#### Entanglement entropy in flat holography

HJE
Entanglement entropy in at holography
Hongliang Jiang 0 1 3 4
Wei Song 0 1 2 4
Qiang Wen 0 1 2 4
Clear Water Bay 0 1 4
Kowloon 0 1 4
Hong Kong 0 1 4
P.R. China 0 1 4
0 dence, Models of Quantum Gravity
1 Beijing , 100084 , China
2 Yau Mathematical Sciences Center, Tsinghua University
3 Department of Physics, The Hong Kong University of Science and Technology
4 connected to @A by two null geodesics
BMS symmetry, which is the asymptotic symmetry at null in nity of at spacetime, is an important input for at holography. In this paper, we give a holographic calculation of entanglement entropy and Renyi entropy in three dimensional Einstein gravity and Topologically Massive Gravity. The geometric picture for the entanglement entropy is the length of a spacelike geodesic which is connected to the interval at null in nity by two null geodesics. The spacelike geodesic is the xed points of replica symmetry, and the null geodesics are along the modular ow. Our strategy is to rst reformulate the Rindler method for calculating entanglement entropy in a general setup, and apply it for BMS invariant eld theories, and nally extend the calculation to the bulk.
Black Holes; Field Theories in Lower Dimensions; Gauge-gravity correspon-
1 Introduction 2
Generalized Rindler method
2.1
Field theory calculation of entanglement entropy
2.1.1
2.1.2
Generalized Rindler method
Holographic entanglement entropy in gravity side
{ i {
The strategy
Poincare coordinate
FSC
Global Minkowski
6
The geometric description for holographic entanglement entropy
Three special curves
The modular ow and its bulk extension
The geometric picture of holographic entanglement entropy
7
Holographic entanglement entropy from
Holographic entanglement entropy for AdS3 with Brown-Henneaux
boundary conditions
Flat limit of AdS3
7.2.1
Transformation to Bondi gauge and the at limit
Holographic entanglement entropy in at limit
8.1
8.2
8.3
8.4
Topologically massive gravity in at space
Thermal entropy formula from
Thermal entropy formula from direct calculation
CS contribution to holographic entanglement entropy
9
Renyi entropy
A \Cardy" formula in BMSFT revisited B
Killing vectors
B.1 FSC
B.2 Poincare patch
B.3 Global Minkowski
C Rindler transformations for BMSFT
C.1 Thermal BMSFT
C.2 BMSFT on the cylinder
30
32
32
33
connected to infra properties of scattering amplitude [28, 29], and memory e ects [30], in
algebra [34], isomorphism between BMS algebra and Galileo conformal algebra [25],
representations and bootstrap [37{45]. Flat holography based on BMS3 symmetry was proposed
in [25, 26] and supporting evidence can be found in [46{48]. The antipodal identi cation
in three dimensions was discussed in [49, 50]. See [51{56] for more discussions for this at
holography and related topics.
One useful probe of holography is the entanglement entropy, which describes the
correlation structure of a quantum system. In the context of AdS/CFT correspondence, Ryu
and Takayanagi [57, 58] (RT) proposed that the entanglement entropy is given by the
area of a codimension-two minimal surface in the bulk, which anchored on the entangling
tween spacetime structure in the bulk and entanglement in the boundary still exist beyond
the context of AdS/CFT, and if so, how it works for non-AlAdS spacetimes. So far, in
the literature, there are three approaches. The rst approach is to start with the RT or
HRT proposal, and study the implications in the holographic dual, see [65{69]. The second
approach is to directly propose a prescription in the bulk, and check its consistency [70, 71].
The third approach, which we will advocated in the current paper, is to derive an analog of
RT proposal using the dictionary of holography, along the lines of [60{64]. In [72, 73],
holographic entanglement entropy in Warped AdS3 spacetime was derived by generalizing the
gravitational entropy [63] and Rindler method [60, 74], respectively. Interestingly, it was
found that the HRT proposal indeed need to be modi ed, and moreover the modi cation
depends on di erent choices of the boundary conditions which determines the asymptotic
symmetry group. Another important lesson is that the Rindler method [60], which maps
entanglement entropy to thermal entropy by symmetry transformations, can be generalized
to non-AlAdS dualities.
In the context of at holography, entanglement entropy for eld theory with BMS3
symmetry was considered in [27] using twist operators.
Using the Chern-Simons
formalism of 3D gravity, [75] took the Wilson line approach [76{78], and found agreement
with [27]. However, a direct calculation in metric formalism is still missing. No geometric
picture has been proposed and it is not clear whether RT (HRT) proposal is applicable
for asymptotic at spacetime. In this paper, we will address this question along the lines
of [72, 73].
In this paper, the Rindler method is formulated in general terms, without referring to
any particular example of holographic duality. We argue that the entanglement entropy
for a subregion A is given by the thermal entropy on B~, if there exists a Rindler
transformation from the causal development of A to B~. Moreover, under such circumstances, the
modular Hamiltonian implements a geometric ow generated by the boost vector kt. Then
we apply this generalized Rindler method to holographic dualities governed by BMS
symmetry, and provide a holographic calculation of entanglement entropy and Renyi entropy in
Einstein gravity and Topologically massive gravity. On the eld theory side, our result of
entanglement entropy agrees with that of [27] obtained using twist operators [79]. On the
{ 2 {
gravity side, by extending the Rinlder method to the bulk, we provide a holographic
calculation of the entanglement entropy in metric formalism and provide a geometric picture
(see gure 1). We also expect a generalization in higher dimensions.
The geometric picture ( gure 1) for holographic entanglement entropy in three
dimensional at spacetime involves three special curves, a spacelike geodesic
, and two null
.
is the set of xed points of the bulk extended modular ow ktbulk (and also
xed points of bulk extended replica symmetry), while
are the orbits of the boundary
is connected to the boundary end points
4G
Length( ) . The main di erence between our picture and the RT (HRT) proposal is that the
. The holographic entanglement entropy for the boundary interval A is given by
spacelike geodesic
eld theory [27] and as well as a Chern-Simons calculation [75]. In this paper, explicit
calculations are done in Bondi gauge at future null in nity. Similar results follows for the
past null in nity. We also expect the techniques and results here can be reinterpreted in
the hyperbolic slicing [50, 80{83].
The paper is organized as follows. In section 2 we explain the generalized Rinder
method and provide a formal justi cation for its validity. In section 3, BMS3 and
asymptotically at spacetimes is reviewed. In section 4, we calculate the EE in the BMS3 invariant
eld theory (BMSFT) by Rindler method. Then we calculate the entanglement entropy
holographically for Einstein gravity in section 5. In section 6, we give geometric picture of
holographic entanglement entropy. In section 7, we calculate the HEE by taking the at
limit of AdS. In section 8 we calculate the holographic entanglement entropy in topological
massive gravity. In section 9, we calculate the Renyi entropy both on the eld theory side
and the gravity side. In appendix A, we rederive the \Cardy formula" for BMSFT using
the BMS symmetries. In appdneix B, we present the Killing vectors of 3D bulk at
spacetime. In appendix C we give the details of Rindler coordinate transformations for BMSFT
in nite temperature and on cylinder.
2
Generalized Rindler method
The Rindler method was developed in the context of AdS/CFT [60] with the attempt to
derive the Ryu-Takayanagi formula. For spherical entangling surfaces on the CFT vacuum,
the entanglement entropy can be calculated as follows. In the CFT side, certain
conformal transformations map the entanglement entropy in the vacuum to the thermodynamic
entropy on a Rindler or hyperbolic spacetime. In the bulk, certain coordinate
transformations map vacuum AdS to black holes with a hyperbolic horizon. Using the AdS/CFT
dictionary, the Bekenstein-Hawking entropy calculates the thermal entropy on the
hyperbolic spacetime, and hence provides a holographic calculation of the entanglement entropy.
