aainstitutetext:  Department of Physics, Florida Atlantic University, 777
Glades Road, Boca Raton, FL 33431-0991, USAbbinstitutetext:  Institut für
Quantengravitation, Universität Erlangen-Nürnberg, Staudtstr. 7/B2, 91058
Erlangen, Germanyccinstitutetext: Shing-Tung Yau Center of Southeast
University, Nanjing 210096, Chinaddinstitutetext: School of Mathematics,
Southeast University, Nanjing 211189, Chinaeeinstitutetext: Graduate School,
China Academy of Engineering Physics, Beijing 100193, Chinaffinstitutetext:
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China

#  Entanglement entropies from entanglement contour: annuli and spherical
shells

Muxin Han c,d,e,f and Qiang Wen<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The partial entanglement entropy $s_{\mathcal{A}}(\mathcal{A}_{i})$ captures
the contribution from the subset $\mathcal{A}_{i}$ of the region $\mathcal{A}$
to the total entanglement entropy $S_{\mathcal{A}}$ of $\mathcal{A}$. The
partial entanglement entropy proposal Wen:2018whg ; Wen:2019ubu  claims that
$s_{\mathcal{A}}(\mathcal{A}_{i})$ equals to a linear combination of the
entanglement entropies of certain relevant subsets in $\mathcal{A}$. We derive
the differential version of this proposal which can directly generate the
scheme independent entanglement contour function. Furthermore we derived the
sufficient condition to apply the differential version of proposal. On the
other way around, the proposal indicates that given the partial entanglement
entropy we can calculate the entanglement entropies for the subsets. Following
this idea, on the field theory side we analytically calculated the
entanglement contour functions and entanglement entropies for annuli and
spherical shells in holographic CFTs in arbitrary dimensions. We also comment
on the phase transition of the mutual information, which is calculated
holographically, across an annulus.

##  1 Introduction

Partition a quantum many body system into two regions $\mathcal{A}$ and
$\mathcal{A}^{c}$, the entanglement entropy captures how much the subset
$\mathcal{A}$ is entangled to its complement $\mathcal{A}^{c}$. The
entanglement entropy can be calculated as the von Newmann entropy of the
reduced density matrix
$\mathcal{\rho}_{\mathcal{A}}=\text{Tr}_{\mathcal{A}^{c}}\rho$,

$\displaystyle
S_{\mathcal{A}}=-\text{Tr}\rho_{\mathcal{A}}\log\rho_{\mathcal{A}}\,.$ (1.1)

In condense matter physics, entanglement entropy has been used to distinguish
new topological phases and characterize critical points, e.g.,
PhysRevLett.96.110405 ; Kitaev:2005dm ; Calabrese:2004eu ; Calabrese:2005zw ;
Hsu:2008af . Furthermore the Ryu-Takayanagi (RT) formula Ryu:2006bv ;
Ryu:2006ef  relates quantum entanglement to spacetime geometry, so
entanglement entropy becomes an important tool to study quantum gravity via
holography (for example the AdS/CFT Maldacena:1997re ; Witten:1998qj ;
Gubser:1998bc ) and also to study holography itself.

However calculating the entanglement entropy is quite formidable. The standard
way to calculate entanglement entropy in quantum field theories is the replica
trick Calabrese:2004eu ; Calabrese:2005zw ; Callan:1994py . Also, for static
spherical $\mathcal{A}$ in the vacuum state of a general dimensional CFT, the
entanglement entropy can be calculated using the Rindler method111The Rindler
method, as well as its bulk extension under holography, is standardized and
generalised to more general theories in Song:2016gtd ; Jiang:2017ecm .
Casini:2011kv . We will come back to this method later. Remarkably the RT
formula Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt  provide a simple and elegant
way to calculate entanglement entropies holographically. More explicitly, for
a subregion $\mathcal{A}$ in the boundary CFT and a minimal (or more generally
extremal) surface $\mathcal{E}_{\mathcal{A}}$ in the dual AdS bulk that
anchored on the boundary $\partial\mathcal{A}$ of $\mathcal{A}$, the RT
formula states that the entanglement entropy of $\mathcal{A}$ is measured by
the area of $\mathcal{E}_{\mathcal{A}}$ in Planck units

$\displaystyle S_{\mathcal{A}}=\frac{Area(\mathcal{E}_{\mathcal{A}})}{4G}\,,$
(1.2)

which provides a simpler routine to calculate entanglement entropy
geometrically. Though we have several ways to calculate entanglement entropy
in quantum field theories, so far we only get analytic results for highly
symmetric subregions, like intervals, balls, strips and so on.

In this paper we propose another strategy to calculate entanglement entropies
for subregions with less symmetries, like annuli and spherical shells. This
strategy is based on the proposal Wen:2018whg ; Wen:2019ubu  of entanglement
contour. The concept of entanglement contour, introduced in Vidal , is a
function that captures how much the degrees of freedom at each site of
$\mathcal{A}$ contribute to the total entanglement entropy $S_{\mathcal{A}}$.
We denote this function by $f_{\mathcal{A}}(x_{1},\cdots,x_{d-1})$, where the
coordinates $x_{1},\cdots,x_{d-1}$ parameterizes the region $\mathcal{A}$ and
$d$ is the dimension of spacetime. By definition it satisfies the following
basic requirements

$\displaystyle
S_{\mathcal{A}}=\int_{\mathcal{A}}f_{\mathcal{A}}(x_{1},\cdots,x_{d-1})dx_{1}\cdots
dx_{d-1}\,,\qquad f_{\mathcal{A}}(x_{1},\cdots,x_{d-1})\geq 0\,.$ (1.3)

Instead of studying the contour function directly, we study the partial
entanglement entropy $s_{\mathcal{A}}(\mathcal{A}_{2})$ for any subset
$\mathcal{A}_{2}$ of $\mathcal{A}$, which captures the contribution from
$\mathcal{A}_{2}$ to $S_{\mathcal{A}}$ and is defined by

$\displaystyle
s_{\mathcal{A}}(\mathcal{A}_{2})=\int_{\mathcal{A}_{2}}f_{\mathcal{A}}(x_{1},\cdots,x_{d-1})dx_{1}\cdots
dx_{d-1}\,.$ (1.4)

In Wen:2018whg ; Wen:2019ubu , one of the author gave a simple proposal for
the partial entanglement entropy in $d=2$, where
$s_{\mathcal{A}}(\mathcal{A}_{2})$ was proposed as a linear combination of the
entanglement entropies of certain subsets in $\mathcal{A}$.

$\displaystyle
s_{\mathcal{A}}(\mathcal{A}_{2})=\frac{1}{2}\left(S_{\mathcal{A}_{1}\cup\mathcal{A}_{2}}+S_{\mathcal{A}_{2}\cup\mathcal{A}_{3}}-S_{\mathcal{A}_{1}}-S_{\mathcal{A}_{3}}\right)\,.$
(1.5)

The first two terms in the bracket are the entanglement entropies of the
unions between $\mathcal{A}_{2}$ and its left and right subsets
$\mathcal{A}_{1}$ and $\mathcal{A}_{3}$. As was pointed out by Wen:2019ubu
this proposal can be extended to higher dimensions with rotational symmetries
and translation symmetry, where all the regions and subsets are balls (disks),
spherical shells (annuli) or infinitely long strips. In these cases the
contour function respecting the rotation symmetry only depends on one
coordinate, thus the partition is effectively similar to the $d=2$ case. The
additivity can be proved without any additional assumptions for entanglement
entropies. We also use $\mathcal{A}_{1}$ and $\mathcal{A}_{3}$ to denote the
subsets that share boundary with $\mathcal{A}$ while $\mathcal{A}_{2}$ is
inside $\mathcal{A}$. See Fig.1 as an example in 3-dimensions. In higher
dimensions, choosing a general subset $\mathcal{A}_{2}$ may lead to a
partition for $\mathcal{A}$ with more than 3 subsets, thus (1.5) should be
generalised to match with the partition and the prove of additivity may need
further assumptions for the subset entanglement entropies, we leave this for
future studies. It is shown in Wen:2019ubu ; Kudler-Flam:2019oru  that the
partial entanglement entropy defined by (1.5) satisfies all the rational
requirements for entanglement contour222By all the rational requirements we
mean all the requirements known to us so far, this includes the set of
requirements proposed in Vidal .. This proposal has been tested in many
2-dimensional cases, like the vacuum state CFT2 with or without a
spatial(thermal) circle Wen:2019ubu  and warped CFT Wen:2018mev , where the
entanglement contour functions333For attempts to construct the contour
functions using condense matter techniques in lattice models, see Botero ;
Vidal ; PhysRevB.92.115129 ; Coser:2017dtb ; Tonni:2017jom . can be obtained
by using some holographic constructions Wen:2018whg ; Wen:2018mev  or
strategies from some lattice models in condense matter theories Vidal ;
Coser:2017dtb .

Note that the partial entanglement entropy in left-hand side of (1.5) is
finite and regularization scheme-independent, while the right-hand side
consists of four entanglement entropies which are divergent and need to be
regularized. If we want to do any calculation based on (1.5) we need to
introduce regularization schemes for the subset entanglement entropies. The
entanglement entropy generally can be written as an expansion in terms of the
UV cutoff. The power-law divergent terms are scheme-dependent, while the non-
trivial physical information is contained in the universal term (or the finite
term). For example, in 3-dimensions let us consider an annulus with inner and
outer radii $R_{0}<R_{e}$, the expansion of its entanglement entropy is given
by

$\displaystyle
S_{annulus}=\alpha\frac{2\pi(R_{0}+R_{e})}{\epsilon}-h(R_{e}/R_{0})+\mathcal{O}(\epsilon)\,,$
(1.6)

where $\epsilon$ is the UV cutoff, $\alpha$ is a constant relating to the
central charge and $h$ is a function of the ratio $R_{e}/R_{0}$. The first
term is the divergent area term while the second term is universal. (1.5) with
finite partial entanglement entropy implies the necessity of cancelling all
area terms on the right-hand side of (1.5). In addition, we find that a higher
order perturbation of the cutoff $O(\epsilon^{2})$ affects the finite terms
drastically although it does not affect the area terms. Therefore we need
further constraints to regularization schemes for making sure the right-hand
side of (1.5) generate the right partial entanglement entropy. In section 2,
we derive a differential version of (1.5) which involves less cutoffs. More
importantly, A sufficient condition for holding (1.5) is derived. These
sufficient/necessary conditions are the key to validate our proposals, and
finding them is one of the main tasks of this paper.

In section 3 we derive the entanglement contour function for a static ball-
shaped region in vacuum CFT in arbitrary dimensions. We firstly derive it via
the Rindler method Casini:2011kv  which involves a conformal transformation
that maps the flat contour of the thermal entropies in the “Rindler” space to
the entanglement contour for balls. Then we derive the contour function via
the fine structure analysis of the entanglement wedge generalised from
Wen:2018whg . A fine correspondence between points on the ball and its RT
surface can be found from the fine structure, and the contour function can be
read from this fine correspondence. This indeed justifies a former derivation
Kudler-Flam:2019oru  based on a proposal Agon:2018lwq  for the Bit threads
configuration Freedman:2016zud .

