USTC-ICTS/PCFT-23-29

Geometrizing the Partial Entanglement Entropy: from PEE Threads to Bit Threads

Jiong Lin1,2 $\ast$, Yizhou Lu3 $\ast$ and Qiang Wen4 $\dagger$

1 Interdisciplinary Center for Theoretical Study, University of Science and
Technology of China, Hefei, Anhui 230026, China

2 Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China

3 Department of Physics, Southern University of Science and Technology,
Shenzhen 518055, China

4  Shing-Tung Yau Center and School of Physics, Southeast University, Nanjing
210096, China

$\ast$ These authors contribute equally

$\dagger$ Corresponding to<EMAIL_ADDRESS>
## Abstract

We give a scheme to geometrize the partial entanglement entropy (PEE) for
holographic CFT in the context of AdS/CFT. More explicitly, given a point x we
geometrize the two-point PEEs between x and any other points in terms of the
bulk geodesics connecting these two points. We refer to these geodesics as the
PEE threads, which can be naturally regarded as the integral curves of a
divergenceless vector field $V_{\textbf{x}}^{\mu}$, which we call _PEE thread
flow_. The norm of $V_{\textbf{x}}^{\mu}$ that characterizes the density of
the PEE threads can be determined by some physical requirements of the PEE. We
show that, for any static interval or spherical region $A$, a unique bit
thread configuration can be generated from the PEE thread configuration
determined by the state. Hence, the non-intrinsic bit threads are emergent
from the intrinsic PEE threads. For static disconnected intervals, the vector
fields describing a divergenceless flow is no longer suitable to reproduce the
RT formula. We weight a PEE thread with the number of times it intersects with
any homologous surface. Instead, the RT formula is perfectly reformulated by
the minimization of the summation of PEE threads with all possible assignment
of weights.

###### Contents

  1. 1 Introduction
  2. 2 A brief introduction to PEE and bit threads
    1. 2.1 Partial entanglement entropy
    2. 2.2 Bit threads
  3. 3 Geometrizing the PEE: from PEE threads to bit threads
    1. 3.1 The scheme
    2. 3.2 PEE threads and bit threads in AdS3
    3. 3.3 PEE threads and bit threads in AdSd+1
  4. 4 PEE threads for multi-intervals
  5. 5 Discussions
  6. A Proof of a statement
  7. B PEE threads for a strip region

##  1 Introduction

In the context of AdS/CFT correspondence [1], the Ryu-Takayanagi (RT) formula
[2, 3, 4] uncovers a relation between bulk geometry and boundary entanglement,
by claiming that the entanglement entropy (EE) $S_{A}$ for a region $A$ in the
boundary CFT is given by the area of a minimal surface $\mathcal{E}_{A}$
homologous to $A$ in the AdS bulk,

$S_{A}=\frac{\text{Area}(\mathcal{E}_{A})}{4G_{N}}.$ (1)

The RT formula was refined to its quantum corrected version, the quantum
extremal surface (QES) formula [5, 6, 7], which recently enlightens a new
understanding on the black hole information paradox [8, 9]and was further
refined towards the so called island formula [10, 11, 12, 13, 14], which has
recently been widely studied [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36].