Going back to vacuum AdS, the image of the hyperbolic horizon then becomes an extremal
surface ending on the entangling surface at the boundary. Recently, the Rindler method
has been generalized to holographic dualities beyond AdS/CFT. The eld theory story
{ 3 {
was generalized to Warped Conformal Field Theories (WCFT) in [74], while the gravity
story was generalized to Warped Anti-de Sitter spacetimes (WAdS) in [73]. The results are
consistent with the WAdS/WCFT correspondence [17].
In this section, we summarized the Rindler method for holographic entanglement
entropy, without referring to the details of the holographic pair. The goal is to provide
a general prescription which could be potentially used in a broader context. Schematic
prescriptions in the eld theory side and the gravity side are as follows.
2.1
2.1.1
Field theory calculation of entanglement entropy
Generalized Rindler method
In the eld theory, the key step is to nd a Rindler transformation, a symmetry
transformation which maps the calculation of entanglement entropy to thermal entropy. Consider a
QFT on a manifold B with a symmetry group G, which act both on the coordinates and on
the elds. The vacuum preserve the maximal subset of the symmetry, whose generators are
formation. The image of R is a manifold B~,1 and the domain is D
denoted by hj . Consider the entanglement entropy for a subregion A with a co-dimension
two boundary @A. Acting on positions, a Rindler transformation R is a symmetry
transB
, with A
D,
supposed to have the following features:
1. The transformation x~ = f (x) should be in the form of a symmetry transformation.
identi cation of the new coordinates x~
referred to as a \thermal" identi cation hereafter.
2. The coordinate transformation x ! x~ should be invariant under some imaginary
i
x~i + i ~i. Such an identi cation will be
3. The vectors @x~i annihilate the vacuum. i.e.
HJEP07(21)4
X bij hj ;
j
{ 4 {
where bij are arbitrary constants.
4. Let kt
the ow x~i(s) = x~i + ~is. A thermal identi cation can be expressed as x~i
x~i(i).
~i@x~i , then kt generates a translation along the thermal circle, and induce
particular, kt can only become degenerate2 at the entangling surface @A,
The boundary of the causal domain @D should be left invariant under the ow. In
1Throughout this paper, we always use tilded variables to describe the B~ spacetimes and their bulk
extensions after the Rindler transformation.
2For Warped conformal
eld theory [73], kt keeps @D invariant, but will not degenerate anywhere
for 6= 0.
(2.1)
(2.2)
Now we argue that the vacuum entanglement entropy on A is given by the thermal entropy
on B~, if such a Rindler transformation with the above properties can be found. Property
2 de nes a thermal equilibrium on B~
. Property 3 implies that the vacuum state on B is
mapped to a state invariant under translations of x~i, which is just the thermal equilibrium
on B~
. The Modular Hamiltonian on B~, denoted by H ~ then implements the geometric
B
ow along kt. With property 1, the symmetry transformation R acts on the operators by
an unitary transformation UR. By reversing the Rindler map, a local operator H
can be de ned by UD(s) = UR U~ (s) UR 1, where U~ (s)
e iH~ s. Note that UD(s) generates
D on D
the geometric ow, and H
D
ln UD(i) is just the conserved charge Qkt up to an additive
D
constant. Property 4 indicates that UD(s) implements a symmetry transformation which
keeps D invariant. Then we can always decompose the Hilbert space of A in terms of
eigenvalues of HD. Therefore, the modular Hamiltonian on D can indeed be written as
H , which again generates the geometric ow along kt. In particular, the density matrix
are related by a unitary transformation
A = UR B~ UR 1 :
given by the thermal entropy on B~.
Since unitary transformations does not change entropy, the entanglement entropy on A is
More explicitly, at the thermal equilibrium, the partition function and density of matrix
on B~ can be written as3
Z(B~) = Tr e ~iQx~i ;
~
B
Tr e H~ = Z(B~) 1e ~iQx~i ;
where Qx~i are the conserved charges associated with the translation symmetries. The
modular
ow is now local, and is generated by kt
positions and elds are given by
x~i(s) = x~i + ~is ;
O~(s) = U~ (s)O~U~ ( s) :
The entanglement entropy which is equivalent to thermal entropy is then given by
SEE(A) = S(B~) = (1
Reyni entropy and the Modular entropy [84] can be calculated in a similar fashion. In
fact,
= P
i 2
x~
i
i
the periodicity boundary condition
parameterizes the Rindler time. The thermal identi cation is just
+ 2 i. The replica trick can be performed by making multiple copies of B and impose
To actually nd the Rindler map, the strategy is to follow the steps below,
3If there are internal symmetries, the partition function should be modi ed accordingly.
{ 5 {
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
Take an arbitrary symmetry transformation, and impose the condition (2.1). This
will give a system of di erential equations, whose solution will depends on the
constants bij .
The temperatures can be read o from the transformations, and will be determined
by certain combinations of bij .
Further solving condition (2.2) will relate the bij to position and size of the entangling
HJEP07(21)4
2.1.2
\Cardy" formula
In conformal eld theory, S-transformation is used to estimate the resulting thermal
entropy, leading to an analog of the Cardy formula [85, 86]. A successful generalization
has been applied to WCFT in [18, 74], and BMSFT in [46, 47].
More generally,
Stransformation can be realized as a coordinate transformation compatible with the
symmetry, which e ectively switches the spatial circle and thermal circle. In some region
of parameters, the partition function is dominated by the vacuum contribution, and the
entropy can hence be estimated.
For our purpose in BMSFT, we revisit the Cary-like formula in appendix A and obtain
the approximated entropy formula for BMSFT on arbitrary torus. Our derivation is based
on the BMS symmetries only, without resorting to at limit of CFT.
2.2
Holographic entanglement entropy
The gravity story is the extension of the eld theory story using holography. There are
two possible routes.
The rst route is to nd the classical solution in the bulk which is dual to thermal
states on B~
be found in [73].
. This can be obtained by extending the boundary coordinate
transformation x~ = f (x) to the bulk, by performing a quotient. More detailed discussion can
The second route is to extend replica symmetry to the bulk along the lines of [63, 64].
The eld theory generator kt has a bulk extension ktbulk via the holographic dictionary.
Since @A is the xed point of kt, we expect a special bulk surface
satisfying
Such a bulk surface will be the analog of RT( HRT ) surface. However, if ktbulk
the homologous condition can not be imposed directly. As we will see later,
is
, which are along the bulk modular ow
ktbulk. We will discuss a local version of this approach in a future work [87].
ktbulk
j = 0 :
{ 6 {
(2.10)
Let J (z) denote the current associated to the reparameterization of z, and let P (z) denotes
the current associated to the z-dependent shift of w. We can de ne charges4 [47]
z~ = f (z) ;
w~ = f 0(z)w + g(z) :
Ln =
Mn =
The conserved charges satisfy the central extended algebra
Ln; Lm] = (n
m)Ln+m +
Ln; Mm] = (n
m)Mn+m +
Mn; Mm] = 0 ;
cL (n3
12
cM (n3
12
n) m+n;0 ;
n) m+n;0 ;
where cL; cM are the central charges. Under the transformation (3.3), the currents
transform as
3.1
BMS invariant eld theory
In this subsection, we review a few properties of two dimensional eld theory with BMS3
symmetry( BMSFT). On the plane [88], BMS symmetries are generated by the
following vectors
Ln =
The nite BMS transformations can be written as [47]
P~(z~) =
J~(z~) =
c1M2 ff; zg ;
1cL2 ff; zg
where the Schwarzian derivative is
In particular, the transformation below maps a plane to a cylinder,
4We believe the analytic continuations of z; w to complex numbers are inessential.
fF; zg =
F 000 (z)
F 0 (z)
F 00(z) 2
F 0(z)
:
z = ei ;
w = iei u :
+
3
2
{ 7 {
(3.1)
(3.2)
(3.3a)
(3.3b)
(3.4)
(3.5)
(3.6a)
(3.6b)
(3.6c)
(3.7)
(3.8)
(3.9)
(3.10)
HJEP07(21)4
The currents transform as
J Cyl( ) =
z2J (z) +
P Cyl( ) =
z2P (z) +
and
Ln
Cyl = Ln
where the generators on the cylinder are de ned as
Mn
Cyl = Mn
cM
24 n;0 ;
cL
24
;
cL
24 n;0 ;
Ln
Cyl =
Mn
Cyl =
2
2
BMS as asymptotic symmetry group
Minkowski spacetime is the playground of modern quantum eld where the Poincare
symmetry gives very stringent constraint on the properties of particles. The Poincare algebra
includes translation and Lorentz transformation, while the Lorentz transformations further
consist of spatial rotation and boost. At the null in nity of at spacetime, the nite
dimensional Poincare isometry group is enhanced to in nitely dimensional asymptotic symmetry
group, called BMS group [20, 21].