In section 4, instead of using the proposal (1.5) to calculate entanglement
contour, given the contour function of $\mathcal{A}$ we propose a prescription
to calculate entanglement entropies for subsets of $\mathcal{A}$ with less
symmetries (compared with balls or strips) via (1.5). This strategy is useful
when we choose the region $\mathcal{A}$ and its partition to be highly
symmetrical so entanglement entropies of the subsets (except the one we want
to derive) are available. More explicitly we consider $\mathcal{A}$ to be a
$(d-1)$-dimensional ball in the vacuum state of CFTd and derive the
entanglement entropy of the spherical shells (or annulus when $d=3$) based on
the proposal for partial entanglement entropy and the entanglement contour
function derived in section 3 for a ball-shaped region.

In section 5, based on our results for the entanglement entropies of annuli
and our proposal, the mutual information across an annulus can be interpreted
as a partial entanglement entropy thus can be calculated analytically. We find
that the mutual information is monogamous and always positive for any ratio
$\frac{R_{e}}{R_{0}}$. This is contradictory with the holographic result which
claims that when the ratio $\frac{R_{e}}{R_{0}}$ is above the critical point
the mutual information will vanish. We think that naively taking the IR
cutoffs for all the RT surfaces at a unified cutoff $z=\epsilon$ in Poincare
coordinates does not give the right mutual information which should be scheme
independent. So our discussion indicates that the phase transition for the
entanglement entropy of annulus (or the mutual information across an annulus)
may be just an illusion.

In section 6, we summarize on the key perspectives in our work which have not
been thoroughly considered before.

##  2 The differential version of the partial entanglement entropy proposal
and regularization scheme independence

Figure 1: In the above figure we set
$\mathcal{A}=\mathcal{A}_{1}\cup\mathcal{A}_{2}\cup\mathcal{A}_{3}$ to be an
annulus whose inner and outer radio is $R_{0}$ and $R_{e}$. It is divided into
three smaller annulues by the two purple circles with radios $R_{1}$ and
$R_{2}$, which satisfies $R_{0}<R_{1}<R_{2}<R_{e}$. The cutoff of the subset
$\mathcal{A}_{i}$ on the boundary circle with radius $R_{j}$ is denoted as
$\epsilon_{iR_{j}}$ while the cutoff of $\mathcal{A}$ on the boundary with
radius $R_{j}$ is denoted as $\epsilon_{R_{j}}$.

We consider an annular region $\mathcal{A}$ and the partition shown in Fig.1.
The region $\mathcal{A}$ and subregions are all chosen to be rotationally
symmetric thus all the arguments for the rationality of our proposal (1.5) in
2-dimensions are still valid without further assumptions on the entanglement
entropies. However unlike in the 2-dimensional cases, (1.5) requires us to
match at the sub-leading order thus a careful handling of the cutoffs is
necessary. As in the following we encounter lots of cutoffs, it is useful to
set the rule for notations of cutoffs. In this paper we denote the cutoffs of
the subset $\mathcal{A}_{i}$ on the boundary $\Sigma_{j}$ as
$\epsilon_{i\Sigma_{j}}$ and denote the cutoffs of $\mathcal{A}$ as
$\epsilon_{\Sigma_{j}}$.

The cancellation of all the area terms on the right hand side of (1.5) only
requires that all these cutoffs equal at the leading order, i.e.,

$\displaystyle\epsilon_{iR_{j}}=\epsilon+g_{iR_{j}}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,,$
(2.7)

where $g_{iR_{j}}$ are constants. We denote that

  * •

the condition (2.7) is the first necessary condition for the validity of the
partial entanglement entropy proposal (1.5).

It is important to note that the sub-leading perturbation of the cutoffs
affects the finite terms drastically, for example

$\displaystyle\frac{2\pi R_{j}}{\epsilon_{iR_{j}}}=\frac{2\pi
R_{j}}{\epsilon}-g_{i\Sigma_{j}}(2\pi R_{j})+\mathcal{O}(\epsilon)\,.$ (2.8)

If we plug regularized entanglement entropies into (1.5), we need to set
constraints on the regularization schemes (or the constants $g_{i\Sigma_{j}}$)
in order that the left hand side of (1.5) is the right partial entanglement
entropy. This is nontrivial. In the following we derive a differential version
of the proposal (1.5) which involves less cutoffs and from which we can
extract the scheme-independent contour function. Then the partial entanglement
entropy can be calculated by integrating the contour function over the
subregions.

###  2.1 Differential version of the proposal

Assuming the existence of the entanglement contour function for $\mathcal{A}$,
the contour function depends on the boundaries of $\mathcal{A}$. We write

$\displaystyle f_{\mathcal{A}}(r)=f(R_{0},R_{e},r)\,,$ (2.9)

and define

$\displaystyle F(R_{0},R_{e},r)=\int f(R_{0},R_{e},r)dr.$ (2.10)

The entanglement entropies and partial entanglement entropies are given by

$\displaystyle S_{\mathcal{A}}$
$\displaystyle=\int_{R_{0}}^{R_{e}}f(R_{0},R_{e},r)\sqrt{g}dr=F(R_{0},R_{e},R_{e})-F(R_{0},R_{e},R_{0})\,,$
(2.11) $\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})$
$\displaystyle=\int_{R_{1}}^{R_{2}}f(R_{0},R_{e},r)\sqrt{g}dr=F(R_{0},R_{e},R_{2})-F(R_{0},R_{e},R_{1})\,,$
(2.12)

where $g$ is the determinant of the induced metric on the spatial slice of the
system. On the other hand, according to the proposal (1.5) we have

$\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})=$
$\displaystyle\frac{1}{2}\left(S_{\mathcal{A}_{1}\cup\mathcal{A}_{2}}-S_{\mathcal{A}_{3}}\right)+\frac{1}{2}\left(S_{\mathcal{A}_{2}\cup\mathcal{A}_{3}}-S_{\mathcal{A}_{1}}\right)\,,$
(2.13)

where these entanglement entropies can be written as

$\displaystyle\frac{1}{2}\left(S_{\mathcal{A}_{1}\cup\mathcal{A}_{2}}-S_{\mathcal{A}_{3}}\right)$
$\displaystyle=\frac{1}{2}\left(F(R_{0},R_{2},R_{2})-F(R_{0},R_{2},R_{0})-F(R_{2},R_{e},R_{e})+F(R_{2},R_{e},R_{2})\right)\,,$
(2.14)
$\displaystyle\frac{1}{2}\left(S_{\mathcal{A}_{2}\cup\mathcal{A}_{3}}-S_{\mathcal{A}_{1}}\right)$
$\displaystyle=\frac{1}{2}\left(F(R_{1},R_{e},R_{e})-F(R_{1},R_{e},R_{1})-F(R_{0},R_{1},R_{1})+F(R_{0},R_{1},R_{0})\right)\,.$
(2.15)

We define the right hand of (2.14) to be a new function
$G(R_{0},R_{e},R_{2})$, then interestingly we find the right hand side of
(2.15) can be written as $-G(R_{0},R_{e},R_{1})$, thus according to (2.13) and
(2.12) we have

$\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})=$ $\displaystyle
G(R_{0},R_{e},R_{2})-G(R_{0},R_{e},R_{1})$ (2.16) $\displaystyle=$
$\displaystyle F(R_{0},R_{e},R_{2})-F(R_{0},R_{e},R_{1})\,.$ (2.17)

Since the above equation holds for any choice of $R_{1}$ and $R_{2}$, we
conclude that these two functions only differ by a constant depending the
region $\mathcal{A}$ (or namely depending on $R_{0}$ and $R_{e}$), i.e.,

$\displaystyle
F(R_{0},R_{e},R_{1})=G(R_{0},R_{e},R_{1})+G_{0}(R_{0},R_{e})\,.$ (2.18)

Note that the constant $G_{0}(R_{0},R_{e})$ may be divergent which are
cancelled in the partial entanglement entropy and the entanglement contour.

Then we choose $\mathcal{A}_{2}$ to be an annulus with an infinitesimal width,
i.e.,

$\displaystyle R_{2}=R_{1}+dr\,,\qquad dr\to 0\,.$ (2.19)

For this choice we have

$\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})=f(R_{0},R_{e},r)\sqrt{g}dr\,,$
(2.20)

and furthermore according to (2.16) we should have

$\displaystyle
f(R_{0},R_{e},r)=\frac{G(R_{0},R_{e},R_{1}+dr)-G(R_{0},R_{e},R_{1})}{\sqrt{g}\,dr}\,,$
(2.21)

or by $dr\to 0$,

$\displaystyle
f(R_{0},R_{e},r)=\frac{1}{2\sqrt{g}}\partial_{R_{1}}\left(S_{\mathcal{A}_{1}}-S_{\mathcal{A}_{3}}\right)|_{R_{1}\to
r}.$ (2.22)

Note that in the above equation, $\mathcal{A}_{2}$ is absorbed by
$\mathcal{A}_{3}$ thus $\mathcal{A}=\mathcal{A}_{1}\cup\mathcal{A}_{3}$.

This derivation of the contour function based on the proposal (1.5) can be
naturally generalized to arbitrary dimensions with rotational or translational
symmetry. (2.22) is the differential version of (1.5) (see also Kudler-
Flam:2019oru  for the related discussion on the differential version of (1.5)
in 2 dimensions). Note that in the differential version (2.22) only two subset
entanglement entropies need to be regularized and the differential helps us to
remove other dependence on cutoffs. So far we have not introduced any
regularization scheme yet. In the next subsection we will plug in regularized
entanglement entropies to derive the regularization independent contour
function.

###  2.2 The scheme-independent entanglement contour function

Recall that the entanglement entropy integrates the contour function, i.e.

$\displaystyle
S_{\mathcal{A}_{1}}=F(R_{0},R_{1},R_{1})-F(R_{0},R_{1},R_{0})\,.$ (2.23)

As the entanglement entropy satisfies the area law, functions like
$F(R_{0},R_{1},r)$ diverge when $r$ approaches the radius of the two
boundaries $R_{0}$ and $R_{1}$. We write the $F$ functions as expansions near
the boundaries, for example,

$\displaystyle F(R_{0},R_{1},r)|_{r\to R_{0}}=$ $\displaystyle\alpha\frac{2\pi
R_{0}}{R_{0}-r}-\mathcal{F}_{1R_{0}}(R_{0},R_{1})+\mathcal{O}(r-R_{0})$ (2.24)
$\displaystyle F(R_{0},R_{1},r)|_{r\to R_{1}}=$ $\displaystyle\alpha\frac{2\pi
R_{1}}{R_{1}-r}+\mathcal{F}_{1R_{1}}(R_{0},R_{1})+\mathcal{O}(R_{1}-r)$ (2.25)

Note that the $\mathcal{F}$ functions are independent of regularization
scheme.