To give a more concrete geometric description for holographic entanglement
entropy, Freedman and Headrick reinterpreted this geometric optimization
problem as a flow optimization problem [37, 38]111Another reformulation of the
RT formula is based on the identification of minimal surfaces in Riemannian
geometry through calibrations [39].. Specifically, the area of the minimal
surface $\mathcal{E}_{A}$ can be given by the maximum flux through the
boundary region $A$ by optimizing over all possible divergenceless vector
fields $v$ whose norm is upper-bounded by $|v|\leq 1/4G_{N}$. This optimized
flow configuration is highly degenerate and is said to _lock_ the boundary
region $A$. On the minimal surface $\mathcal{E}_{A}$, an optimized flow should
be normal to $\mathcal{E}_{A}$ with the norm saturating the bound $1/4G_{N}$.
In other words, $\mathcal{E}_{A}$ is the bottleneck of the flow configuration.
The unoriented integration curves of this optimized flow configuration is
known as _bit threads_ [37]. Later in [40, 41], the flow is generalized to
_multiflow_ to simultaneously lock all the non-overlapping multi-regions.
Explicit bit thread configurations in pure AdS were first constructed by using
the bulk geodesics in [42]. Then, in [43], the perturbations of bit threads
configurations around pure AdS were considered by using bulk geodesics or
Iyer-Wald formalism [44]. Bit threads have been also generalized to its
quantum corrected version [45, 46] and the covariant version [47]. For other
recent progresses on bit threads, please refer to [48, 49, 50, 51, 52, 53,
54].

Despite the equivalence between bit thread description and the RT formula in
computing holographic entanglement entropy, the bit thread configuration is
considered to be non-intrinsic for its non-uniqueness and dependence on the
choice of the region. In other words, no bit threads configuration can lock
all the regions simultaneously. For this reason, the explicit distribution of
a bit thread configuration has no physical meaning. It is very important to
give particular physical meaning to the distribution of the bit threads, hence
clarify the rules that uniquely determine the bit threads configurations. A
natural interpretation for bit threads distribution could be the _entanglement
contour_ $s_{A}(\textbf{x})$ [55], which captures the contribution from the
local degree of freedom at each site x to $S_{A}$. For a given bit thread
configuration, a unique entanglement contour $s_{A}(\textbf{x})$ function can
be read from the configuration by computing the bit thread flux emanating from
x to the complement of $A$ (see [56] for an explicit example). Nevertheless,
even the entanglement contour is uniquely determined by the reduced density
matrix $\rho_{A}$, the bit thread configuration consistent with
$s_{A}(\textbf{x})$ is still not unique as the thread distribution on the
complement $\bar{A}$ can not be clarified. In addition, the dependence on the
choice of $A$ persists.

In this paper, we will show that the _partial entanglement entropy_ (PEE) [57,
56, 58, 59, 60, 61] 222See [62, 63, 64, 65, 66, 67, 16, 68] for other recent
progress on PEE. can give rise to a specific and unique bit threads
configuration that locks any static intervals in two dimensions and spherical
regions in general dimensions in the context of AdS/CFT. Like the mutual
information, the PEE measures the correlation between two regions in a certain
way. The key feature of the PEE is satisfying the property of additivity. The
PEE satisfies a set of physical requirements [55, 57], and in Poincaré
invariant theories the PEE can be uniquely determined by these requirements.
So far, there are already several proposals to construct the PEE, including
the geometric construction that works in holographic theories with a local
modular Hamiltonian [57, 60, 69], the additive linear combination (ALC)
proposal that works in general two dimensional theories, the solution of the
set of requirements, or the extensive mutual information (EMI) proposal that
works for theories with conformal symmetries, and the Gaussian formula for
Gaussian states of many-body systems [55, 70]. In the regimes where different
proposals applies, these proposals generate highly consistent results [57, 60,
70]. The PEE encodes all the information of _entanglement contour_. For a
region $A$, the PEE not only characterizes the contributions from each site
inside $A$, but also clarifies the the different roles played by the sites
outside $A$.

Due to the property of additivity, any PEE $\mathcal{I}(A,B)$ between two non-
overlapping regions $A$ and $B$ can eventually decompose into a summation of
two-point PEEs,

$\displaystyle\mathcal{I}(A,B)=\sum_{\textbf{x},\textbf{y}}\mathcal{I}(\textbf{x},\textbf{y}),\qquad\textbf{x}\in
A,~{}~{}~{}~{}\textbf{y}\in B\,.$ (2)