In three spacetime dimensions, the topology of the boundary at null in nity is S1
R
where R is the null direction. Under proper boundary conditions [24, 34], the general
solution to Einstein equation in the Bondi gauge is
ds2 =
( ) du2
2dudr + 2
h ( ) +
u
2
(3.15)
where the null in nity is at r ! 1. The asymptotic symmetry group is the three
dimensional BMS group, whose algebra is given in the previous subsection.
BMS3 group is generated by the super-translation and super-rotation. The
supertranslation can be thought as the translation along the null direction which may vary
from one point to another in S1, while the super-rotation is the di eomorphism of S1.
Furthermore, the corresponding conserved charges generates BMS3 group on the phase
space. The in nitely dimensional algebra now has central extensions, which also coincide
with (3.6). For Einstein gravity, cL = 0; cM = 3=G [34].
3.3
Global Minkowski, null-orbifold and FSC
The zero mode solutions in (3.15), describing some classical background of spacetime, are
of particular interest. With the standard parameterization of the S1
general classical solutions of Einstein gravity without cosmology constant takes the
following form [24]
ds2 = 8GM du2
2dudr + 8GJ dud + r2d 2
! M du2
2dudr + J dud + r2d 2;
where in the second line we have used the convention 8G = 1 which will be adopted
throughout the paper. We will spell it out whenever it is necessary to restore G.
Via holography, the identi cation (3.16) speci es a canonical spatial circle where the
BMSFT is de ned on. These classical at-space backgrounds (3.17) can be classi ed into
three types:
M =
1; J = 0: Global Minkowski. The solution (3.17) (3.16) cover the full three
dimensional Minkowski. The holographic dual is the BMSFT de ned on the cylinder.
M = J = 0: Null-orbifold. It was rst constructed in string theory [89]. They are
supposed to play the role of zero temperature BTZ, being the holographic dual of
BMSFT on a torus with zero-temperature and a xed spatial circle.
M > 0: Flat Space Cosmological solution (FSC). This was previously studied in
string theory as the shifted-boost orbifold of Minkowski spacetimes [
90, 91
]. Their
boundary dual is the thermal BMSFT at nite temperature.
The at-space metric (3.17) can also be written in the ADM form5
This indicates that the at-space (3.17) admit a Cauchy horizon [37] at
J 2
4r2
dr2
J2
4r2
M
ds2 =
M
dt2 +
+ r2 d' +
J
2r2 dt
2
:
rH = jrcj;
rc
2
J
p
M
:
The thermal circle of (3.17) is given by
with
thermal circle : (u; )
boundary eld was shown in [46, 47]
5The coordinates are related by
SFSC = SBMSFT =
2 rc
4G
:
t = u
Mr + 2rMc log rr + rrcc
;
' =
> 0 can also be brought into the Cartesian
coordinate locally6
If we decompactify the angular direction , the boundary theory will be put on the plane
instead of cylinder. In particular, the resulting spacetime with M = J = 0 is more like a
at version of AdS in Poincare patch. The coordinate transformation from the Poincare
coordinate and the Cartesian coordinate is
It is easy to check that
t =
x =
y =
ds2 =
l
4
l
4
=
r +
l
lu + r
r
2
l
2
l
u +
;
u +
will diverge except when x ! llu .
3.5
Solutions with general spatial circle
More generally, we can consider solutions locally with the same metric as (3.17), but with
a spatial circle di erent from (3.16). More precisely,
6We hope that the same symbol \t" with di erent meanings in (3.18) and (3.24) will not cause any
confusions.
When we study the bulk extension of the Rindler transformations in section 5, we will
encounter the bulk extensions of B~, which are usually this kind of spacetime. Hereafter we will
refer to (3.31) as FgSC. The proper length on Cauchy horizon is ds2 =
Integration along this proper length will give the length of the horizon, and the
BekensteinpM~ du~ + r~cd ~ 2.
Hawking entropy
SBH =
`horizon =
4G
pM~
u~ + rc
4G
~
;
where
~ represent the extension of the spacetime along the u~ and ~ direction.
4
Entanglement entropy in eld theory side
In this section we apply the generalized Rindler method to a general BMS eld theory
with arbitrary cL and cM. As we will show below, our results in this section agree with the
previous calculation using twist operators [27].
4.1
Rindler transformations and the modular ow in BMSFT
In this section, following the guidelines we give in section 2.1.1, we derive the most
general Rindler transformations in BMSFT. According to property 1 in subsection 2.1.1, the
Rindler transformations should be a symmetry of the eld theory, thus for BMSFT it
should be the following BMS transformations [47]
(3.33)
HJEP07(21)4
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
where we use @ to denote derivative with respect to . Furthermore, this indicates the
theory after Rindler transformation is also a BMS invariant eld theory, which we call
B^MSFT. The inverse transformation can be written as
where
and @~ denotes derivative with respect to ~. The property 4 of the Rindler transformation
indicates that the vectors @ ~ and @u~ have to be linear combinations of the global BMS3
~ = f ( ) ;
= f~( ~) ;
g~ =
X (bnLn + dnMn) ;
which implies that the condition for @u~ is automatically satis ed, as
Note the BMS generators have the following general form (see [47] or consider the u;
components of (B.1) in the limit of r ! 1)
then we can get two di erential equations
1
n= 1
It is interesting to note
Furthermore by noting the relation in (4.5), the previous di erential equations can be
simpli ed as
Ln =
Mn =
2
b1 = ~ l
;
b 1 =
l
2 ~ ;
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
HJEP07(21)4
Rindler transformation on the plane.
Now we consider the BMSFT on the plane
with zero temperature. In this case the symmetry generators of BMS3 are
where the n =
1; 0; 1 part form a subalgebra, and generate the global symmetries. By
matching the general form of Killing vectors in (4.9), one can easily see that
Y =
b 1
b0
b1 2;
T = d 1 + d0 + d1 2
:
Substituting them into (4.13), one gets
f =
2
p
b20 + 4b1b 1
arctan
p
b0 + 2b1
b20 + 4b1b 1
!
+ c1 :
Note that c1 and b0 shift the origin of
losing generality. Taking
or ~, and therefore we can set c1 = b0 = 0 without
we get
Taking
then,
~
f ( ) =
:
2
l
Plugging (4.17) and b 1;0 into (4.14), we get
g =
~ =
The two parameters d0 and c2 can be absorbed by a shift of u~ and u. Finally, up to some
trivial shifts, we get the most general coordinate transformations
The above Rindler transformation satis es the property 2 as it induces a thermal circle
thermal circle : (u~; ~)
(u~ + i ~u; ~
i ~ ) :
Note that the subregion D of B which maps to B~ is a strip bounded by
and
= l =2.
Rewritten in the original coordinate system, we get
(4.20)
(4.22)
2 ~ l
2 ~ l
l
2
2
Thus, the generator of modular ow is
lul + 4 lu 2
l
8u + ~
~
u 2
l
~
~
!
!
;
kt =
=
~
2l
u
!
l
2
X
n= 1
~
bnLn +
~
u
dn + ~ bn
!
!
Mn
;
+ lul + 4 lu 2
l
8u
It is easy to verify that at the points ( lu=2; l =2), kt = 0. Following our
prescription (2.2), this implies that ( lu=2; l =2) is that endpoints of the interval. Thus, we can
naturally interpret lu; l as the extension of the interval along u and
direction.