We now introduce cutoffs. As was pointed out in Wen:2018whg , the regularized
entanglement entropy should be interpreted as the partial entanglement entropy
of the regularized region. In other words cutoffs are just the limits of
integration for the contour function (later we call the cutoffs defined in
this way as the direct cutoffs). So we should have, for example

$\displaystyle S_{\mathcal{A}_{1}}^{reg}=$ $\displaystyle
F(R_{0},R_{1},R_{1}-\epsilon_{1R_{1}})-F(R_{0},R_{1},R_{0}+\epsilon_{1R_{0}})$
(2.26) $\displaystyle=$ $\displaystyle\alpha\left(\frac{2\pi
R_{0}}{\epsilon_{1R_{0}}}+\frac{2\pi
R_{1}}{\epsilon_{1R_{1}}}\right)+\mathcal{F}_{1R_{1}}(R_{0},R_{1})+\mathcal{F}_{1R_{0}}(R_{0},R_{1})+\mathcal{O}(\epsilon)\,.$
(2.27)

We also perform the similar expansion for $S_{\mathcal{A}_{3}}$ and write the
cutoffs as in (2.7) with their higher order perturbations $g_{iR_{j}}$
unsettled. Put all these expansions together we find

$\displaystyle S_{\mathcal{A}_{1}}-S_{\mathcal{A}_{3}}=$
$\displaystyle\frac{2\pi(R_{e}-R_{0})}{\epsilon}-2\pi R_{e}g_{1R_{e}}+2\pi
R_{0}g_{3R_{0}}+\mathcal{O}(\epsilon)$ (2.28)
$\displaystyle+\mathcal{F}_{1R_{0}}(R_{0},R_{1})+\mathcal{F}_{1R_{1}}(R_{0},R_{1})-\mathcal{F}_{3R_{1}}(R_{1},R_{e})-\mathcal{F}_{3R_{e}}(R_{1},R_{e})$
(2.29) $\displaystyle-2\pi R_{1}g_{1R_{1}}+2\pi R_{1}g_{3R_{1}}\,.$ (2.30)

It is easy to see terms in the first line are all independent of $R_{1}$. Then
we perform the partial derivative with respect to $R_{1}$ thus get rid of the
cutoffs on $R_{0}$ and $R_{e}$, and find

$\displaystyle\partial_{R_{1}}\left(S_{\mathcal{A}_{1}}-S_{\mathcal{A}_{3}}\right)=$
$\displaystyle\partial_{R_{1}}\left(\mathcal{F}_{1R_{0}}(R_{0},R_{1})+\mathcal{F}_{1R_{1}}(R_{0},R_{1})-\mathcal{F}_{3R_{1}}(R_{1},R_{e})-\mathcal{F}_{3R_{e}}(R_{1},R_{e})\right)$
(2.31) $\displaystyle+2\pi(g_{3R_{1}}-g_{1R_{1}})\,.$ (2.32)

We see that the first line consists of scheme-independent functions while the
second line depend on the cutoffs on the boundary between $\mathcal{A}_{1}$
and $\mathcal{A}_{3}$. Since the contour function should also be scheme-
independent, it is natural to conclude that,

  * •

the sufficient condition for the validity of the differential version of the
partial entanglement entropy proposal (2.22) is that, the cutoffs on $\Sigma$
which divide $\mathcal{A}$ into $\mathcal{A}_{1}\cup\mathcal{A}_{3}$ satisfy
$\epsilon_{1\Sigma}=\epsilon_{3\Sigma}$ (or up to a perturbation of order
$\mathcal{O}(\epsilon^{3})$ since we ignore the $\mathcal{O}(\epsilon)$ terms
on the right hand side of (2.22)).

Here $\Sigma$ denote the boundary between $\mathcal{A}_{1}$ and
$\mathcal{A}_{3}$ that determines the partition
$\mathcal{A}=\mathcal{A}_{1}\cup\mathcal{A}_{3}$. Note that

  * •

the condition $\epsilon_{1\Sigma}=\epsilon_{3\Sigma}$ is also the second
necessary condition for the validity of the original proposal (1.5) .

We also would like to point out that requiring all the cutoffs to be equal
$\epsilon_{i\Sigma_{j}}=\epsilon$ (the first and second necessary conditions
for (1.5) are both satisfied) does not guarantee the validity of (1.5). When
we write the right hand side of (1.5) as expansions of the $F$ functions and
set $\epsilon_{i\Sigma_{j}}=\epsilon$, the area terms will cancel thus we get
a linear combination of the $\mathcal{F}$ functions which is also scheme-
independent. However it is possible to differ from the partial entanglement
entropy on the left hand side by a constant. We will show this with an
explicit example later in section 5.

##  3 Entanglement contour for balls in holographic CFTs

###  3.1 Entanglement contour from the Rindler method

The Rindler method is developed in Casini:2011kv  to calculate holographic
entanglement entropies for static ball-shaped regions in the context of
AdSd+1/CFTd, and justify the RT formula in these cases. Here we focus on the
story on the field theory side, where a static spherical entangling surface
$S^{d-2}$ is considered in the vacuum state of a CFTd. We denote the ball-
shaped region as $\mathcal{A}$, the conformal transformation $\mathcal{R}$
that maps the causal development $\mathcal{D}_{\mathcal{A}}$ of $\mathcal{A}$
to a Rindler space $\tilde{B}$ (which is a hyperbolic space) can be found.
Therefore the entanglement entropy $S_{\mathcal{A}}$ is mapped to the thermal
entropy $S_{\tilde{\mathcal{B}}}$ of $\tilde{\mathcal{B}}$.

Here we would like to propose that

  * •

the Rindler transformation $\mathcal{R}$ is not only a mapping between thermal
entropy $S_{\tilde{\mathcal{B}}}$ and the entanglement entropy
$S_{\mathcal{A}}$, but also a mapping between the contour function of
$S_{\tilde{\mathcal{B}}}$ and the entanglement contour function
$f_{\mathcal{A}}(r)$ of $S_{\mathcal{A}}$.

We believe that this idea is equivalent to the idea Cardy:2016fqc ;
Coser:2017dtb  of identifying the inverse of the local weight function, which
multiplies the local operator $T_{00}$ in the corresponding modular
Hamiltonian $K_{\mathcal{A}}$, as the entanglement contour function, i.e.,
$K_{\mathcal{A}}\propto\int_{x\in\mathcal{A}}\frac{T_{00}}{f_{\mathcal{A}}(x)}dx^{d-1}$.

In the following we list the metrics on $\mathcal{B}$ and
$\tilde{\mathcal{B}}$, and the Rindler transformation444To map from
$\mathcal{B}$ to $\tilde{\mathcal{B}}$, the Rindler transformation
$\mathcal{R}$ should be followed by a Weyl transformation that eliminate the
overall prefactor of the metric in $\tilde{\mathcal{B}}$. We didnot write it
down because it is spurionic Jensen:2013lxa  thus does not change the thermal
partition function in $\tilde{\mathcal{B}}$. $\mathcal{R}$ between them
Casini:2011kv ,

$\displaystyle\mathcal{B}:~{}~{}ds^{2}$
$\displaystyle=-dt^{2}+dr^{2}+r^{2}d\Omega_{d-2}^{2}\,,$ (3.33)
$\displaystyle\tilde{\mathcal{B}}:~{}~{}ds^{2}$
$\displaystyle=-d\tau^{2}+R^{2}\left(du^{2}+\sinh^{2}ud\Omega_{d-2}^{2}\right)\,,$
(3.34) $\displaystyle\mathcal{R}:~{}~{}~{}\Big{\\{}t$
$\displaystyle=R\frac{\sinh(\tau/R)}{\cosh u+\cosh(\tau/R)}\,,\quad
r=R\frac{\sinh u}{\cosh u+\cosh(\tau/R)}\Big{\\}}\,.$ (3.35)

Since we only consider static regions, we focus on the time slice $t=\tau=0$,
where the Rindler transformation reduces to

$\displaystyle\mathcal{R}:~{}r=R\tanh(\frac{u}{2})\,.$ (3.36)

It is clear from (3.36) that the region $0<u<\infty$ in the Rindler space
$\tilde{\mathcal{B}}$ covers the ball $0<r<R$ in the original spacetime
$\mathcal{B}$. Also the UV cutoff $\epsilon$ of the ball and the cutoff
$u_{\text{max}}$ for the size of the Rindler space $\tilde{\mathcal{B}}$ is
related by $R-\epsilon=R\tanh(u_{\text{max}})\,.$

The contour function $f_{\tilde{\mathcal{B}}}(u)$ is just a constant (the
contour of the thermal entropy in $\tilde{\mathcal{B}}$ is flat)

$\displaystyle
f_{\tilde{\mathcal{B}}}(u)=a=\frac{S_{\tilde{\mathcal{B}}}}{Volume(\tilde{\mathcal{B}})}.$
(3.37)

According to the AdS/CFT, the thermal entropy $S_{\tilde{\mathcal{B}}}$ equals
to the thermal entropy of a $(d+1)$-dimensional hyperbolic black hole

$\displaystyle
ds^{2}=\frac{d\rho^{2}}{\rho^{2}/L^{2}-1}-(\rho^{2}/L^{2}-1)d\tilde{\tau}^{2}+\rho^{2}(du^{2}+\sinh^{2}ud\Omega_{d-2}^{2})\,,$
(3.38)

where $L=Re^{\beta}$ is the AdS radius, $\tilde{\tau}=e^{-\beta}\tau$ and
$\beta$ is a constant. Using Wald’s entropy formula the thermal entropy of the
hyperbolic black hole is then given by (see CHMbase4 ; Casini:2011kv  for
details)

$\displaystyle
S_{\tilde{\mathcal{B}}}=\frac{2\Gamma(d/2)}{\pi^{d/2-1}}\frac{a^{*}_{d}}{L^{d-1}}\int_{0}^{u_{\text{max}}}du~{}L^{d-1}(\sinh
u)^{d-2}\Omega_{d-2}$ (3.39)

where $a^{*}_{d}$ is a central charge that characterizes the degrees of
freedom in the boundary CFT CHMbase4 . Note that

$\displaystyle
Volume(\tilde{\mathcal{B}})=\int_{0}^{u_{\text{max}}}du~{}R^{d-1}(\sinh
u)^{d-2}\Omega_{d-2}\,,$ (3.40)

thus we have

$\displaystyle a=\frac{2\Gamma(d/2)}{\pi^{d/2-1}}\frac{a^{*}_{d}}{R^{d-1}}$
(3.41)

Since the thermal entropy $S_{\tilde{\mathcal{B}}}$ equals to the entanglement
entropy $S_{\mathcal{A}}$, we have

$\displaystyle
S_{\mathcal{A}}=\int_{0}^{R-\epsilon}drf_{\mathcal{A}}(r)r^{d-2}\Omega_{d-2}=a\int_{0}^{u_{\text{max}}}duR^{d-1}(\sinh
u)^{d-2}\Omega_{d-2}\,.$ (3.42)

Plugging (3.36) and (3.41) into the above equation we get the contour function
for $(d-1)$-dimensional balls with radius $R$,

$\displaystyle
f_{\mathcal{A}}(r)=\frac{c}{6}\left(\frac{2R}{R^{2}-r^{2}}\right)^{d-1}$
(3.43)

where we define the new constant
$\frac{c}{6}=a^{*}_{d}\frac{2\Gamma(d/2)}{\pi^{d/2-1}}$. In the case where
$d$, the dimension of the CFT, is even, $a_{d}^{*}$ is precisely equal to the
coefficient of the A-type trace anomaly in the CFT. For example when $d=2$, we
have $a_{2}^{*}=\frac{c}{12}$, as expected we have

$\displaystyle
a^{*}_{d}\frac{2\Gamma(d/2)}{\pi^{d/2-1}}\Big{|}_{d=2}=\frac{c}{6}\,.$ (3.44)

In (3.43) we have abused the above relation to arbitrary dimensions.