The two-point PEEs then fully describe the PEE structure of the state. Our
scheme to geometrize the PEE is to represent the two-point PEEs as the bulk
geodesics connecting these two points, which we call the PEE threads. The PEE
threads emanating from a single boundary point x can be further regarded as
the integral curves of a divergenceless vector field $V_{\textbf{x}}^{\mu}$ in
the bulk, which we call the _PEE thread flow_. Then by superposing these PEE
thread flows associated with all points inside the interval,we explicitly show
how to get a unique bit threads configuration from the PEE threads in AdS3.
Interestingly, the resulting bit thread configuration respects the symmetries
of the theory, and coincides with the bit threads configuration previously
constructed in [42]. This prescription also works for static spherical regions
in higher dimensional CFTs ($d\geq$ 3). Nevertheless, this scheme fails when
applied to the non-spherical boundary region in higher dimensions. We also
study the PEE threads for disconnected intervals and show how PEE threads
picture can interpret the phase transition of the RT surfaces in AdS3/CFT2.

In Sec.2, we will give a brief introduction to the PEE and bit threads. The
scheme to geometrize the PEE is explicitly presented in Sec.3. Also, we will
show how a bit thread configuration emerges from the PEE threads
configuration. Explicit calculations will be carried out for intervals and
spherical boundary regions in the pure AdS spacetime in general dimensions. In
Sec.4, we study the reformulation of the RT formula based on the PEE thread
configuration. In this case the entanglement entropy is reproduced by a
summation of weighted PEE threads. The discontinuous phase transition of the
RT surfaces for multi-intervals is reproduced by the switching the assignment
of the weight of the PEE threads that gives the minimal value of the
summation. We give a discussion in Sec.5.

##  2 A brief introduction to PEE and bit threads

###  2.1 Partial entanglement entropy

The partial entanglement entropy $\mathcal{I}(A,B)$ [57, 56, 58, 59, 60, 61]
is a measure of the correlation between two spacelike separated region $A$ and
$B$. It is defined to satisfy a set of physical requirements [55, 59]
including all those satisfied by the mutual information $I(A,B)$333Note that,
we should not mix between the mutual information $I(A,B)$ and the PEE
$\mathcal{I}(A,B)$. and the feature of being additive. For non-overlapping
regions $A$, $B$ and $C$, the physical requirements for the PEE can be briefly
summarized in the following444For more details about the well (or uniquely)
defined scope of the PEE and the ways to construct the PEEs in different
situations, the readers may consult [57, 56, 59, 58, 60, 61, 71]. These
details are also summarized in the background introduction sections of [65,
66].:

  1. 1.

Additivity: $\mathcal{I}(A,B\cup C)=\mathcal{I}(A,B)+\mathcal{I}(A,C)$;

  2. 2.

Permutation symmetry: $\mathcal{I}(A,B)=\mathcal{I}(B,A)$;

  3. 3.

Normalization: $\mathcal{I}(A,\bar{A})=S_{A}$;

  4. 4.

Positivity: $\mathcal{I}(A,B)>0$;

  5. 5.

Upper bounded: $\mathcal{I}(A,B)\leq\text{min}\\{S_{A},S_{B}\\}$;

  6. 6.

$\mathcal{I}(A,B)$ should be Invariant under local unitary transformations
inside $A$ or $B$;

  7. 7.

Symmetry: For any symmetry transformation $\mathcal{T}$ under which
$\mathcal{T}A=A^{\prime}$ and $\mathcal{T}B=B^{\prime}$, we have
$\mathcal{I}(A,B)=\mathcal{I}(A^{\prime},B^{\prime})$.

The Additivity and Permutation symmetry properties indicates that, the PEE
structure is fully described by the two-point PEEs
$\mathcal{I}(\textbf{x},\textbf{y})$ [59]. In other words, any PEE
$\mathcal{I}(A,B)$ can be evaluated by the integration (or summation for
discrete systems) of certain class of two-point PEEs,

$\mathcal{I}\left(A,B\right)=\int_{A}\mathrm{d}\sigma_{\textbf{x}}\int_{B}\mathrm{d}\sigma_{\textbf{y}}~{}\mathcal{I}(\textbf{x},\textbf{y}).$
(3)

where $\sigma_{\textbf{x}}$ and $\sigma_{\textbf{y}}$ are infinitesimal area
elements located at x and y inside $A$ and $B$ respectively.