The global BMS3 generators Ln; Mn have bulk extensions, which are just the Killing
vectors (B.11) of the
at-space (3.17) in Poincare patch. Substitute (B.11) into (4.7)
and (4.8), the modular generator (4.28) can also be extended to the bulk and is given by
l
8lu +8r
Following the similar analysis, we give the construction of Rindler transformations in
BMSFT with a thermal or spatial circle in appendix C. We also calculate the modular ow
and its bulk extension in these two cases.
In this subsection we consider zero temperature BMSFT on the plane, nite temperature
BMSFT, and BMSFT on a cylinder respectively, and calculate the entanglement entropy
of the following interval in these BMSFTs,
(4.30)
HJEP07(21)4
A :
( lu=2; l =2) ! (lu=2; l =2) ;
(4.31)
where the arrow means a line connecting the two endpoints. After the Rindler
transformation, the entanglement entropy equals to the thermal entropy of the B^MSFT on B~, which
can be calculated via a Cardy-like formula. Since the extension of B~ is essentially in nite,
we need to introduce the cuto s to regulate the interval
In [46, 47], the Cardy-like formula for BMSFT is derived from the modular invariance of
the theory, which is inherited from the modular invariance of CFT2 under at limit. In
appendix A we re-derive the Cardy-like formula using the BMS symmetry only.
The manifold B~ can be considered as a torus with the following identi cations
where (a; a) parametrize a thermal circle and (b; b) parametrize a spatial circle. We nd
that (see appendix A), under some regime (A.12), the thermal entropy can be calculated by
Sbjb (aja) =
2
3
b
a
cL
+ cM
(ab
ab)
a2
:
4.2.1
Zero temperature BMSFT on the plane
After performing the Rindler transformation (4.24), the image of this regularized interval
Areg is
(4.33)
(4.34)
(4.35)
Ireg :
u~
;
2
~ =
~
2
~!
log
!
l
;
u~
2
;
u~ =
2
~!
;
~ lu
l
~
u
~
u log
l
;
2 b =
~
;
2 b =
u~ :
^ =
log l ;
^u =
^2
l
u
u
:
;
thus satisfy the regime (A.12) for (4.34).
obtain the entropy
Substituting these quantities into the entropy formula (4.34) derived before, we can
where we have neglected irrelevant terms O( u; ) and terms O( 2u)= , but keep terms of
order u . The endpoint e ects are expected to be negligible for this large interval, thus we
can identify the endpoints to form a spatial circle. The same story happens in the later
two cases. Together with the thermal circle induced by the Rindler transformation (4.24),
B~ can be considered as a torus parametrized by
The canonical torus parameters de ned by (A.5) are
This agrees with the result in literature [27] when we set u = 0. In general, we would
like to keep the cuto related terms. It is interesting to note that cL term is the same as
that in CFT. This is not surprising since the Ln generators satisfy the chiral part of the
Virasoro algebra.
4.2.2
Finite temperature BMSFT
The temperature of the quantum
eld theory is dictated by the periodicity along the
imaginary axis of time, namely u
u + i . More generally, we can have the following
thermal identi cation
(u; )
The Rindler transformation for thermal BMSFT is given in (C.8), which in particular
respects the above thermal circle. Following the similar steps, we
parametrized by
nd the torus B~ is
~
2 b =
;
2 b =
~
u
+
~
lu +
u
l
coth
l
u ;
where
= log
sinh l
.
thus the EE can be obtained from (4.34)
coinciding with the result in [75].
One can check that the canonical torus parameters (A.5) also satisfy the regime (A.12),
Consider the cylinder with periodicity
+ 2 , we should apply the Rindler
transformation (C.9). Similarly we nd B~ can be considered as a torus parametrized by
where
= log
2 sin l2 . When l < , again the canonical parameters de ned in (A.5)
satis es the regime (A.12) for (4.34). Thus the entropy can be calculated by (4.34),
When the interval is very small, i.e. l
! 0, SEcyEl ! SEplEane, which is just the EE on
the plane. This is reasonable, since when the interval is very small, whether the space is
compact or not is expected to be irrelevant.
5
5.1
The strategy
Holographic entanglement entropy in gravity side
In this section we extend the eld side story to the bulk. On the eld theory side, the
Rindler transformations map the entanglement entropy for BMSFT to the thermal entropy
of B^MSFT. According to section 3, this story should have a bulk description. We start from
BMSFT and its gravity dual, the at-space (3.17). Then our task is to nd a transformation
from (3.17) to a new spacetime, which is usually a FgSC (3.31) in tilde coordinates, such that
its dual boundary eld theory is just the B^MSFT in the eld theory side story. According to
of B^MSFT, thus, reproduces the entanglement entropy for BMSFT.
the at holography, the Bekenstein-Hawking entropy of FgSC equals to the thermal entropy
To make sure that the gravity side story reproduces the eld theory side story on the
boundary, we expect FgSC to satisfy the following requirements.
The bulk transformations from
at-space (3.17) to FgSC should be a bulk extension
of the Rindler transformation, thus reproduce the eld side story on the boundary.
The asymptotic structures of FgSC should satisfy the BMS3 boundary conditions
locally, and have the same thermal and spatial periodicities as the boundary
eld
theory B^MSFT.
More explicitly we expect the metric of FgSC to be in the form of (3.17). The second
requirement is necessary, since we expect FgSC to be the gravity dual of B^MSFT.
Based on our discussions on the Rindler transformations in the eld theory side story,
we nd that the vectors @u~ and @ ~ should be the following linear combinations of the global
BMSFT generators
1
X (bnLn + dnMn) ;
This relation can be naturally extended to the bulk by simply replacing the global
generators in BMSFT with their bulk counterparts, i.e. the Killing vectors in at-space. These
Killing vectors are explicitly given by (B.11) for Poincare, and by (B.10) for FSC and
global Minkowski. The rst requirement indicates that, in order to reproduce the Rindler
transformation on the boundary, we should choose the same coe cients b0; 1 and d0; 1 as
in the eld theory side. The second requirement gives the Bondi gauge conditions for the
HJEP07(21)4
new coordinates
M~ ;
J~;
r~2 ;
where the inner products are calculated with the old metric of at-space. The constants
M~ and J~ are determined by the coe cients in (5.1) and can be regarded as the mass and
angular momentum of the new spacetime FgSC. Note that, in the bulk @u~ and @ ~ are two
commuting Killing vectors, the new metric only depend on the third metric r~. So the Bondi
gauge conditions (5.2) (5.3) are consistent with (5.1). We de ne the new radial coordinate
r~ with the third equation in (5.2), and the third equation in (5.3) requires gu~r~ only depends
Jacobian matrix between the old and new coordinate systems. Solving all these conditions
will give the bulk coordinate transformation, as well as the unknown metric component
gu~r~(r~). As expected, we always get gu~r~(r~) =
1, so the new metric is in the form of (3.17).
The bulk transformation can be regarded as a quotient on
at-space without doing
identi cation for the new coordinates.
Similar strategy has been successfully applied
on warped AdS3 and AdS3 spacetimes with certain boundary conditions in [73] to
calculate holographic entanglement entropy, and both the results ful ll their
eld theory
side expectations.
We will use the method elaborated above to perform quotient for Poincare coordinate
in subsection 5.2. Then a short-cut method of quotient for FSC and global Minkowski in
Einstein gravity will be used for calculating HEE in section 5.3 and section 5.4, respectively.
Note that, in Einstein gravity, the central charges of the dual BMSFT are given by cL =
0; cM = 3=G, with cL vanished. To investigate the holographic entanglement entropy
contributed from the cL term, we will consider the topologically massive gravity (TMG) in
section 8.