As a consistency check, we consider the $d=2$ case. According to (3.43) and
(3.44) we have

$\displaystyle
S_{\mathcal{A}}=2\int_{r=0}^{r=R-\epsilon}f_{\mathcal{A}}(r)dr=\frac{c}{3}\log\frac{2R}{\epsilon}+\mathcal{O}(\epsilon)$
(3.45)

which as expected is the entanglement entropy for an interval with length
$l=2R$ in the vacuum CFT2 on a plane.

So far our results are valid for holographic CFTs because we used holography
to calculate the thermal entropy $S_{\tilde{\mathcal{B}}}$. When $d=2$,
$S_{\tilde{\mathcal{B}}}$ can be calculated by Cardy-formula without using
holography. In addition, the Rindler method has been generalized to some
2-dimensional field theories like the warped CFT and field theories invariant
under the BMS3 group (BMSFT) in Song:2016gtd ; Jiang:2017ecm . In these two
cases, the thermal entropy of the Rindler space, as well as their entanglement
contour functions, can be calculated using some Cardy-like formulas (see also
Detournay:2012pc ; Barnich:2012xq ; Bagchi:2012xr ; Castro:2015csg ) without
holography.

###  3.2 Entanglement contour from the fine structure of entanglement wedge

We can also read the contour function for ball-shaped regions from the fine
structure analysis of the entanglement wedge Wen:2018whg . In the context of
AdS3/CFT2, a holographic picture for the entanglement contour of a single
interval is given in Wen:2018whg . It is shown that the entanglement wedge can
be sliced by the modular planes, which are defined as the orbits of the
boundary modular flow lines under the bulk modular flow. Each modular plane
will intersect with the interval $\mathcal{A}$ on a point $\mathcal{A}(x_{0})$
and its RT surface $\mathcal{E}_{\mathcal{A}}$ on
$\mathcal{E}_{\mathcal{A}}(\bar{x}_{0})$. Furthermore, cyclically gluing the
point $\mathcal{A}(x_{0})$ on $\mathcal{A}$ turns on the nonzero contribution
to the $S_{\mathcal{A}}$ on $\mathcal{E}_{\mathcal{A}}(\bar{x}_{0})$, thus
gives a fine correspondence between the points on $\mathcal{A}$ and
$\mathcal{E}_{\mathcal{A}}$. In the same sense, the partial entanglement
entropy $s_{\mathcal{A}}(\mathcal{A}_{i})$ is just given by

$\displaystyle
s_{\mathcal{A}}(\mathcal{A}_{i})=\frac{Area\left(\mathcal{E}_{i}\right)}{4G}\,.$
(3.46)

where $\mathcal{E}_{i}$ is just the part of $\mathcal{E}_{\mathcal{A}}$ that
correspond the subset $\mathcal{A}_{i}$ under this fine correspondence. See
Fig.2 for a graphical description of this construction. Also the similar
construction is conducted for WCFT in Wen:2018mev  in the context of AdS3/WCFT
correspondence Detournay:2012pc ; Compere:2013bya .

Figure 2: The left figure shows the slicing of the entanglement wedge with
modular planes (the blue surface), while the left figure shows the
correspondence between the partial entanglement entropies and chords on
$\mathcal{E}_{\mathcal{A}}$.

We would like to generalize the above construction to higher dimensions and
read the contour functions for ball-shaped regions holographically. The key of
the Rindler method is the construction of the Rindler transformation that maps
the entanglement wedge $\mathcal{W}_{\mathcal{A}}$ of a $d$-dimensional static
ball $\mathcal{A}$ in Poincare AdSd+1 to a hyperbolic black hole
$\widetilde{AdS}_{d+1}$ (3.38), which can be written as

$\displaystyle\widetilde{AdS}_{d+1}=AdS_{2}\times\mathcal{H}_{d-1}\,.$ (3.47)

where $\mathcal{H}_{d-1}$ is just the $(d-1)$-dimensional hyperbolic plane.
The Rindler transformation is a conformal symmetry transformation of the field
theory. The static ball $\mathcal{A}$ is mapped to a time slice
$\tilde{\mathcal{A}}$ on the boundary of the $\widetilde{AdS}_{d+1}$, while
the RT surface $\mathcal{E}_{\mathcal{A}}$ is mapped to the black hole horizon
$\rho=L$ of the hyperbolic black hole, which we call
$\tilde{\mathcal{E}}_{\mathcal{A}}$. Since $\widetilde{AdS}_{d+1}$ corresponds
to a thermal state, the modular Hamiltonian is just the ordinary Hamiltonian
in $\widetilde{AdS}_{d+1}$. In other words the bulk modular flow in
$\mathcal{W}_{\mathcal{A}}$ maps to the time evolution in
$\widetilde{AdS}_{d+1}$. The modular planes in $\widetilde{AdS}_{d+1}$ are
simply the AdS2 planes with a fixed point on the hyperbolic plane
$\mathcal{H}_{d-1}$.

Unlike the strategy in Wen:2018whg  which construct modular planes directly in
$\mathcal{W}_{\mathcal{A}}$, here we study the much simpler fine structure in
the Rindler $\widetilde{AdS}_{d+1}$. This gives a fine correspondence between
the points on $\tilde{\mathcal{A}}$ and the points on
$\tilde{\mathcal{E}}_{\mathcal{A}}$: A point on $\tilde{\mathcal{A}}$
corresponds to another point on $\tilde{\mathcal{E}}_{\mathcal{A}}$ when they
are on the same AdS2 plane, in other words, they have the same coordinates in
$\mathcal{H}_{d-1}$. This fine correspondence in $\widetilde{AdS}_{d+1}$ is
mapped to the fine correspondence between points on $\mathcal{A}$ and the
points on $\mathcal{E}_{\mathcal{A}}$ in the original AdSd+1.

Figure 3: Here we have chosen a time slice and fixed all the angular
coordinates in the original Poincare AdSd+1 as well as the hyperbolic black
hole $\widetilde{AdS}_{d+1}$ (3.38). The dashed purple lines are the
intersection between modular planes and the time slice, they are also
geodesics normal to the RT surface $\mathcal{E}_{\mathcal{A}}$ (the horizon of
the hyperbolic black holes $\tilde{\mathcal{E}}_{\mathcal{A}}$). The two end
points of these dashed lines are related by the fine correspondence (3.48)
between the points on $\mathcal{A}$ ($\tilde{\mathcal{A}}$) and
$\mathcal{E}_{\mathcal{A}}$ ($\tilde{\mathcal{E}}_{\mathcal{A}}$).

Since the region $\mathcal{A}$ is static has the rotation symmetry, the
picture can be simplified by only considering a time slice $t=0$ and fixing
all the angle coordinates. The homology surface $\mathcal{R}_{\mathcal{A}}$
satisfies
$\partial\mathcal{R}_{\mathcal{A}}=\mathcal{A}\cup\mathcal{E}_{\mathcal{A}}$.
The $\mathcal{R}_{\mathcal{A}}$ at the time slice $t=0$ is mapped to the time
slice $\tilde{\tau}=0$ in $\widetilde{AdS}_{d+1}$, where the image is called
$\tilde{\mathcal{R}}_{\mathcal{A}}$. The modular planes in
$\widetilde{AdS}_{d+1}$ intersect with $\tilde{\mathcal{R}}_{\mathcal{A}}$ on
lines along the $\rho$ coordinate. These lines end on the horizon
$\tilde{\mathcal{E}}_{\mathcal{A}}$ orthogonally (see the dashed purple lines
in the right figure of Fig.3). The fine structure in $\widetilde{AdS}_{d+1}$
also implies that the contour function for the thermal entropy of the
hyperbolic black hole is flat. It is useful to note that these intersection
lines are also geodesics in $\widetilde{AdS}_{d+1}$ and normal to the horizon
$\tilde{\mathcal{E}}_{\mathcal{A}}$. This implies that when mapping back to
$\mathcal{W}_{\mathcal{A}}$, the pair of points that corresponded to each
other are connected by geodesics that start from one point on $\mathcal{A}$
and normally anchored on the other points on $\mathcal{E}_{\mathcal{A}}$ (see
the dashed purple lines in the left figure of Fig.3). We set the center of
$\mathcal{A}$ at $r=0$, then after some simple calculations we find that the
points with radius $r$ in $\mathcal{A}$ correspond to the points with radius
$\bar{r}$ on the RT surface $\mathcal{E}_{\mathcal{A}}$, via the following
relation

$\displaystyle r=\frac{\bar{r}R}{\sqrt{R^{2}-\bar{r}^{2}}+R}$ (3.48)

Note that the above relation does not depend on the spacetime dimension.

The RT surface is given by

$\displaystyle\mathcal{E}_{\mathcal{A}}:~{}~{}z^{2}=R^{2}-r^{2}\,.$ (3.49)

According to the fine correspondence, we have

$\displaystyle
f_{\mathcal{A}}(r)r^{d-2}\Omega_{d-2}dr=\frac{1}{4G}\sqrt{\frac{R^{2}\bar{r}^{2(d-2)}}{(R^{2}-\bar{r}^{2})^{d}}}\Omega_{d-2}d\bar{r}\,,$
(3.50)

where the left-hand side is the partial entanglement entropy of a thin annulus
at $r$ with the infinitesimal width $dr$ while the right hand side is the area
of the subregion on the RT surface that correspond to the annulus. Plug the
fine correspondence relation (3.48) into the above equation, we immediately
get

$\displaystyle
f_{\mathcal{A}}(r)=\frac{1}{4G}\left(\frac{2R}{R^{2}-r^{2}}\right)^{d-1}$
(3.51)

which reproduces the result (3.43) we get via the Rindler method.

The entanglement contour functions (3.43) for static balls in $d$-dimensional
CFTs has been proposed in an earlier paper Kudler-Flam:2019oru  based on
certain explicit constructions Agon:2018lwq  of bit thread configurations
Freedman:2016zud  on the gravity side. In Agon:2018lwq  the bit threads in
$\mathcal{W}_{\mathcal{A}}$ of static balls are assumed to follow the bulk
geodesics normal to the RT surface. In our construction, these bit threads are
simply the lines where the modular planes intersect with the homology surface
$\mathcal{R}_{\mathcal{A}}$ at the time slice $t=0$. Our fine structure
analysis is consistent with the constructions in Kudler-Flam:2019oru ;
Agon:2018lwq .