The entanglement contour [55] is a special type of PEE hence can also be
generated from the two-point PEEs. The entanglement contour function
$s_{A}(x)$ is assumed to be the density function of the entanglement entropy,
which captures the contributions from the local degrees of freedom to $S_{A}$.
Although it is hard to clarify what it means by the contribution from each
degrees of freedom, it is clear that the entanglement entropy $S_{A}$ is just
the collection of the contributions from all the degrees of freedom inside
$A$,

$S_{A}=\int_{A}s_{A}(\textbf{x})\mathrm{d}\sigma_{\textbf{x}}.$ (4)

The physical interpretation for the entanglement contour perfectly matches
with the physical requirements of the PEE, which makes the PEE a natural
proposal for the entanglement contour. According to the normalization property
$S_{A}=\mathcal{I}(\bar{A},A)=\int_{A}\mathcal{I}(\bar{A},\textbf{x})\mathrm{d}\sigma_{\textbf{x}}$,
it is straightforward to propose that

$\displaystyle s_{A}(\textbf{x})=\mathcal{I}(\bar{A},\textbf{x})\,.$ (5)

Similarly, the contribution from a subset $A_{i}$ of $A$ to $S_{A}$ can also
be expressed as a PEE,

$s_{A}(A_{i})=\int_{A_{i}}s_{A}(\textbf{x})\mathrm{d}\sigma_{\textbf{x}}=\mathcal{I}(\bar{A},A_{i})\,.$
(6)

Here we only introduce one particular proposal to construct the PEE in generic
two-dimensional theories with all the degrees of freedom settled in a unique
order (for example settled on a line or a circle), which we call the additive
linear combination (ALC) proposal [57, 60, 59].

  * •

The ALC proposal: Consider a boundary region $A$ which is partitioned in the
following way, $A=\alpha_{L}\cup\alpha\cup\alpha_{R}$, where $\alpha$ is some
subregion inside $A$ and $\alpha_{L}$ ($\alpha_{R}$) denotes the regions left
(right) to it. The proposal claims that:

$\displaystyle
s_{A}(\alpha)=\mathcal{I}(\alpha,\bar{A})=\frac{1}{2}\left(S_{\alpha_{L}\cup\alpha}+S_{\alpha\cup\alpha_{R}}-S_{\alpha_{L}}-S_{\alpha_{R}}\right)\,.$
(7)

Using ALC formula, the entanglement contour can be settled down. Then the two-
point PEE can be derived by differentiating the contour function. For a static
spherical region $A=\\{\textbf{x}||\textbf{x}|^{2}\leq R^{2}\\}$, the contour
function for a $(d-2)$-dimensional sphere with radius $r$ is by [58, 56]555 In
[58], this expression is derived using the ALC proposal and the trick in [72].
In [56], this contour function is read from a specific construction of the bit
threads [42].

$s_{A}(r)=\frac{c}{6}\left(\frac{2R}{R^{2}-r^{2}}\right)^{d-1}.$ (8)

We first determine $\mathcal{I}(0,\textbf{x})$ between the origin point $r=0$
and the point $\textbf{x}=(r,\phi_{i})$ with $r>R$. Due to spherical symmetry,
we have

$\int_{R}^{\infty}\mathcal{I}(0,r)r^{d-2}\Omega_{d-2}\mathrm{d}r=s_{A}(0)=\frac{c}{6}\frac{2^{d-1}}{R^{d-1}},$
(9)

where

$\Omega_{d-2}=\frac{2\pi^{\frac{d-1}{2}}}{\Gamma\left(\frac{d-1}{2}\right)}\,,$
(10)

is the area of the $(d-2)$-dimensional spherical surface with unit radius.
Then we have

$\mathcal{I}(0,\textbf{x})=\frac{c}{6}\frac{2^{d-1}(d-1)}{\Omega_{d-2}|\textbf{x}|^{2(d-1)}}.$
(11)

From eq. (11), we can conclude that
$\mathcal{I}(\textbf{x}_{1},\textbf{x}_{2})$ between two points only depends
on their Euclidean distance, i.e.