5.2
Poincare coordinate
In Poincare coordinates, the coe cients in (5.1) are given by (4.19) and (4.22), while the
bulk Killing vectors are given by (B.11). We also need to use the following relations
~ =
2
pM~
;
~
~
u =
~
J
2M~
;
(5.1)
(5.2)
(5.3)
(5.4)
r~ =
~ =
u~ =
s
~
M
16l2
1
pM~
log
pM~ 0
8u
4l
r~ + J~= 2
pM~
r~
M
~ +
1
4l pM~
On the boundary, i.e. r ! 1, the coordinate transformation becomes
~ =
u~ =
2
pM~
4 (ul
pM~ (l2
;
2
l
lu )
4 2)
which reproduces the Rindler transformation (4.24) in the eld theory side story after using
the identi cation (5.4).
By virtue of the at holography, we can calculate the thermal entropy of B^MSFT by
the Bekenstein-Hawking entropy SBH of FgSC,
SBH =
pM~
u~ + J~=(2pM~ ) ~
4G
where
~ are the extension of the new coordinates. Since these two quantities
are essentially in nity, we need to introduce two cuto s u and
, as was discussed on the
eld theory side before, and consider the regularized interval
l
2
lu + u < u < lu
2
u :
Then it is easy to nd that
which is just (3.22) in FgSC. Following the steps outlined in section 5.1, we get the
coordinate transformation between the \Poincare" coordinates and FgSC,
8u
4lu + r (l
+
4M~
;
Very straightforwardly, we get the holographic entanglement entropy for BMSFT,
SHEE = SBH =
1
2G
lu
l
u
:
As expected, this result agrees with the eld theory side result (4.39) after inserting the
central charges of BMSFT dual to Einstein gravity cL = 0; cM = G3
.
~ =
l
2
2
2
pM~
pM~
;
lu
l
l
log
:
u
~
J
2M~
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
only extremal (or saddle) curve among all the curves that connect the null rays
orbits of @A1;2 under bulk modular ow. This can be considered as a generalized version
of the RT (HRT) proposal when the boundary entangling surface @A is not xed under
the bulk modular ow (or bulk extended replica symmetry).
The geometric picture of holographic entanglement entropy
We observe that the holographic entanglement entropy is given by the length of the
spacelike geodesic
which is connected to the end points at the boundary by two null
,
HJEP07(21)4
Alternatively, we can de ne a curve
A homologous to the boundary interval A,
Length( A) :
The holographic entanglement entropy can also be given by the total length of
A
We hope to further clarify the choice between
and
A in the future.
7
Holographic entanglement entropy from
In this section, we re-derive the HEE from a at limit of AdS3. As summarized in [60, 73],
the logic of Rindler method for AdS3 is the following: we do a quotient on AdS3 and get a
Rindler AgdS3 black string, with the boundary of the later covers the a causal development
of an interval on the boundary of the former. Accordingly the entanglement entropy of the
interval equals to the Bekenstein-Hawking entropy of AgdS3.
the holographic entanglement entropy for the BMSFT.7
Under the at limit ` ! 1, this picture provides a way to calculate the holographic
entanglement entropy for BMSFT. The quotient on AdS3 to Rindler AgdS3 under the at
limit become a quotient on at-space (3.17) to Rindler FgSC as described in section 5. Note
that the outer horizon of the Rindler AgdS3 is now pushed to an in nitely far away location,
while the inner horizon becomes the Cauchy horizon of FgSC. Thus, under the at limit,
the Bekenstein-Hawking entropy of the inner horizon of AgdS3 is the quantity that gives
In this section we rst revisit the Rindler method for AdS3 in section 7.1. Then in
section 7.2, we change to Bondi gauge and discuss how to take
at limit. We re-derive
the holographic entanglement entropy for BMSFT from a at limit of AdS3 for Einstein
gravity in section 7.3 and TMG in section 8 respectively.
7This di ers from the prescription in [92], which takes at limit directly on the CFT results. This equals
to taking at limit on the Bekenstein-Hawking entropy of the outer horizon in AgdS3.
TU~ U~ =
TV~ V~ =
~
TU~ TV~
=
0
1
4
1
4
log
log
(1 + (2U
(1 + (2V
" (1 + (2U + lU ) V )2
" (1 + (2V + lV ) U )2
2 lU2 l
2
V
4V 2
lU ) V )2
lV ) U )2
4U 2l2
V
4 lU lV
2lV2 (lU =2 + U )2 #
2lV2 (lU =2
2lU2 (lV =2 + V )2 #
2lU2 (lV =2
;
;
U )2
V )2
:
+ 4 (2 U V + 1)2
Then we get the metric of the Rindler AgdS3
ds2 = `2 @TU~2 dU~ 2 + 2 ~ dU~ dV~ + TV~2 dV~ 2 +
The asymptotic behavior of bulk quotient is given by
boundary conditions
Holographic entanglement entropy for AdS3
with
Brown-Henneaux
Poincare AdS3. For simplicity we rst consider Poincare AdS3
The dual boundary
eld theory is a CFT2 de ned on a plane with zero temperatures.
Following the guide lines to construct Rindler transformations in section 2, we de ne @ ~
U
and @V~ as a combination of global generators in either copy of the Virasoro algebras (see
appendix B in [73] for an explicit example). Then we nd the Rindler transformation for
CFT2 on a plane, and extend it to the bulk. The bulk transformation is a quotient on (7.1),
and is given by
which is just the Rindler transformation and, as expected, a conformal mapping. It shows
that the boundary of AgdS3 (7.3) covers the causal development of an interval
A :
( lU =2; lV =2) ! (lU =2; lV =2) ;
on the boundary of the original Poincare AdS3. We introduce two in nitesimal parameters
U and V and regulate the interval as
These two parameters also regulate the extension of the U~ ; V~ coordinates of the AgdS3,
TU~ U~ = ArcTanh
TV~ V~ = ArcTanh
2U
lU
2V
l
V
+ O
+ O
1
1
;
;
U~ =
1
log lU ;
U
V~ =
1
T ~
V
l
log V :
V
Hence the Bekenstein-Hawking entropy of the outer horizon of the AgdS3 is given by
Following the logic of the Rindler method, Souter gives the holographic entanglement
entropy for the single interval (7.6) in a CFT2 with zero temperature. This result is also
consistent with the HRT [59] formula.
AdS3 with a thermal (spatial) circle.
To consider CFT2 with nite temperatures,
we take the dual bulk spacetime to be a BTZ black string8
with the thermal circle
The mapping from Poincare AdS3 (7.1) to (7.12) is given by
4( 2
TU2 TV2 )
;
i
TU
i
TV
:
U ! e2TU U
V ! e2TV V
s
s
1
1
2TU TV
+ TU TV
2TU TV
+ TU TV
;
;
!
( + TU TV )e 2(TU U+TV V )
4TU TV
:
Accordingly, in terms of the coordinates of BTZ black string (7.12) we have
8Under the following transformations
(7.1) can be written in the ADM formula
log lU
U
! log
sinh (TU lU )
TU U
;
log V
l
V
! log
sinh (TV lV )
TV V
:
U !
`'2`+2 t ; V !
`' t
2`2 ;
! 2r2
r
2
r+2 ;
TV ! r+
r ; TU ! r+ + r ;
ds2 =
`2r2
r
2 r
2
r
2 r+ dt2 +
2
r2 r2
`2r2
r2 r2 dr2 + r2 d' + r`rr2+ dt :
2
+
(7.8)
(7.12)
(7.13)
(7.14)
(7.15)
(7.9)
(7.10)
(7.11)
It is easy to see that the quotient on (7.12) to AgdS3 (7.3) is just given by the combination
of (7.2) and (7.14). Also the Bekenstein-Hawking entropy of the outer horizon of (7.3)
changes to
temperatures.
log
`
4G
sinh (TU lU ) sinh (TV lV )
TU TV U V
:
This gives the holographic entanglement entropy for a single interval in a CFT2 with nite
When we consider imaginary temperatures TU !
LUi ; TV !
LVi , the thermal
circle (7.13) changes to a spatial circle
Correspondingly, the boundary eld theory becomes a zero temperature CFT2 on a
cylin
Accordingly, the Bekenstein-Hawking entropy of the outer horizon of AgdS3 (7.3)
then becomes
Souter =
`
4G
log 4
2 LU LV sin
LU
lU sin
lV
LV
3
5 :
As expected, we get the holographic entanglement entropy for a single interval in a CFT
de ned on the cylinder (7.17).