###  3.3 An exact relation between the bulk and boundary cutoffs

An important proposal in Wen:2018whg  is that, the regularized entanglement
entropy is the partial entanglement entropy of the regularized region. This
means the UV cutoff of the boundary field theory and the IR cutoff in the bulk
are related by the fine correspondence between the points in $\mathcal{A}$ and
$\mathcal{E}_{\mathcal{A}}$. In the cases of static spherical regions,
consider $\mathcal{A}$ is cut off at $R-\epsilon$ and
$\mathcal{E}_{\mathcal{A}}$ is cut off at $z=\bar{\epsilon}$, then according
to the fine correspondence (3.48) we have

$\displaystyle
R-\epsilon=\frac{\bar{r}R}{\sqrt{R^{2}-\bar{r}^{2}}+R}\,,\qquad\bar{r}=\sqrt{R^{2}-\bar{\epsilon}^{2}}\,.$
(3.52)

This gives an exact relation between the boundary and bulk cutoffs

$\displaystyle\epsilon$
$\displaystyle=\frac{R\left(R+\bar{\epsilon}-\sqrt{(R^{2}-\bar{\epsilon}^{2})}\right)}{R+\bar{\epsilon}}$
(3.53)
$\displaystyle=\bar{\epsilon}-\frac{\bar{\epsilon}^{2}}{2R}+\mathcal{O}\left(\bar{\epsilon}^{3}\right)\,.$
(3.54)

When we calculate the regularized entanglement entropy from both the field
theory side and the bulk gravity side, the
$\mathcal{O}\left(\bar{\epsilon}^{2}\right)$ correction does not affect the
leading contribution (the area law divergent term) to the two entanglement
entropies. However it will affect the finite term contributions. When we match
the entanglement entropy calculated by the RT formula to the entanglement
entropy integrated from the contour function, the UV and IR cutoffs should
satisfy the fine correspondence (3.53) and the
$\mathcal{O}\left(\bar{\epsilon}^{2}\right)$ correction should be considered.

##  4 Entanglement entropies and entanglement contour for annuli

###  4.1 The strategy

Figure 4: On the left figure we consider the region
$\mathcal{A}=\mathcal{A}_{1}\cup\mathcal{A}_{2}$ is a disk, which is divided
into a smaller concentric disk $\mathcal{A}_{2}$ and a concentric annulus
$\mathcal{A}_{2}$. On the right figure we consider the partition of another
region
$\mathcal{A}^{\prime}=\mathcal{A}_{1}^{\prime}\cup\mathcal{A}_{2}^{\prime}$,
and $\mathcal{A}_{1}^{\prime}$ equals to the annulus $\mathcal{A}_{1}$ in the
right figure of Fig.4.

In this section we firstly consider the region $\mathcal{A}$ to be a disk and
its partition into a concentric smaller disk $\mathcal{A}_{1}$ and an annulus
$\mathcal{A}_{2}$, see the left figure in Fig. 4. Since there are only two
relevant subsets the proposal (1.5) reduces to the following equation

$\displaystyle
S_{\mathcal{A}_{1}}=S_{\mathcal{A}}+S_{\mathcal{A}_{2}}-2s_{\mathcal{A}}(\mathcal{A}_{2})\,.$
(4.55)

Similarly for the configuration in the right figure of Fig. 4 we have

$\displaystyle S_{\mathcal{A}^{\prime}_{1}}=$ $\displaystyle
S_{\mathcal{A}^{\prime}}+S_{\mathcal{A}^{\prime}_{2}}-2s_{\mathcal{A}^{\prime}}(\mathcal{A}^{\prime}_{2})\,,$
(4.56)

Our strategy to derive the entanglement entropy for annuli is in the following

  1. 1.

Calculate the partial entanglement entropies
$s_{\mathcal{A}}(\mathcal{A}_{2})$ and
$s_{\mathcal{A}^{\prime}}(\mathcal{A}^{\prime}_{2})$.

  2. 2.

Consider the left figure in Fig.4 and calculate the entanglement entropy
$S_{\mathcal{A}_{1}}$ of the annulus using (4.55). Note that the
$S_{\mathcal{A}_{1}}$ we obtained in this way is indirectly regularized by the
cutoffs (we call them indirect cutoffs of $\mathcal{A}_{1}$) of the two disks
$\mathcal{A}$ and $\mathcal{A}_{2}$. The second necessary condition for the
partial entanglement entropy proposal allows us to replace the cutoff of
$\mathcal{A}_{2}$ with the direct cutoff of $\mathcal{A}_{1}$ on its inner
boundary. So we can write the entanglement entropy for an annulus with a
direct cutoff at the inner boundary.

  3. 3.

Similarly consider the configuration in the right figure we can write the
entanglement entropy of an annulus with a direct cutoff at the outer boundary.

  4. 4.

Consider the configuration in Fig.5 where an annulus $A$ is subdivided by a
circle $\Sigma$ into two smaller annuli $A_{1}$ and $A_{3}$. Then based on
results from previous two steps, we get the $S_{A_{3}}$ with a direct cutoff
on the inner boundary and write $S_{A_{1}}$ with a direct cutoff on its outer
boundary. Set these two direct cutoffs on $\Sigma$ to be equal so that the
sufficient condition for the differential version of our proposal (2.22) is
satisfied, then we can derive the entanglement contour function for annuli.

  5. 5.

Integrating the contour function of $A$ to get the entanglement entropy for
annuli with direct cutoffs on both the inner and outer boundaries.

###  4.2 Indirectly regularized entanglement entropies for annuli

Consider the left figure in Fig.4, and set the radius of $\mathcal{A}$ to be
$R_{e}$ and the radius of $\mathcal{A}_{2}$ to be $R_{0}$. The entanglement
contour function for disks is given by (3.43) with $d=3$. So we can easily get
the regularized entanglement entropy for disks $\mathcal{A}$ and
$\mathcal{A}_{2}$,

$\displaystyle S_{\mathcal{A}}=$
$\displaystyle\int_{0}^{R_{e}-\epsilon_{R_{e}}}\frac{c}{6}\left(\frac{2R_{e}}{R_{e}^{2}-r^{2}}\right)^{2}2\pi
r\,dr=\frac{2\pi
c}{3}\frac{(R_{e}-\epsilon_{R_{e}})^{2}}{\epsilon_{R_{e}}(2R_{e}-\epsilon_{R_{e}})}\,,$
(4.57) $\displaystyle S_{\mathcal{A}_{2}}=$
$\displaystyle\int_{0}^{R_{0}-\epsilon_{2R_{0}}}\frac{c}{6}\left(\frac{2R_{0}}{R_{0}^{2}-r^{2}}\right)^{2}2\pi
r\,dr=\frac{2\pi
c}{3}\frac{(R_{0}-\epsilon_{2R_{0}})^{2}}{\epsilon_{2R_{0}}(2R_{0}-\epsilon_{2R_{0}})}\,.$
(4.58)

The partial entanglement entropy is calculated by

$\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})$
$\displaystyle=\int_{0}^{R_{0}}\frac{c}{6}\left(\frac{2R_{e}}{R_{e}^{2}-r^{2}}\right)^{2}2\pi
r\,dr=\frac{2\pi cR_{0}^{2}}{3\left(R_{e}^{2}-R_{0}^{2}\right)}\,.$ (4.59)

Then according to (4.55) we can straightforwardly get

$\displaystyle S_{\mathcal{A}_{1}}=\frac{2\pi
c}{3}\left(\frac{(R_{e}-\epsilon_{R_{e}})^{2}}{(2R_{e}-\epsilon_{R_{e}})\epsilon_{R_{e}}}+\frac{(R_{0}-\epsilon_{2R_{0}})^{2}}{(2R_{0}-\epsilon_{2R_{0}})\epsilon_{2R_{0}}}-\frac{2R_{0}^{2}}{R_{e}^{2}-R_{0}^{2}}\right)\,.$
(4.60)

Note that the above $S_{\mathcal{A}_{1}}$ is regularized indirectly by the
cutoffs $\epsilon_{R_{e}}$ and $\epsilon_{2R_{0}}$ of the two disks
$\mathcal{A}_{2}$ and $\mathcal{A}$. The second necessary condition for the
validity of (4.55) is $\epsilon_{2R_{0}}=\epsilon_{1R_{0}}$ thus we have

$\displaystyle S_{\mathcal{A}_{1}}=\frac{2\pi
c}{3}\left(\frac{(R_{e}-\epsilon_{R_{e}})^{2}}{(2R_{e}-\epsilon_{R_{e}})\epsilon_{R_{e}}}+\frac{(R_{0}-\epsilon_{1R_{0}})^{2}}{(2R_{0}-\epsilon_{1R_{0}})\epsilon_{1R_{0}}}-\frac{2R_{0}^{2}}{R_{e}^{2}-R_{0}^{2}}\right)\,.$
(4.61)

where $\epsilon_{1R_{0}}$ is a direct cutoff of $\mathcal{A}_{1}$ on its inner
boundary, while $\epsilon_{R_{e}}$ is not.

Then similarly the entanglement entropy can be derived for annuli with a
direct cutoff on its outer boundary. This is archived by considering the
configuration in the right figure of Fig.4, where
$\mathcal{A}^{\prime}=\mathcal{A}_{1}^{\prime}\cup\mathcal{A}_{2}^{\prime}$.
We set the inner and outer boundaries of $\mathcal{A^{\prime}}_{1}$ to be
$R_{0}$ and $R_{e}$ respectively so
$\mathcal{A}^{\prime}_{1}=\mathcal{A}_{1}$. The calculation of the partial
entanglement entropy is easy when we consider the total system to be pure
state. So we have

$\displaystyle s_{\mathcal{A}^{\prime}}(\mathcal{A}_{2}^{\prime})=$
$\displaystyle\frac{1}{2}\left(S_{\mathcal{A}_{2}^{\prime}}+S_{\mathcal{A}^{\prime}}-S_{\mathcal{A}_{1}^{\prime}}\right)$
(4.62) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(S_{\mathcal{A}}+S_{\mathcal{A}_{2}}-S_{\mathcal{A}_{1}}\right)$
(4.63) $\displaystyle=$ $\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})\,.$
(4.64)

Then the partial entanglement entropy
$s_{\mathcal{A}^{\prime}}(\mathcal{A}_{2}^{\prime})$ is given by (4.59).
According to (4.56) and similarly using the second necessary condition for
(4.56), we get the entanglement entropies for annuli with a direct cutoff on
its outer boundary

$\displaystyle S_{\mathcal{A}_{1}^{\prime}}=\frac{2\pi
c}{3}\left(\frac{2R_{0}^{2}}{R_{0}^{2}-R_{e}^{2}}+\frac{(R_{e}-\epsilon_{1R_{e}})^{2}}{(2R_{e}-\epsilon_{1R_{e}})\epsilon_{1R_{e}}}+\frac{(R_{0}-\epsilon_{R_{0}})^{2}}{(2R_{0}-\epsilon_{R_{0}})\epsilon_{R_{0}}}\right)\,,$
(4.65)

where $\epsilon_{1R_{e}}$ is a direct cutoff of the annulus on its outer
boundary, while $\epsilon_{R_{0}}$ is not.

###  4.3 Entanglement contour and entanglement entropies for the annuli

Figure 5: The partition of an annulus $A=A_{1}\cup A_{3}$ with rotation
symmetry. The purple circle is the partition boundary with $r=R$.