$\mathcal{I}(\textbf{x}_{1},\textbf{x}_{2})=\frac{c}{6}\frac{2^{d-1}(d-1)}{\Omega_{d-2}|\textbf{x}_{2}-\textbf{x}_{1}|^{2(d-1)}}.$
(12)

Taking $d=2$, we get the two-point PEE for vacuum CFT2666See also the
adjacency matrix defined in [71].,

$\mathcal{I}(x,y)=\frac{c}{6}\frac{1}{(x-y)^{2}},$ (13)

From (13), we see that $\mathcal{I}(x,y)$ is proportional to the two-point
function of a primary field with scaling dimension $1$, i.e.
$\mathcal{I}(x,y)\propto\innerproduct{\mathcal{O}(x)\mathcal{O}(y)}{\mathcal{O}(x)\mathcal{O}(y)}$,
and therefore we may further relate this two-point PEE to the length
$\mathcal{L}(x,y)$ of a bulk geodesic connecting $x$ and $y$ as
$\mathcal{I}(x,y)\propto e^{-\mathcal{L}(x,y)/\ell_{AdS_{3}}}$, where
$\ell_{\rm AdS_{3}}$ is the AdS radius. This is an important observation that
motivates us to geometrize PEE in terms of geodesics in Sec.3.

###  2.2 Bit threads

The _bit threads_ configuration [37] is a reformulation of the RT formula to
characterize the holographic entanglement entropy. Using the max-flow min-cut
(MFMC) theorem [73, 74, 75], the computation of entanglement entropy of a
boundary region $A$ is translated into constructing a divergenceless bulk
vector field $v_{A}$ with maximum flux through any codimension-2 surface
$\Sigma_{A}$ homologous to $A$,

$S(A)=\max\int_{A}\mathrm{d}\Sigma\sqrt{h}v^{\mu}_{A}n_{\mu}.$ (14)

The requirements for a vector field $v_{A}$ to describe bit threads are
classified in the following:

  1. 1.

the vector field $v_{A}$ should be divergenceless.

  2. 2.

the vector field $v_{A}$ should satisfy the inequality
$|v_{A}|\leq\frac{1}{4G}$.

  3. 3.

on the RT surface $\mathcal{E}_{A}$, the vector field should be normal to
$\mathcal{E}_{A}$ and saturates the above inequality.

Based on the above three requirements, it is easy to conclude that the flux of
the vector field $v_{A}$ through any homologous surface should recover the
entanglement entropy $S_{A}$, i.e. (14). Intuitively, any flow from $A$ is
clearly bounded by the minimal-area of bottleneck the flow has to pass
through, which is just the RT surface $\mathcal{E}_{A}$, and the main content
is that an optimal flow achieves this bound. As a result, an optimal flow
$v_{A}$ should be normal to $\mathcal{E}_{A}$ and its norm should saturate
$|v_{A}|=1/4G_{N}$ on $\mathcal{E}_{A}$. Such an optimal flow is said to lock
the region $A$ and its unoriented integral curves are known as the bit
threads.