Flat limit of AdS3
Transformation to Bondi gauge and the at limit
It is more convenient to take the at limit in the Bondi gauge. Following [37], we nd that,
under the following coordinate transformations
the BTZ black string (7.12) becomes
U =
V =
=
r
2
`2
ds2 =
8GM
du2
;
with M and J de ned by
TU = 2
TV = 2
r
r
G` `M
p`2M 2
J 2 +
G` `M + p`2M 2
G` `M + p`2M 2
J 2
G` `M
p`2M 2
J 2
J 2
!
!
;
:
r
r
(7.16)
(7.17)
(7.18)
(7.19)
(7.20)
(7.21)
In terms of the new coordinates (u; ), the regulated interval (7.6) now becomes
which is regulated by u and
the parameters
. According to (7.19), we nd the relationships between
where lU and lV depicts the interval on the boundary of the Poincare AdS3. Using (7.23)
and (7.24), we nd the Bekenstein-Hawking entropy of the inner horizon under at limit
SHEE = SC = Sinnerj at limit =
cM lu
6 l
u
+ O
1
`
;
where cM = G3
. This agrees with our previous result (4.39) with cL = 0.
FSC. FSC can be considered as a at limit of a BTZ solution. Note that under the at
limit, the outer horizon is pushed to in nity. Thus the thermal circle in the bulk is actually
associated to the inner horizon.9 In other words, we have
(7.22)
(7.23)
(7.24)
(7.25)
(7.26)
(7.27)
(7.28)
lU =
U =
`l + lu
2`2
+
;
u ;
2`2
2`
lV =
V =
`l
2`
2`2
lu
;
u :
2`2
Sinner =
=
4G
`
`
4G
T ~
U
log
U
T ~ V~ =
V
`l + lu + log V
lu
`
4G
;
U
l
V
log U
V
We see that under the transformations (7.19) and (7.21), the physical quantities in
AdS3 are functions of the physical quantities in Bondi gauge and the AdS radius `. The
right way to take the at limit is to take ` ! 1 while keeping all the physical quantities
in Bondi gauge xed.
Holographic entanglement entropy in at limit
Poincare coordinate system. The Bekenstein-Hawking entropy of the inner horizon
TU~ TV~ of the Rindler AgdS3 (7.3) is given by
thermal circle : (U; V )
U +
U i ; V +
V i
`
U
` ; TV =
`
V
where U and V are nite quantities de ned by TU =
circle in the (u; ) coordinates becomes
. Then the thermal
thermal circle : (u; )
(u + ui;
i) ;
with
u = `( U
V ) ;
=
( U + V ). Here we choose u and
as the physical
quantities in Bondi gauge and keep them
xed when taking at limit. Note that, the order
of U and V in terms of ` are chosen such that we can get nite u and
.
9The thermal circle associated to the outer horizon is given by (U; V )
(U + U` i ; V
V` i ).
`
LV
`
:
Compared with the nite temperatures case, this equals to replace ( U ; V ) with (iLU ; iLV )
and replace ( u;
) with (iLu; iL ). Accordingly the spatial circle in the (u; ) coordinates
is given by
spatial circle : (u; )
Lu; + L ) :
The Bekenstein-Hawking entropy of the inner horizon of AgdS3 (7.3) then becomes
Sinner =
log
`
4G
LU sin ( lU =LU )
U
log
LV sin ( lV =LV )
V
;
which, after taking the at limit, reduces to
u :
As expected, when we set L
= 2 and Lu = 0 the at space is just global Minkowski.
Then we have
l
2
u ;
The Bekenstein-Hawking entropy of the inner horizon of the Rindler AgdS3 is given by
Then we use (7.23) (7.24) and take the at limit. We nd
`
log
cM
6
` U
lu +
U sinh( `lU = U )
V sinh( `lV = V )
log
:
` V
l u
coth
u
which agrees with (4.43) when cL = 0.
Global
Minkowski. Similarly when we consider imaginary temperatures TU
LiU` ; TV !
LiV` , the thermal circle (7.27) changes to a spatial circle
HJEP07(21)4
which agrees with our previous result (5.33) when cL = 0.
8
8.1
Topologically massive gravity
Topologically massive gravity in at space
In the previous sections, we considered the Einstein gravity and calculated the holographic
entanglement entropy. The results are in agreement with eld theory calculations as well
as other methods in literatures. However, the asymptotic symmetry algebra of Einstein
gravity has only one non-vanishing central charge cM. In order to go beyond and incorporate
the cL e ects, we can consider the topologically massive gravity (TMG) [93, 94].
(7.29)
(7.30)
!
(7.31)
(7.32)
(7.33)
(7.34)
(7.35)
and Chern-Simons term
STMG =
1
d3x p
g R +
2
`2 +
1
2
Einstein gravity is recovered in the limit
algebra with left and right central charges [
95
]
+
cTMG =
3`
2G
1 +
1
`
; cTMG =
1
In the at limit10 ` ! 1, the cosmological constant disappears and we come to the
TMG in
at spacetime. Asymptotic symmetry group analysis at null in nity yields the
BMS3 algebra (3.6) with central charge [35]
3`
2G
+
2
3
1
`
:
! 1. The dual CFT is described by Virasoro
cL =
3
G
;
cM =
3
G
:
Alternatively, these central charges can be obtained from AdS by taking Wigner-Inonu
contraction: cL = cTMG
+
cTMG; cM = (cT+MG + cTMG)=`.
The TMG also admits BTZ black hole solutions as in the Einstein gravity case
r
2
r
2
r
2
r
2
`2r2
`2r2
r2
r
2
dr2 + r2 d'
r+r
`r2 dt
2
; (8.4)
ds2 =
with
BTZ in TMG get shifted [
95, 97
]
dt2 +
M^ =
r+2 + r2
`2
;
J^ =
2r+r
`
:
The action of TMG in AdS includes Einstein-Hilbert term, cosmological constant term
:
(8.1)
(8.2)
(8.3)
(8.5)
(8.6)
(8.7)
(8.8)
(8.9)
(8.10)
But due to the presence of CS term, the physical conserved charges associated with
M = M^ +
J
`2
;
J = J^ +
M
:
Fixing the physical charges M; J and taking the at limit ` ! 1, we have
and the BTZ metric reduces to the FSC metric after transforming to the Bondi gauge.
The line element on the inner and outer horizon are
^
M ! M = M;
r+ ! `
p
M = `
M ;
p
r
!
^
J ! J = M
J + M=
2
pM^
=
J
p
2
M
J
;
dso2uter =
dsi2nner =
r+d' +
r d' +
r dt 2
r+dt 2
`
`
`!1
`!1
!
!
p
M `d' + rcdt=`
rcd' +
M dt
p
2
:
rc ;
2
;
10Here we are simply choosing the at limit as ` ! 1. A di erent double scaling limit yields the so-called
at-space chiral gravity [35] which is argued to be unitary and ghost-free. For TMG in AdS, it generally
admits ghost
uctuations except at some critical point [96]. Hence, the simple large ` limit of TMG is
expected to admit non-unitary
uctuations as well. But here we assume that this is not relevant for our
entropy calculation in both eld theory and gravity sides.
8.2
4G
4G `
where the length of horizon comes from integration.
The thermal entropy associated with the inner and outer horizons of the BTZ black hole
can be obtained as [98{101]
Sinner = `inner horizon + `outer horizon ;
`outer horizon + `inner horizon ;
(8.11)
4G
4G `
In the at limit, the thermal entropy of the FSC comes from the inner horizon of BTZ
black hole, so the entropy should be
where in the second line, we have transformed to Bondi gauge. The CS correction to
thermal entropy is
SFSC TMG = Sinnerj`!1 =
=
M
u + rc
4G
SCS =
M
p
M
4G
t + rc '
4G
M
4G
:
+
M ` '
4G `
SBH + SCS ;
Thermal entropy formula from direct calculation
In the previous subsection, we derived the contribution of Chern-Simons term to the entropy
by taking the
at limit of BTZ black hole. This approach is physically more intuitive.