In this subsection we derive the contour function for an annulus $A$ via the
differential version of the partial entanglement entropy proposal (2.22).
Again we chose the annulus whose inner and outer boundaries are circles with
radius $R_{0}$ and $R_{e}$. We divide the annulus $A$ into two thinner annuli
$A_{1}$ and $A_{3}$ by a circle $r=R$. (see Fig.5). According to the
discussion in section 2, the sufficient condition for the validity of (2.22)
is that, $S_{A_{1}}$ and $S_{A_{3}}$ are directly regularized at the partition
boundary $r=R$ with equal cutoffs $\epsilon_{1R}=\epsilon_{3R}$. In other
words we apply (4.61) to $A_{3}$ so its inner boundary is directly regularized
by the cutoff $\epsilon_{3R}$, then we apply (4.65) to $A_{1}$ so its outer
boundary is directly regularized by the cutoff $\epsilon_{1R}$. Note that the
parameters $R_{0}$ and $R_{e}$ in (4.61) and (4.65) are not what we defined in
Fig.5. Then we have,

$\displaystyle S_{A_{1}}=$ $\displaystyle\frac{2\pi
c}{3}\left(\frac{2R_{0}^{2}}{R_{0}^{2}-R^{2}}+\frac{(R-\epsilon_{1R})^{2}}{(2R-\epsilon_{1R})\epsilon_{1R}}+\frac{(R_{0}-\epsilon_{R_{0}})^{2}}{(2R_{0}-\epsilon_{R_{0}})\epsilon_{R_{0}}}\right)\,,$
(4.66) $\displaystyle S_{A_{3}}=$ $\displaystyle\frac{2\pi
c}{3}\left(\frac{2R^{2}}{R^{2}-R_{e}^{2}}+\frac{(R_{e}-\epsilon_{R_{e}})^{2}}{(2R_{e}-\epsilon_{R_{e}})\epsilon_{R_{e}}}+\frac{(R-\epsilon_{3R})^{2}}{(2R-\epsilon_{3R})\epsilon_{3R}}\right)\,.$
(4.67)

Although in the above equations $\epsilon_{R_{0}}$ and $\epsilon_{R_{e}}$ are
not direct cutoffs, the entanglement contour function is independent of them.
Then we set $\epsilon_{1R}=\epsilon_{3R}$ and plug the above regularised
entanglement entropies into (2.22), then we find the scheme-independent
contour function for the annulus $A$,

$\displaystyle f(R_{0},R_{e},r)=\frac{1}{4\pi
r}\partial_{R}\left(S_{\mathcal{A}_{1}}-S_{\mathcal{A}_{3}}\right)|_{R\to
r}=\frac{2c}{3}\left(\frac{R_{e}^{2}}{\left(r^{2}-R_{e}^{2}\right)^{2}}+\frac{R_{0}^{2}}{\left(r^{2}-R_{0}^{2}\right){}^{2}}\right)\,.$
(4.68)

The above entanglement contour function is one of our main results in this
article. In the following we give some consistency checks on this result. The
entanglement entropy $S_{A}$ with direct cutoffs is then given by

$\displaystyle S_{A}=$
$\displaystyle\int^{R_{e}-\epsilon_{R_{e}}}_{R_{0}+\epsilon_{R_{0}}}2\pi
rf(R_{0},R_{e},r)dr$ (4.69) $\displaystyle=$ $\displaystyle\frac{2\pi
c}{3}\Big{[}\frac{R_{e}^{2}}{2R_{e}\,\epsilon_{R_{e}}-\epsilon^{2}_{R_{e}}}-\frac{R_{e}^{2}}{R_{e}^{2}-\left(R_{0}+\epsilon_{R_{0}}\right){}^{2}}$
(4.70)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{R_{0}^{2}}{2R_{0}\epsilon_{R_{0}}+\epsilon^{2}_{R_{0}}}-\frac{R_{0}^{2}}{\left(R_{e}-\epsilon_{R_{e}}\right){}^{2}-R_{0}^{2}}\Big{]}\,.$
(4.71)

The first consistency check is that when we take the limit $R_{0}\to 0$, the
annulus becomes a disk thus we recover the contour function (3.43) for disks
with $d=3$. This is obvious when we set $R_{0}=0$ in (4.68).

The second consistency check is that the entanglement entropy from (4.68)
satisfies the area law as expected. Using the entanglement contour function
$f(R_{0},R,r)$ (4.68), we get the entanglement entropy for annuli under a
direct regularization scheme. We regularize $S_{\mathcal{A}}$ with the cutoff
$\epsilon_{R_{e}}$ on the out boundary and $\epsilon_{R_{0}}$ on the inner
boundaries, then we obtain

$\displaystyle S_{A}=$ $\displaystyle\frac{c}{6}\left(\frac{2\pi
R_{e}}{\epsilon_{R_{e}}}+\frac{2\pi
R_{0}}{\epsilon_{R_{0}}}-4\pi\frac{\left(R_{e}^{2}+R_{0}^{2}\right)}{\left(R_{e}^{2}-R_{0}^{2}\right)}+\mathcal{O}(\epsilon)\right)\,.$
(4.72)

As expected the leading contribution is proportional to the area of the inner
and outer boundaries of $\mathcal{A}$ thus the entanglement entropy satisfies
the area law.

The third consistency check is based on the behavior of (4.69) under the limit
$R_{0}\to R$. When $\epsilon_{R_{e}}=\epsilon_{R_{0}}$, we fix $R+R_{0}$ and
let $R_{0}$ approach $R$. The leading contribution from the area terms should
be fixed during the approaching. However the entanglement entropy should go to
zero when $R-R_{0}=0$ because the annulus vanishes555Note that (4.69) is not
accurate when $R_{e}-R_{0}\sim\epsilon$ so we can not get the vanished $S_{A}$
when $R_{0}=R_{e}$.. This means during the approaching the entanglement
entropy should decrease. In other words the sub-leading contribution should
increase and grow to the same order as the area terms as
$R-R_{0}\sim\epsilon$. This is indeed true for (4.69) because the sub-leading
term is proportional to $\frac{1}{R^{2}-R_{0}^{2}}$ which increases faster and
faster as $R_{0}$ approaches $R$.

We compare (4.69) with the holographic entanglement entropy calculated by the
RT formula Fonda:2014cca 666We also briefly introduce how to calculate the
minimal surface anchored on the boundary of a annulus in AdS4 in the
appendix.. Note that the boundary and bulk cutoffs do not exactly equal to
each other and are related by the fine correspondence between the points of
$\mathcal{A}$ and the RT surface $\mathcal{E}_{\mathcal{A}}$. See (3.53) for
an explicit example where the fine correspondence is known. More explicitly,
they have the same leading order thus generate the same area term contribution
to the entanglement entropy while differ at the sub-leading order, i.e.,

$\displaystyle\epsilon_{R_{0}(R_{e})}=\bar{\epsilon}_{R_{0}(R_{e})}+g_{R_{0}(R_{e})}(R_{0},R_{e})\bar{\epsilon}_{R_{0}(R_{e})}^{2}+\mathcal{O}\left(\bar{\epsilon}^{3}\right)\,.$
(4.73)

where $\epsilon_{R_{0}(R_{e})}$ is the field theory cutoff at the inner (or
outer) boundary of the annulus, while $\bar{\epsilon}_{R_{0}(R_{e})}$ is the
corresponding bulk cutoff taken at $z=\bar{\epsilon}_{R_{0}(R_{e})}$. The
function $g_{R_{0}(R_{e})}(R_{0},R_{e})$ is determined by the fine
correspondence. The difference of order $\bar{\epsilon}_{in(out)}^{2}$ will
not affect the area term but will affect the sub-leading contribution to the
entanglement entropy drastically. Plugging (4.73) into (4.69) we have

$\displaystyle S_{A}=\frac{c}{6}\left(\frac{2\pi
R_{0}}{\bar{\epsilon}_{R_{0}}}+\frac{2\pi
R_{e}}{\bar{\epsilon}_{R_{e}}}-2\pi\left(\frac{2(R_{e}^{2}+R_{0}^{2})}{R_{e}^{2}-R_{0}^{2}}+g_{R_{0}}(R_{e},R_{0})R_{0}+g_{R_{e}}(R_{e},R_{0})R_{e}\right)+\mathcal{O}\left(\bar{\epsilon}\right)\right)\,.$
(4.74)

Then the above result should match the entanglement entropy calculated by the
RT formula if the fine correspondence is known to us. However in this case we
can not find the fine correspondence via the Rindler method or the fine
structure analysis by modular planes, because the Rindler transformations as
well as the modular flows are not explicitly constructed yet. One possible way
is to follow the steps of Coser:2017dtb  to impose reasonable assumptions for
the bit threads, then the fine correspondence can be read from the
configuration of the bit threads (or the modular planes at a time slice). Our
result (4.60) can only match the holographic entanglement entropy (A.90) when
the UV and IR cutoffs satisfy the fine correspondence. We leave this exact
matching for future investigations. If we naively take
$\epsilon_{R_{0}}=\epsilon_{R_{e}}=\bar{\epsilon}_{R_{0}}=\bar{\epsilon}_{R_{e}}$,
then (4.69) and (A.90) differ by a value that depend on $\frac{R_{0}}{R}$.
Also we show that they have similar behaviors when the value of $R_{0}$
approaches $R$ while $R_{0}+R$ is fixed, see Fig.6.

Figure 6: Here we have set
$\epsilon_{R_{0}}=\epsilon_{R_{e}}=\bar{\epsilon}_{R_{0}}=\bar{\epsilon}_{R_{e}}=10^{-6}$
and $c=6$. When we fix $R_{e}+R_{0}=2$ and let $R_{0}$ approach $R_{e}$, both
of the entanglement entropy $S_{A}$ (4.69) and the holographic entanglement
entropy $S_{A}^{h}$ (A.90) calculated by the RT formula decrease as $R_{0}\to
R_{e}$.$R_{e}-R_{0}$

###  4.4 Entanglement entropies and entanglement contours for spherical shells
in general dimensions

In this subsection we extend the discussion for annuli to higher dimensions,
and calculate the entanglement contour function and entanglement entropy for
spherical shell with rotation symmetry. Based on the contour function (3.43)
for balls in arbitrary dimensions, we straightforwardly get the entanglement
entropy for a $(d-1)$-dimensional ball $\mathcal{A}$ with radius $R_{e}$ in
CFTd,

$\displaystyle S_{\mathcal{A}}=$
$\displaystyle\int_{0}^{R_{e}-\epsilon_{R_{e}}}f_{\mathcal{A}}(r)\Omega_{d-2}r^{d-2}dr$
(4.75) $\displaystyle=$
$\displaystyle\frac{c}{6}\pi^{\frac{d-1}{2}}\left(2-\frac{2\epsilon_{R_{e}}}{R_{e}}\right)^{d-1}\,_{2}\tilde{F}_{1}\left(\frac{d-1}{2},d-1;\frac{d+1}{2};\frac{(R_{e}-\epsilon_{R_{e}})^{2}}{R_{e}^{2}}\right)\,,$
(4.76)

where
${}_{2}\tilde{F}_{1}\left(a,b,c,z\right)=~{}_{2}F_{1}\left(a,b;c;z\right)/\Gamma(c)$
is the regularized hypergeometric function, $\Omega_{d-2}$ is the surface area
of the $(d-2)$-sphere and $\epsilon$ is the UV-cutoff.