Note that the optimal flow $v_{A}$ that satisfies the above requirements
admits an enormous degeneracy and in general depends on the entangling surface
we chose, hence the distribution of the bit threads has no physical meaning.
Nevertheless, it will be interesting to construct explicit configurations for
the flow or bit threads and endow the configuration with a physical
interpretation. For example, in [42] the authors considered static spherical
boundary regions with radius $R$ in Poincaré AdSd+1

$\mathrm{d}s^{2}=\frac{1}{z^{2}}\left(\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{d-2}^{2}+\mathrm{d}z^{2}\right).$
(15)

In their construction, the optimal flow is assumed to flow along the bulk
geodesics. Regarding the divergenceless condition and the saturation of the
bound $|v_{A}|=1/4G_{N}$ on $\mathcal{E}_{A}$, the vector field can be
determined as

$v^{\mu}_{A}=\left(\frac{2Rz}{\sqrt{\left(R^{2}+r^{2}+z^{2}\right)^{2}-4R^{2}r^{2}}}\right)^{d}\left(\frac{rz}{R},\frac{R^{2}-r^{2}+z^{2}}{2R}\right),$
(16)

where all the angular coordinate components are suppressed due to spherical
symmetry. Such an optimal flow is perhaps the most natural one as it respect
the symmetry of the configurations under considerations.

The relation between the entanglement contour and the bit thread configuration
was first pointed out in a talk by Erik Tonni [76]. And in [56], the
entanglement contour for static spherical regions in states dual to Poincaré
AdSd+1 was read from the explicit bit threads configuration (16),

$s_{A}(r)=\frac{c}{6}\left(\frac{2R}{R^{2}-r^{2}}\right)^{d-1}\,,$ (17)

where the center of the sphere is located at $r=0$. As we can see, the contour
function respect the symmetry of the configuration and only depends on the
radius coordinate $r$. This contour function also coincide with the one
derived in [58] based on the constant contour function measured by a Rindler
observer. It is reasonable to require the bit thread configuration to
reproduce the entanglement contour $s_{A}(\textbf{x})$ by computing the bit
thread flux that emanating from the site x and anchor at the complement
$\bar{A}$. Nevertheless, this additional requirement is still not enough to
determine the flow configuration.

There is no way to construct a bit thread configuration that lock all the
regions we choose. It may be more realistic to take the bit thread
configuration as some emergent concepts from certain intrinsic structure of
the state. In the following section we will show that, the bit thread
configurations that lock any static interval in holographic CFT2 can be
generated from the two-point PEE structure of the state, which is an intrinsic
structure independent from the regions we consider.

##  3 Geometrizing the PEE: from PEE threads to bit threads

Figure 1: PEE threads on a static time slice of global AdS3/CFT2.

In this section, we give a scheme to geometrize the two-point PEEs with the
bulk geodesics anchored on the two points. We refer these geodesics bundles as
the PEE threads. The PEE threads emanating from any site x can be described by
a divergenceless vector fields along the geodesics emanating from x. The PEE
threads emanating from different sites will intersect with each other. We will
show that, the superposition of these vector fields will generate a flow in
the bulk, which is just the natural optimal bit thread configuration (16) for
static intervals or spherical regions in higher dimensions.

###  3.1 The scheme

$R_{\Sigma}$$\Sigma$$x$ Figure 2: The flow $V_{x}^{\mu}$ is tangent to the PEE
threads (the red dashed curves). $\Sigma$ (the blue curve) denotes any co-
dimension two surface, which is homologous to a boundary region $R_{\Sigma}$.
The norm of $V_{x}^{\mu}$ is determined by requiring the flux of $V_{x}^{\mu}$
on $\Sigma$ be equal to the entanglement contour $f_{R_{\Sigma}}(x)$.

