom.

As the corner function a

(θ) is central to our investigation, let us summarize a few of

its key properties: if eq. (1.2) is evaluated for the vacuum state, reﬂection positivity [34, 35]

constrains a

(θ) to be a positive convex function of θ, i.e.,

(θ) ≥ 0 , ∂

(θ) ≤ 0 , ∂

(θ) ≥ 0 , (1.3)

in the range 0 ≤ θ ≤ π. In fact, reﬂection positivity gives rise to an inﬁnite tower of

nonlinear higher-derivative constraints as well [34].

Further, this function satisﬁes a

(θ) =

(2π −θ) if eq. (1.2) is evaluated for a pure state, e.g., in the vacuum state of the CFT.

The form of a

(θ) is also constrained on general grounds in the limits where the corner

becomes very sharp (θ → 0) and where it becomes almost smooth (θ → π, ﬁgure 1b) [8–10]:

(θ → 0) =

, a

(θ → π) = σ

(π −θ)

. (1.4)

This behaviour is schematically illustrated in ﬁgure 2a. Hence these limits deﬁne two

regulator-independent coeﬃcients, κ

and σ

, which are representative of the CFT.

We thank Horacio Casini for explaining these points. We discuss the nonlinear constraints further in

section 6.

– 2 –

JHEP09(2015)091

(a) (b)

Figure 1. a) Region V whose boundary contains a sharp corner with opening angle θ. b) The

contribution to the R´enyi entropy S

from a corner in the almost smooth limit yields a great deal

of insight into the degrees of freedom of the CFT via the coeﬃcient σ

In studying the corner contribution in the entanglement entropy [36, 37], we recently

conjectured that the smooth-corner coeﬃcient σ

has a universal form in general three-

dimensional CFTs,

, (1.5)

where C

is the central charge appearing in the two-point function of the stress tensor

— see eq. (1.12). We have veriﬁed that this relation holds for a free conformally coupled

scalar and a free massless fermion, as well as for an eight-parameter family of strongly

coupled holographic CFTs [36, 37]. A more general holographic proof appears in [38, 39].

Our primary result here is the generalization of eq. (1.5) to arbitrary values of the R´enyi

index n > 0:

n − 1

, (1.6)

where h

is the scaling dimension of the twist operator τ

appearing in calculations of

(V ). It is deﬁned in the n-fold replicated theory as the surface operator on the boundary

of V which acts to permute the n copies of the original QFT — see section 2 for more details.

A ﬁrst test of eq. (1.6) is to verify that we recover eq. (1.5) from this new relation in

the limit n → 1. To do so, we make two observations: ﬁrst, the twist operator becomes

trivial at n = 1 and hence the scaling dimension h

vanishes in this limit. Second, at n = 1

the ﬁrst derivative of h

with respect to n is proportional to the central charge C

in any

d-dimensional CFT [40, 41]. In d = 3, the precise relation is

∂

n=1

. (1.7)

Therefore in the vicinity of n = 1, the scaling dimension can be expanded as

n→1

(n − 1) + O



(n − 1)



. (1.8)

Now it is straightforward to see that substituting this expression into eq. (1.6) and taking

the limit n → 1 precisely reproduces the original relation (1.5).

Further, we have been able to verify eq. (1.6) for a free conformally coupled scalar ﬁeld

and for a free massless fermion. In particular, as we discuss in the following, we can evaluate

– 3 –

JHEP09(2015)091

the corner coeﬃcient σ

using the results of [8–10, 42] while the scaling dimension h

can

be determined with the results of [40, 41]. We demonstrate that these two independent

calculations yield complete agreement with eq. (1.6) for any integer values of n > 1. Since

the methods involved in these two calculations are so completely disparate, we ﬁnd this

agreement to be very strong evidence for the new conjecture.

Before moving to detailed discussions, let us draw an interesting parallel with two-

dimensional CFTs where twist operators are well understood. In the case of d = 2, the

twist operator is a local primary operator with scaling dimension [6, 7, 43]:

(2)



n −



, (1.9)

where c is the Virasoro central charge of the theory. The R´enyi entropy of a single interval

is calculated by ﬁrst evaluating the correlator of two twist operators inserted at each of the

endpoints and the ﬁnal result can be written as

d=2

= 2 σ

(2)

log( `/δ ) + ··· , where σ

(2)

≡

(2)

n − 1

. (1.10)

In this expression, ` is the length of the interval and δ, the UV cut-oﬀ. The factor of two

here comes from having two endpoints [43]. Hence, there is a striking similarity between

the coeﬃcient of the logarithmic contribution in d = 2 and in d = 3 — in the limit θ → π

for the latter. The parallel extends to the n → 1 limit, where one recovers the well-

known result S

d=2

= (c/3) log(`/δ) by applying the two-dimensional analog

of eq. (1.7):

∂

n=1

= c/6. While it calls for some deeper physical insight, this interesting connection

serves as a stepping stone from the well-known results for d = 2 to new considerations of

R´enyi entropies in higher dimensions — see further discussion in sections 4 and 7.

1.1 Main results

Eq. (1.6) is our main result. In particular, we conjecture this to be a relation valid for

general CFTs in d = 3. We thus predict that the corner coeﬃcient of the R´enyi entropies

in the smooth limit σ

is proportional to the scaling dimension h

of the twist operator.

Most of the remainder of the paper is devoted to supporting this new conjecture, and to

extract various consequences from it.

Eq. (1.6) incorporates our previous conjecture (1.5) when n = 1, as we discussed above.

Using independent computations of σ

[8–10, 42] and h

[40, 41], we have veriﬁed that (1.6)

holds exactly for a free scalar and for a free massless Dirac fermion at integer values of n

(up to n = 500) as well as in the n → ∞ limit. Further, since we have a simple expression

for h

that holds for any real n, we can predict the behavior of σ

for non-integer n. This

is plotted in ﬁgure 3 for the free complex scalar and Dirac fermion. Figure 3 also includes

for a holographic CFT dual to Einstein gravity. This is a prediction of our conjecture

as h

is known for these holographic theories, but direct access to σ

is currently limited

Eq. (2.5) provides the general equation relating ∂

n=1

and C

in d dimensions. The above result

follows from simply substituting d = 2 and also C

= c/(2π

– 4 –

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