We consider $\mathcal{A}_{2}$ as the ball in the center with radius
$R_{0}<R_{e}$, and $\mathcal{A}_{1}$ as the spherical shell characterized by
$R_{0}<r<R_{e}$ whose entanglement entropy we are calculating. Then
$S_{\mathcal{A}_{2}}$ is also given by (4.75) with $R$ replaced by $R_{0}$ and
$\epsilon$ replaced by $\epsilon_{0}$ which is denoted as the cutoff of the
smaller ball $\mathcal{A}_{2}$. The partial entanglement entropy for
$\mathcal{A}_{2}$ is given by

$\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})=$
$\displaystyle\int_{0}^{R_{0}}f_{\mathcal{A}}(r)\Omega_{d-2}r^{d-2}dr$ (4.77)
$\displaystyle=$
$\displaystyle\frac{c}{6}\pi^{\frac{d-1}{2}}\left(\frac{2R_{0}}{R_{e}}\right)^{d-1}\,_{2}\tilde{F}_{1}\left(\frac{d-1}{2},d-1;\frac{d+1}{2};\frac{R_{0}^{2}}{R_{e}^{2}}\right)$
(4.78)

According to (4.55) we get the indirectly regularized $S_{\mathcal{A}_{1}}$,

$\displaystyle S_{\mathcal{A}_{1}}=\frac{c}{6}\pi^{\frac{d-1}{2}}\Big{[}$
$\displaystyle\left(2-\frac{2\epsilon_{R_{e}}}{R_{e}}\right)^{d-1}\,_{2}\tilde{F}_{1}\left(\frac{d-1}{2},d-1;\frac{d+1}{2};\frac{(R_{e}-\epsilon_{R_{e}})^{2}}{R^{2}}\right)$
(4.79)
$\displaystyle~{}~{}~{}~{}+\left(2-\frac{2\epsilon_{R_{0}}}{R_{0}}\right)^{d-1}\,_{2}\tilde{F}_{1}\left(\frac{d-1}{2},d-1;\frac{d+1}{2};\frac{(R_{0}-\epsilon_{R_{0}})^{2}}{R_{0}^{2}}\right)$
(4.80)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-2^{d}\left(\frac{R_{0}}{R_{e}}\right)^{d-1}\,_{2}\tilde{F}_{1}\left(\frac{d-1}{2},d-1;\frac{d+1}{2};\frac{R_{0}^{2}}{R_{e}^{2}}\right)\Big{]}$
(4.81)

Apply a similar strategy in the 3-dimensional CFT case, we calculate the
entanglement entropies for shells with direct cutoffs for inner and outer
boundaries via (4.55), (4.56) and the second necessary condition for the
validity of (1.5). The differential version of the proposal (2.22) can be
generalized to

$\displaystyle
f(R_{0},R_{e},r)=\frac{1}{2\Omega_{d-2}r^{d-2}}\partial_{R}\left(S_{A_{1}}-S_{A_{3}}\right)|_{R\to
r}\,,$ (4.82)

in arbitrary dimensions, where similarly the shells $A_{1}$ and $A_{3}$ are
defined by $A_{1}:R_{0}<r<R$ and $A_{3}:R<r<R_{e}$. Then we can calculated the
scheme-independent entanglement contour functions for spherical shells in
arbitrary dimensions using the sufficient condition for (2.22), and get

$\displaystyle
f_{shell}(R_{0},R_{e},r)=\frac{2^{d-2}c}{3}\left(\left(\frac{R_{e}}{R_{e}^{2}-r^{2}}\right)^{d-1}+\left(\frac{R_{0}}{r^{2}-R_{0}^{2}}\right)^{d-1}\right)\,.$
(4.83)

The above result is obviously consistent with our previous result (4.68) when
$d=3$. One can see from Fig.7 that, as the spacetime dimension increase, the
contribution of entanglement entropy away from the entangling surface
decreases.

Figure 7: This figure shows the entanglement contour function
$f_{\mathcal{A}}(r)$ for a spherical shell with $c=1,R_{0}=3,R_{e}=10$ in
different spacetime dimensions.

##  5 Comments on the phase transition of the holographic mutual information

As we have shown the entanglement contour and partial entanglement entropy are
regularization independent quantities. Another important regularization
independent quantity is the mutual information. The mutual information between
the regions $A$ and $B$ are defined by

$\displaystyle I(A:B)=S_{A}+S_{B}-S_{A\cup B}\,.$ (5.84)

For example we consider the left configuration in Fig.4 and assume the total
system is in a pure state thus
$S_{\mathcal{A}_{1}}=S_{\mathcal{A}^{c}\cup\mathcal{A}_{2}}$. It has been
pointed out by Wen:2019ubu  that the partial entanglement entropy for the
subset $\mathcal{A}_{2}$ is indeed half of the mutual information
$I(\mathcal{A}_{2}:\mathcal{A}^{c})/2$. More explicitly we can relate the
partial entanglement entropy to the mutual information

$\displaystyle
I(\mathcal{A}_{2}:\mathcal{A}^{c})=2s_{\mathcal{A}}(\mathcal{A}_{2})=S_{\mathcal{A}_{2}}+S_{\mathcal{A}}-S_{\mathcal{A}_{1}}\,.$
(5.85)

Since the contour function $f_{\mathcal{A}}(r)$ for the disk $\mathcal{A}$ is
non-vanishing everywhere, the mutual information is positive for any value of
of the ratio $\frac{R_{e}}{R_{0}}$,

$\displaystyle I(\mathcal{A}_{2}:\mathcal{A}^{c})=2\int_{r=0}^{r=R_{0}}2\pi
r\frac{2cR_{e}^{2}}{3\left(R_{e}^{2}-r^{2}\right)^{2}}dr=\frac{4\pi
cR_{0}^{2}}{3R_{e}^{2}-3R_{0}^{2}}>0\,.$ (5.86)

Like the contour function, mutual information integrated from the contour
function is scheme-independent. As expected this mutual information is always
positive and approaches zero when $R_{e}/R_{0}\to\infty$. It is monotonic by
satisfying the following inequalities,

$\displaystyle\frac{\partial I(\mathcal{A}_{2}:\mathcal{A}^{c})}{\partial
R_{0}}\leq 0\,,\qquad\frac{\partial
I(\mathcal{A}_{2}:\mathcal{A}^{c})}{\partial R_{e}}\geq 0\,.$ (5.87)

Previously in section 2 we pointed out that, the partial entanglement entropy
(or mutual information) calculated by the right-hand side of (1.5) with all
the subset entanglement entropies regularized by an identical cutoff, is
scheme-independent. But it may differ from the right partial entanglement
entropy by a constant. Here we call the mutual information calculated in such
a way as $I^{\prime}(\mathcal{A}_{2}:\mathcal{A}^{c})$. We apply

$\displaystyle
I^{\prime}(\mathcal{A}_{2}:\mathcal{A}^{c})=S_{\mathcal{A}_{2}}+S_{\mathcal{A}}-S_{\mathcal{A}_{1}}$
(5.88)

where the all the terms in the right hand side are regularized by an identical
cutoff $\epsilon$. Then we find that

$\displaystyle I^{\prime}(\mathcal{A}_{2}:\mathcal{A}^{c})=\frac{\pi
c\left(5R_{0}^{2}-R_{e}^{2}\right)}{3\left(R_{e}^{2}-R_{0}^{2}\right)}=I(\mathcal{A}_{2}:\mathcal{A}^{c})-\frac{\pi
c}{3}$ (5.89)

This is not right because when $R_{e}/R_{0}\to\infty$,
$I^{\prime}(\mathcal{A}_{2}:\mathcal{A}^{c})\to-\frac{\pi c}{3}$, which is
negative.

On the other hand, based on the RT formula we can calculate the regularized
entanglement entropies on the right hand side of (5.85) thus calculate the
mutual information $I(\mathcal{A}_{2}:\mathcal{A}^{c})$ across the annulus
$\mathcal{A}_{1}$. For the case of an annulus $\mathcal{A}_{1}$, it has been
believed that the mutual informations undergoes a phase transition between
finite value to zero value. Because in the context of AdS/CFT there are two
extremal surfaces for a given annulus,

  * •

Two disk phase: the extremal surface is disconnected and is the union of the
RT surfaces of the two disk $\mathcal{A}$ and $\mathcal{A}_{2}$, i.e.,
$\mathcal{E}_{\mathcal{A}_{1}}$=$\mathcal{E}_{\mathcal{A}}\cup\mathcal{E}_{\mathcal{A}_{2}}$.

  * •

Hemi-torus phase: the extremal surface is a connected surface of a hemi-torus.
The area of this surface is calculated in the appendix.

The holographic entanglement entropy $S_{\mathcal{A}_{1}}$ is proportional to
the area of the smaller one. However both of the two areas are infinite, it
doesn’t make sense to compare their size unless they are regularized. The
regularization scheme used previously (for example in Fonda:2014cca ;
Nakaguchi:2014pha ) is to take a uniform cutoff $z=\bar{\epsilon}$. Under this
scheme we can adjust the parameter $\frac{R}{R_{0}}$, the minimal surfaces
switch between these two phases at a critical value of the ratio
$\frac{R}{R_{0}}$. When $\frac{R}{R_{0}}$ is larger than the critical value
the mutual information is finite in the hemi-torus phase. Otherwise
$S_{\mathcal{A}_{1}}=S_{\mathcal{A}_{2}}+S_{\mathcal{A}}$ and the mutual
information will vanish.

Here we suspect that the inconsistency of the mutual information comes from
the inappropriate choice of the regularization scheme for the RT surfaces. As
we have discussed previously, the cancellation of the area term in the mutual
information can be satisfied when all the cutoffs have the same leading order,
which is satisfied by taking the uniform IR cutoff $z=\bar{\epsilon}$. However
this does not mean that the quantity we get is scheme-independent. Firstly the
IR cutoff in the bulk and UV cutoff on the field theory side is related by the
fine correspondence (which is not known to us in the case of annulus) and in
general they differ at the sub-leading order (for the case of balls see the
relation (3.53)). This difference can affect the entanglement entropy at the
sub-leading order (or mutual information) drastically. Secondly even the UV
and IR cutoffs are identical, as we have just shown, taking a uniform cutoff
for all the entanglement entropies will not lead to the right mutual
information. There is still a constant remains to be determined.

So here we propose that the phase transition of the holographic entanglement
entropy for annuli and the mutual information crossing that annulus come from
a misunderstanding of the regularization scheme independence, and may not
exist at all.

##  6 Conclusion

The generalization of the partial entanglement entropy proposal (1.5) to
higher dimensions with rotation symmetries is natural, because all the reasons
Wen:2019ubu ; Kudler-Flam:2019oru  why the proposal is right in 2-dimensions
still hold without any additional requirements. The further generalization
would be to claim that (1.5) hold for an arbitrary partition of the region
$\mathcal{A}$ into three subsets. This general situation turns out to admit a
different version of the proposal which is derived in appendix B. Also in
higher dimensions $d\geq 3$, there exist subsets that divide $\mathcal{A}$
into more than three subsets thus a more general formula for partial
entanglement entropy should be proposed. In these cases the additivity of the
partial entanglement entropies will need additional assumptions for the
entanglement entropies.