Let us first consider the vacuum state of the holographic CFT2 that duals to
the Poincaré AdS3. The motivation that we associate two-point PEEs with the
bulk geodesics is based on an interesting observation that, the two-point PEE
$\mathcal{I}(\textbf{x},\textbf{y})$ can be related to the length of bulk
geodesic that connects two boundary points x and y. Specifically, by inserting
the geodesic length

$\mathcal{L}(\textbf{x},\textbf{y})=\ell_{\rm
AdS_{3}}\log\frac{(\textbf{x}-\textbf{y})^{2}}{\delta^{2}},$ (18)

into (13), we have

$\mathcal{I}(\textbf{x},\textbf{y})=\frac{\ell_{\rm
AdS_{3}}}{4G_{N}\delta^{2}}e^{-\mathcal{L}(\textbf{x},\textbf{y})/\ell_{\rm
AdS_{3}}},$ (19)

where $\delta$ is the UV cutoff. Also, the geodesic is the most natural
geometric object connecting two boundary points. Hence, it is a good starting
point to geometrize the two-point PEEs in terms of their corresponding bulk
geodesics. We name these geodesics bundles as the _PEE threads_. In Fig.1, we
illustrate the PEE threads on a static time slice of global AdS3.

For any boundary point x, the PEE threads emanating from it can be understood
as the integral curves of a divergenceless vector field
$V^{\mu}_{\textbf{x}}$, and the norm of the vector field characterizes the
density of the PEE threads (see Fig.2 for an illustration). We refer to
$V^{\mu}_{\textbf{x}}$ as the _PEE thread flow_ vector field, which can be
written as

$\displaystyle
V_{\textbf{x}}^{\mu}=|V_{\textbf{x}}|\tau_{\textbf{x}}^{\mu}\,,$ (20)

where $\tau_{\textbf{x}}^{\mu}$ represents the tangent unit vectors of the PEE
threads emanating from x. Our main task is to determine the norm
$|V_{\textbf{x}}|$ of this vector field. Since the entanglement entropy is a
collection of the two-point PEEs, it is reasonable to require that the
contribution for site x to $S_{A}$, or the value of $s_{A}(\textbf{x})$,
should be captured by the number of the PEE threads that connecting x and
$\bar{A}$, i.e., the flux of the PEE thread flow $V_{\textbf{x}}^{\mu}$ on any
co-dimension two bulk hypersurface $\Sigma$ homologous to a boundary region
$A$.

$\text{Requirement:}\quad\int_{\bar{A}}\mathrm{d}\sigma_{\textbf{y}}~{}\mathcal{I}(\textbf{x},\textbf{y})=s_{A}(\textbf{x})=\int_{\Sigma}\mathrm{d}\Sigma~{}\sqrt{h}V_{\textbf{x}}^{\mu}n_{\mu},$
(21)

where $n^{\mu}$ is the unit vector normal to $\Sigma$. Since the PEE thread
flow is divergenceless, the flux is independent from the choice of the co-
dimension two homologous surface.

Note that, our scheme to geometrize the two-point PEEs as bulk geodesics gives
us a geometric picture for the fine structure of entanglement, hence contains
much more information than the above requirement. For any point x inside $A$,
the PEE threads emanating from x give a one-to-one mapping between the points
on $\Sigma$ and the points on $\bar{A}$. Indeed, the differential version of
the requirement (21) can fully characterize the feature of our scheme, which
is just

$\displaystyle\mathrm{d}\sigma_{\textbf{y}}~{}\mathcal{I}(\textbf{x},\textbf{y})=\mathrm{d}\Sigma~{}\sqrt{h}V_{\textbf{x}}^{\mu}n_{\mu},$
(22)

where the area element $\mathrm{d}{\Sigma}$ on $\Sigma$ is mapped to the area
element $\mathrm{d}\sigma_{\textbf{y}}$ on $\bar{A}$ via the one-to-one
mapping determined by the PEE threads (See Fig.3). We will see that, when
choosing an proper $\Sigma$, the norm $|V_{\textbf{x}}|$ of the vector field
can be easily settled down by the above requirement (22).

$x$$\mathrm{d}\sigma_{y}$$\mathrm{d}\Sigma$ Figure 3: For a fixed point $x$ in
$A$, the PEE threads give a one-to-one mapping from $\mathrm{d}\Sigma$ on
$\Sigma$ to $\mathrm{d}\sigma_{y}$ on boundary.