However, considering the possible subtlety of at limit, a more direct calculation without
involving BTZ is also very desirable. This can be done by performing a symplectic analysis.
As elaborated in [101], the shift of thermal entropy due to the Chern-Simons term in 3D
can be calculated from the following formula:
SCS =
4G
N ;
with
N =
1
2
dx ;
where the bifurcation horizon
generated by the Killing vector
is a co-dimension 2 surface at radius r = rc
The binormal vector
is de ned as
where n is null vector normal to , given by
= n
2M
J
;
(8.12)
(8.13)
(8.14)
(8.15)
J=2p
M ,
(8.16)
(8.17)
(8.18)
(8.19)
(8.20)
Then, we can easily nd that ur =
ru =
1; r =
2JM and
Therefore,
SCS =
u =
4G
M
d =
=
r =
2
p
M :
p
M
4G
;
where the
is the extension of
along
direction. This agrees with the previous
results (8.14) obtained by taking at limit.
8.4
In previous sections, we have derived the holographic entanglement entropy in Einstein
gravity by considering Bekenstein-Hawking entropy in the Rindler spacetime. As expected,
it reproduces the results in dual eld theory with cL = 0. Now by considering the TMG,
we get a non-vanishing central charge cL in the dual eld theory. The e ects of cL on
holographic entanglement entropy can be similarly calculated by employing the CS correction
to the thermal entropy SCS.
Using Rindler method, the contribution of CS term to entanglement entropy boils
down to the quantity
~ in Rindler spacetime FgSC. Since we have calculated
~ in
di erent cases (see (5.15), (5.26), (5.32)), it is straightforward to get entanglement entropy
corrections due to the CS term
Poincare: SCS =
FSC: SCS =
Global Minkowski: SCS =
1
1
1
log
log
log
l
;
2 sinh
M
2 sin l2 :
pMl
2
;
Together with the Bekenstein-Hawking part considered previously, it is easy to check that
the total SHEE exactly agrees with eld theory results.
9
Renyi entropy
In this section, we will use Rindler method to derive the Renyi entropy. Under the Rindler
transformation, the density matrix is transformed unitarily. Thus, the entanglement
entropy and Renyi entropy in the original space are the same as the thermal entropy and
thermal Reyni entropy in the Rindler space respectively. Therefore, we only need to
consider the thermal Renyi entropy in the Rindler space and make a connection between
thermal Renyi entropy and thermal entropy. Then, the relation between Renyi entropy
and entanglement entropy in the original space follows directly.
Field theory side.
The thermal Renyi entropy on arbitrary torus is de ned as
Sb(nb) (aja) =
1
1
n
log
Zbjb (najna) !
Zbjb (aja)n
:
By performing a BMS coordinate transformation, the Renyi entropy on arbitrary torus is
the same as the Renyi entropy on the canonical torus
Sb(nb) (aja) = S(n) ^uj ^
0j1
1
1
n
log
Z0j1(n ^u; n ^ ) !
j
Z0 1( ^u; ^ )n
;
where the partition on the canonical torus, as justi ed in appendix A, can be
approximated by
Z0 1
^uj ^
exp
cM
2 ^u
6 ^2
(8.21)
(8.22)
(8.23)
(9.1)
(9.2)
(9.3)
1+
cL +
1+
j
Sb(1b) (aja)
1
2
1+
1
Sbjb (aja) :
Therefore, for all the cases we considered before, one can obtain the following relation
between Renyi entropy and entanglement entropy by using the Rindler method
SB(nM) SFT =
1
2
1
1 +
SEE :
HJEP07(21)4
Obviously, the Renyi entropy is alway proportional to entanglement entropy by a xed
coe cient and reduces to the entanglement entropy in the limit n ! 1. In addition, the
coe cient in BMSFT in the same as that in CFT. This is expected from the fact that
BMSFT is a speci c limit of CFT.
Gravity side.
Consider the periodic identi ed FSC ~
~ + 2 ,11 the relevant
thermal dynamic quantities associated with the Cauchy horizon, including thermal entropy,
Hawking temperature and angular velocity, are given by12
SC = p
J
2GM
2GM
G
;
TC =
2M p
2GM
J
;
C =
2M
J
;
where the second term in SC has been derived in section 8. The physical conserved
charges are
Then, it is easy to show that the system satis es the rst law [47]
M = M;
J = J + M= :
dM =
TC dSC +
C dJ :
SC
2
2G
(9.4)
(9.5)
(9.6)
(9.7)
(9.8)
(9.9)
(9.10)
(9.11)
Note that the rst law is not the conventional one due to the minus sign before TC . The
partition function is thus de ned as
ln ZC ( C ; C )
The Renyi entropy is de ned as
ln ZC ( C ; C ) =
C2 +
C
1
C
:
1
1
n
S(n) =
log Tr n
;
11For general periodic identi cation, one can discuss similarly by calculating the conserved charges more
carefully. Or one can consider a further linear coordinate transformation to arrive at the torus ^
^ + 2 .
12Here we restore the G-dependence of all the quantities, especially M; J in the previous sections should
Then, we can obtain the thermal Renyi entropy
Sb(nu)lk =
1
1
log
ZC (n C ; n C )
ZC ( C ; C )
n =
1 +
ln ZC ( C ; C ) :
This yields the following relations
Sb(nu)lk =
1
2
1 +
1
Sb(1u)lk
1 +
Sthermal ;
where the thermal entropies include the Bekenstein-Hawking term and the Chern-Simons
term contribution.
After performing a Rindler transformation, (9.14) is translated to the relations between
holographic Renyi entropy(HRE) and holographic entanglement entropy(HEE)
1
2
1
i
:
1
1
2
1 +
SHEE :
SH(nR)E = SB(nM) SFT :
where we put an additional minus sign to take account of the unconventional rst law,
otherwise the entropy obtained below is negative. The normalized density matrix is
exp
Tr exp
h
h
C M +
C J
i
C M +
C J
A
\Cardy" formula in BMSFT revisited
Consider a BMSFT living on an arbitrary torus with the following identi cation
u~; ~
u~ + ia; ~
ia
u~ + 2 b; ~
2 b :
The partition function of BMSFT on such torus is de ned as
Zbjb (aja)
Trbjb e aMb0jb eaLb0jb
;
Comparing the results of gravity and eld theory, we can nd that the Renyi entropy
agrees on both sides
Acknowledgments
We thank for helpful discussions with A. Castro, C. Chang, G. Compere, B. Czech,
M. Guica, T. Hartman, R. Miao, R. Myers, M. Rangamani, and J. Xu. This work was
supported in part by start-up funding from Tsinghua University. W.S. is also supported
by the National Thousand-Young-Talents Program of China. H.J. is supported by grants
HKUST4/CRF/13G and ECS 26300316 issued by the Research Grants Council (RGC) of
Hong Kong. H.J. would like to thank the Yau Mathematical Sciences Center in Tsinghua
University for kind hospitality.
(9.12)
(9.13)
(9.14)
(9.15)
(9.16)
(A.1)
(A.2)
where in the a-circle is viewed as the thermal circle, and Mb0jb; Lb0jb are the charges
generating the translation along u and
directions, de ned on the spatial b-circle.
The spectrum of a eld theory is usually discussed on certain canonical circle which is
a 2 spatial circle along . Under the BMS transformation
^ =
;
u^ =
bb2 ~ ;
the new torus has a canonical spatial circle,
Again, the Schwarzian derivative for the BMS transformation (A.7) vanishes, thus, no
anomalous terms appear and we have
j
^uj ^
j
Note that the Schwarzian derivative for the BMS transformation (A.3) vanishes, thus, the
new charges from (3.8) will not acquire any anomalous terms, and the partition function
is invariant under this transformation
Zbjb (aja) = Z0j1
^uj ^
;
= Tr0j1 e ^uM0Cyl e ^ L0Cyl
;
= Tr0j1 e ^u(M0 c2M4 )e ^ (L0 c2L4 ) ;
spatial cycle, and on the third line we have used (3.12) to express the partition function in
terms of the charges on the plane.