In this paper we have introduced some new perspectives which are not
thoroughly studied before. The first perspective is how to understand the
regularization of the entanglement entropy. In this paper, following
Wen:2018whg ; Wen:2019ubu  we consider the regularized entanglement entropy as
the partial entanglement entropy for the regularized region, where cutoffs are
bounds for integrating contour functions. The second perspective is that, the
UV and IR cutoffs are related by the fine correspondence between points in
$\mathcal{A}$ and its RT surface $\mathcal{E}_{\mathcal{A}}$ by holography.
Furthermore they are equal at the leading order but in general differ at the
sub-leading order. This is crucial in higher dimensions where the sub-leading
contribution for entanglement entropy contains physical information. The third
perspective is that, we give more careful derivation for scheme-independent
quantities like the entanglement contour function, partial entanglement
entropy and mutual information. This leads to our proposal that the phase
transition of the mutual information crossing an annulus may not exist.

## Acknowledgments

The authors acknowledge Hongguang Liu, Zhuo-yu Xian and Gang Yang for helpful
discussions. QW would also like to thank Chang-pu Sun and Chuan-jie Zhu for
support and is supported by NSFC Grant No.11805109. MH receives support from
the US National Science Foundation through grant PHY-1602867 and PHY-1912278,
and a start-up grant at Florida Atlantic University, USA.

##  Appendix A Entanglement entropies for annuli calculated by the RT formula

Holographic calculations for entanglement entropies of annuli via the RT
formula are given in Fonda:2014cca  (see also Drukker:2005cu ; Dekel:2013kwa ;
Nakaguchi:2014pha  for related discussions). The area of the connected RT
surface which looks like the surface of a hemi-torus is given by the following
integration,

$\displaystyle S_{A}^{h}=$ $\displaystyle\frac{c}{6}\times
2\pi\left(\int^{\tilde{z}_{m}}_{\bar{\epsilon}_{R_{e}}/R_{e}}\frac{d\tilde{z}}{\tilde{z}^{2}\sqrt{1+\tilde{z}^{2}-\tilde{z}^{4}/K}}+\int^{\tilde{z}_{m}}_{\bar{\epsilon}_{R_{0}}/R_{0}}\frac{d\tilde{z}}{\tilde{z}^{2}\sqrt{1+\tilde{z}^{2}-\tilde{z}^{4}/K}}\right)$
(A.90) $\displaystyle=$ $\displaystyle\frac{c}{6}\times\left(\frac{2\pi
R_{0}}{\bar{\epsilon}_{R_{0}}}+\frac{2\pi
R}{\bar{\epsilon}_{R_{e}}}-\frac{4\pi}{\sqrt{2\kappa^{2}-1}}\left(\mathbb{E}(\kappa^{2})-(1-\kappa^{2})\mathbb{K}(\kappa^{2})\right)+\mathcal{O}(\epsilon^{b})\right)$
(A.91)

where the parameters $K$, $\kappa$ and $\tilde{z}_{m}$ are determined by
$\frac{R_{0}}{R_{e}}$ in the following way

$\displaystyle\log\frac{R_{0}}{R_{e}}=$
$\displaystyle\frac{\left(K+\sqrt{K(K+4)}\right)\left(\mathbb{K}\left[\frac{K+\sqrt{K(K+4)}+4}{2K+8}\right]-\Pi\left[\frac{2}{K+\sqrt{K(K+4)}+4}|\frac{K+\sqrt{K(K+4)}+4}{2K+8}\right]\right)}{\sqrt[4]{K(K+4)}}\,,$
(A.92) $\displaystyle\tilde{z}_{m}=$
$\displaystyle\frac{\sqrt{K+\sqrt{K(K+4)}}}{\sqrt{2}}\,,$ (A.93)
$\displaystyle\kappa=$ $\displaystyle\sqrt{\frac{K+\sqrt{K(K+4)}+4}{2K+8}}\,.$
(A.94)

Here the RT surface is cut off at $z=\bar{\epsilon}_{R_{0}}$ and
$z=\bar{\epsilon}_{R_{e}}$ near the inner and outer boundaries of the annulus
respectively. The incomplete elliptic integrals of the first, second and third
kind are defined respectively by

$\displaystyle\mathbb{F}(x|m)\equiv$
$\displaystyle\int_{0}^{x}\frac{d\theta}{\sqrt{1-m\sin^{2}\theta}}\,,$ (A.95)
$\displaystyle\mathbb{E}(x|m)\equiv$
$\displaystyle\int_{0}^{x}\sqrt{1-m\sin^{2}\theta}d\theta\,,$ (A.96)
$\displaystyle\Pi(n,x|m)\equiv$
$\displaystyle\int_{0}^{x}\frac{d\theta}{(1-n\sin^{2}\theta)\sqrt{1-m\sin^{2}\theta}}\,.$
(A.97)

And the complete elliptic integrals of the first, second and third kind are
given by

$\displaystyle\mathbb{K}(m)=\mathbb{F}(\frac{\pi}{2}|m)\,,\qquad\mathbb{E}(m)=\mathbb{E}(\frac{\pi}{2}|m)\,,\qquad\Pi(n,m)=\Pi(n,\frac{\pi}{2}|m)\,.$
(A.98)

##  Appendix B Differential version of the proposal for general regions

Figure 8: In the above figure we set
$\partial\mathcal{A}=\Sigma_{0}\cup\Sigma_{e}$,
$\partial\mathcal{A}_{1}=\Sigma_{0}\cup\Sigma$ and
$\partial\mathcal{A}_{3}=\Sigma^{\prime}\cup\Sigma_{e}$. The small square in
the middle of $\mathcal{A}$ is the $\mathcal{A}_{2}$ we choose.

Assuming that the proposal (1.5) hold for an arbitrary partition of an
arbitrary connected region $\mathcal{A}$ into three subsets (see Fig.8) and
the entanglement contour function is unique for any $\mathcal{A}$. Then when
the theory is given the contour function should be totally determined by the
the boundaries of $\mathcal{A}$. In other words we can write, for example in
3-dimensions,

$\displaystyle f_{\mathcal{A}}(x,y)=f(\Sigma_{0},\Sigma_{e},\\{x,y\\})\,.$
(B.99)

where $x$ and $y$ parametrize the region $\mathcal{A}$. Similarly we define
the function $F$ as the integral of the contour function. Then we can write
the entanglement entropy and partial entanglement entropy in the following way

$\displaystyle S_{\mathcal{A}}$
$\displaystyle=\int_{\Sigma_{0}}^{\Sigma_{e}}f(\Sigma_{0},\Sigma_{e},\\{x,y\\})\sqrt{g}dxdy=F(\Sigma_{0},\Sigma_{e},\Sigma_{e})-F(\Sigma_{0},\Sigma_{e},\Sigma_{0})\,,$
(B.100) $\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})$
$\displaystyle=\int_{\Sigma}^{\Sigma^{\prime}}f(\Sigma_{0},\Sigma_{e},\\{x,y\\})\sqrt{g}dxdy=F(\Sigma_{0},\Sigma_{e},\Sigma^{\prime})-F(\Sigma_{0},\Sigma_{e},\Sigma)\,.$
(B.101)

Similar to the discussions in section 2, we have

$\displaystyle\frac{1}{2}\left(S_{\mathcal{A}_{1}\cup\mathcal{A}_{2}}-S_{\mathcal{A}_{3}}\right)$
$\displaystyle=\frac{1}{2}\left(F(\Sigma_{0},\Sigma^{\prime},\Sigma^{\prime})-F(\Sigma_{0},\Sigma^{\prime},\Sigma_{0})-F(\Sigma^{\prime},\Sigma_{e},\Sigma_{e})+F(\Sigma^{\prime},\Sigma_{e},\Sigma^{\prime})\right)\,,$
(B.102)
$\displaystyle\frac{1}{2}\left(S_{\mathcal{A}_{2}\cup\mathcal{A}_{3}}-S_{\mathcal{A}_{1}}\right)$
$\displaystyle=\frac{1}{2}\left(F(\Sigma,\Sigma_{e},\Sigma_{e})-F(\Sigma,\Sigma_{e},\Sigma)-F(\Sigma_{0},\Sigma,\Sigma)+F(\Sigma_{0},\Sigma,\Sigma_{0})\right)\,.$
(B.103)

We define the right hand of (2.14) to be a new function
$G(\Sigma_{0},\Sigma_{e},\Sigma^{\prime})$, then interestingly we find the
right hand side of (2.15) can be written as
$-G(\Sigma_{0},\Sigma_{e},\Sigma)$, thus according to (2.13) and (2.12) we
have

$\displaystyle s_{\mathcal{A}}(\mathcal{A}_{2})=$ $\displaystyle
G(\Sigma_{0},\Sigma_{e},\Sigma^{\prime})-G(\Sigma_{0},\Sigma_{e},\Sigma)$
(B.104) $\displaystyle=$ $\displaystyle
F(\Sigma_{0},\Sigma_{e},\Sigma^{\prime})-F(\Sigma_{0},\Sigma_{e},\Sigma)\,.$
(B.105)

Since the above equation holds for any choice of $\Sigma$ and
$\Sigma^{\prime}$, we can conclude that the two functions only differ by a
constant determined by the region $\mathcal{A}$ (or determined by $\Sigma_{0}$
and $\Sigma_{e}$), i.e.,

$\displaystyle
F(\Sigma_{0},\Sigma_{e},\Sigma)=G(\Sigma_{0},\Sigma_{e},\Sigma)+G_{0}(\Sigma_{0},\Sigma_{e})\,.$
(B.106)

Then we choose $\mathcal{A}_{2}$ as the small square settled at a point
$(x,y)$ and covers the infinitesimal region $x\to x+dx,~{}y\to y+dy$. For this
choice we have

$\displaystyle
s_{\mathcal{A}}(\mathcal{A}_{2})=f(\Sigma_{0},\Sigma_{e},\\{x,y\\})\sqrt{g}dxdy\,,$
(B.107)

and furthermore according to (2.16) we should have

$\displaystyle
f(\Sigma_{0},\Sigma_{e},\\{x,y\\})=\frac{G(\Sigma_{0},\Sigma_{e},\Sigma^{\prime})-G(\Sigma_{0},\Sigma_{e},\Sigma)}{\sqrt{g}\,dxdy}\,,$
(B.108)

where $\Sigma^{\prime}$ is just a small perturbation of $\Sigma$ at $(x,y)$
according to our specific choice of $\mathcal{A}_{2}$ (see Fig.8). It maybe
more understandable to write the above equation in the following way

$\displaystyle
f(\Sigma_{0},\Sigma_{e},\\{x,y\\})=\frac{1}{2\sqrt{g}}\partial_{x}\partial_{y}\left(S_{\mathcal{A}_{1}}-S_{\mathcal{A}_{3}}\right).$
(B.109)

Note that $\mathcal{A}_{2}$ is absorbed by $\mathcal{A}_{1}$ thus
$\mathcal{A}=\mathcal{A}_{1}\cup\mathcal{A}_{3}$. The point $(x,y)$ is on the
boundary between $\mathcal{A}_{1}$ and $\mathcal{A}_{3}$ and the differential
is with respect to a infinitesimal change of the boundary at $(x,y)$ in the
sense of Fig.8. This derivation of the contour function based on the proposal
(1.5) can be naturally generalized to arbitrary dimensions.

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