Next we perform the following BMS transformation (or S-transformation) that
exchanges the spatial and thermal circle of the torus,
The torus under this transformation become
with
^0 = 2 i
u^0 =
2 i
u^ + ^
u ^ :
u^0; ^0
u^0; ^0 + 2
u^0 + i ^u0; ^0
i ^0 ;
^u0 =
4 2 u
^2 ;
^0 =
4 2
^
:
^ =
b
^u =
ab
:
i ^
u^; ^
u^ + i ^u; ^
u^; ^
2
;
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
This can be regarded as the modular invariance of BMSFT. This modular invariance agrees
with the modular invariance inherited from CFT2 under at limit [46, 47].
For a CFT, the Cardy formula is valid in certain parameter region [85, 86] where
the partition function can be approximated by the ground state contribution after the
S-transformation. Similarly, in the \Cardy region" of GCFT, vacuum contribution is
expected to dominate the partition function, so that
^uj ^
;
exp
cM
2 ^u
6 ^2
2 !
(A.11)
;
(A.12)
(A.13)
(A.14)
(A.15)
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
where the vacuum charges on the plane have been taken to be zero. In this paper, we
will not attempt to give a necessary condition for (A.11), due to some unusual properties
of BMSFT including the abnormal rst law of thermodynamics [46, 47] and the generic
non-unitarity of the highest weight representation [88]. A su cient condition for the last
line of (A.11) is that the charges L0; M0 are bounded from below, and
^
! 0 ;
^u= ^2
! R :
Then the thermal entropy is approximately
Sbjb (aja) = S0j1
^uj ^
log Z0 1
uj ^
2
2
3
3
= 1
cL +
b
+ cM
ab !
:
B
Killing vectors
We focus on the local isometries of the solutions (3.17), which can be obtained from the
Killing equations L g
= 0. The general solutions have the following form
where Y and T satisfy the following equations
;
a2
u
u
r =
J
2r
{ 37 {
Li; Lj ] = (i
Li; Mj ] = (i
Mi; Mj ] = 0 ;
X biLi + diMi ;
1
i= 1
j)Li+j ;
j)Mi+j ;
where bi; ci are arbitrary constants and Li; Mi are the normalized to satisfy the sub-algebra
of the GCA
The explicit form of the generators depend on the background, which will be displayed
below.
B.1
FSC
The most general solutions can be found
Y ( ) =
T ( ) =
1
X
j= 1
1
X
j= 1
p
M
j
j e jpM ;
J
2M 2
1)e jpM
p
M
dj e jpM
:
More explicitly, the Killing vectors are
( u; r; )
p
M rc +M u+rc (r+rc) p
M rc +M u r
p
M rc +M u r !
A general Killing vector can be obtained and written in the following form
p
M rc +M u rc (r rc) p
M rc +M u r
p
M rc +M u r !
i; j = f 1; 0; 1g ;
HJEP07(21)4
(B.6)
(B.7a)
(B.9)
; (B.10a)
(B.10b)
; (B.10c)
(B.10d)
(B.10e)
(B.10f)
L1 =
L0 =
L 1 = epM
M1 =
M0 =
M 1 = epM
L1 =
M1 =
2u ; 2r ;
2; 2;
2
2u
r
Mrc ; 0; p1
M
p
M
1 ; 0; 0 ;
p
1 ;
M
p
1 ;
M
M (r+rc) ; 1
p
M (r rc) ; 1
;
M
p
M
r
r
r
:
B.2
Poincare patch
In the Poincare patch, M = J = 0, the Killing vectors are
2
L0 = ( u; r;
L 1 = (0; 0; 1) ;
M0 =
; 0;
M 1 = (1; 0; 0) :
(B.11)
r
r
1
r
) ;
p
M r
p
M r
C
Rindler transformations for BMSFT
We follow the procedure in section 4.1 to derive the Rindler transformation for BMSFT
with a thermal or spatial circle.
C.1
Thermal BMSFT
(B.12a)
(B.12b)
(B.12c)
(B.12d)
(B.12e)
(B.12f)
(C.1)
(C.2)
(C.3)
For nite temperature BMSFT dual to FSC, the symmetry generators that preserves
vacuum can be obtained from the bulk Killing vectors (B.10) by keeping the u;
components
and then taking limit r ! 1. By matching with the general form of generators (4.9), we
can get the following two functions
+e
b1 u + d1
2 b1 u
T =
2
1
2
b0 + b1e
2
2
2
+ b 1e
2
b 1 u + d 1
;
2 b 1 u
+ b0 u + d0
One can even show that the above Killing vectors can also be obtained by taking the
at limit of the SL(2; R)
SL(2; R) algebra of Poincare AdS: expressing the AdS Killing
vectors in terms of BMS coordinate and then taking Wigner-Inonu contraction.
B.3
This is the global Minkowski spacetime and the isometry group is the Poincare group which
is composed of translation, spatial-rotation and boost.
L1 = e i
L 1 = ei
u; u
r; i
+ 1
;
M1 = e i
i; i;
u; u
r; i
+ 1
;
r
M0 = (i; 0; 0) ;
M 1 = ei
i; i;
1
1
Solving (4.13), we get
f = p4b 1b1
2
0
0 (b 1
b0 + b1) tanh
+ b 1
b1 1
p4b 1b1
2
A + c1 :
We set c1 = 0, since c1 shift the origin of ~. Also we set b1 = b 1 such that the center
point of the interval is at
= 0. Furthermore, we de ne
b0 = ~
tanh
2
+ coth
; b1 =
~ csch
l
;
(C.4)
2
l
then, the solution (C.3) can be written as
h
f =
0 tanh
tanh 2
A :
Solving (4.14) with Y given in (C.2), we can get the solution of g. Similarly we x
the translation symmetry of u~ and set the center points of the interval to be at u = 0.
Furthermore we de ne the new parameters ~u; ~ and express d1; d0; d 1 as
d1 = d 1 =
d0 =
then the solution becomes
g =
2 2
~ 2 u sinh( l = )
(l u + lu
cosh( l = )
) sinh(2
~u f :
csch( l = )
~u 2 + (l u + lu
) ~ coth( l = )
2 ~2
i
;
csch2( l = )
h
~u 2 sinh(2 l = ) + 2 (l u + lu
2 ~2
) ~ i
vectors (B.10).
C.2
BMSFT on the cylinder
The BMSFT with a spatial circle
through analytical continuation by setting
~ =
tan
tan l
2
4
u ~ =
~ =
0 tanh
tanh 2
sinh ( l = )
1
A ;
u~ + ~
u ~ =
~ sin (l =2)
2 (cos
cos (l =2))
u
1
2 lu csc
l
2
sin
cosh ( l = )
cosh (2
= )
u +
l u + lu
2
csch ( l = ) sinh (2
= ) :
With the Rindler transformation coe cients (C.6) and (C.4) known, one can also write
down the modular ow and its bulk extension using (4.7), (4.8), (4.28) and the Killing
{ 40 {
+ 2 , can be obtained from thermal BMSFT
i2 ; u = 0,
(C.5)
(C.7)
(C.9a)
The modular ow is given by
kt =
csc
+ 2
csc
2
cot l
2
csc
2
2
cos( ) csc
2
lu sin( ) csc l
2
+ 2r
r
which vanishes at the boundary end points ( lu ;
l2 ) and ( l2u ; l2 ). Substitute (B.12)
into (4.7) and (4.8), the bulk extension of the modular ow in global Minkowski is given by
2
(C.10)
lu csc
+ lu cos( ) cot
HJEP07(21)4
lu cos( ) cot
lu csc
+ lu cos( ) cot
Open Access.
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any medium, provided the original author(s) and source are credited.
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