$A$$\mathcal{E}_{A}$$Q$

$A$$\mathcal{E}_{A}$$Q$

Figure 4: Superposition of the PEE thread flow in Poincaré AdS with a
spherical boundary region $A$. For the field point $Q$ on the RT surface
$\mathcal{E}_{A}$ (see the left figure), only _outer threads_ pass through
$Q$. For $Q$ inside the entanglement wedge of $A$ (see the right figure), the
_inner threads_ (green dashed curves) also pass through $Q$. Physically, the
_bit thread flow_ $V_{A}^{\mu}$ should only count the contributions from
_outer threads_. However, since the _inner threads_ have the zero net
contributions to $V_{A}^{\mu}$, it is safe to integrate over both outer and
_inner threads_ to obtain $V_{A}^{\mu}$.

The PEE threads emanating from different boundary points could intersect with
each other. Since the evaluation of the entanglement entropy $S_{A}$ is
equivalent to counting all the PEE threads coming out from $A$, it is natural
to consider the superposition of all the PEE thread flows $V_{\textbf{x}}$
with x inside $A$. This results in a divergenceless vector field $V_{A}$ which
we call the _bit thread flow_ vector field. Soon, we will see that such a _bit
thread flow_ satisfies all the requirements for the vector field representing
the bit threads, and thus the name.

Given a connected boundary region $A$, the PEE threads can always be
classified into two classes:

  * •

the _inner threads_ of $A$ that emanate and terminate inside $A$,

  * •

the _outer threads_ of $A$ that emanate inside $A$ and terminate outside $A$.

Only the _outer threads_ of $A$ (see Fig.4) contribute to the entanglement
entropy. One may intend to construct a vector field, which is the
superposition of all the _outer threads_. Nevertheless, this is not necessary
since that, for any _inner thread_ representing
$\mathcal{I}(\textbf{x},\textbf{y})$, the PEE thread will be counted twice
(multiplied by corresponding integral measures) with inverse directions (see
the green dashed curves in the right of Fig. 4 for an illustration). According
to the permutation property
$\mathcal{I}(\textbf{x},\textbf{y})=\mathcal{I}(\textbf{y},\textbf{x})$ of the
PEE and the divergenceless property of the _PEE thread flow_ , the _inner
threads_ cancel with each other under superposition. Therefore, there is no
need to distinguish between the _inner_ and _outer threads_ and the _bit
thread flow_ $V_{A}^{\mu}$ can be achieved by integrating all the _PEE thread
flow_ $V_{\textbf{x}}$ with $\textbf{x}\in A$, i.e.

$V_{A}^{\mu}=\int_{\text{outer threads of
$A$}}\mathrm{d}^{d-1}\textbf{x}V_{\textbf{x}}^{\mu}=\int_{\text{all threads of
$A$}}\mathrm{d}^{d-1}\textbf{x}V_{\textbf{x}}^{\mu}.$ (23)

According to the requirement (21), since the flux of the _PEE thread flow_
$V_{\textbf{x}}$ gives the entanglement contour $s_{A}(\textbf{x})$, the flux
of the _bit thread flow_ $V_{A}^{\mu}$ should recover the entanglement entropy
$S_{A}$. Soon we will check that, for static intervals or spherical regions
the inequality $|V_{A}|\leq 1/4G_{N}$ is satisfied by $V_{A}^{\mu}$ inside the
entanglement wedge, and is saturated only on the RT surface. Together with the
divergenceless property of $V_{A}$, the requirements for the vector field to
describe bit threads are all satisfied by $V_{A}$.

In the next sub-sections, we will explicitly construct vector fields $V_{A}$
for static intervals and spherical regions from the PEE structure and show
that $|V_{A}|=\frac{1}{4G}$ is satisfied on the RT surface $\mathcal{E}_{A}$.

###  3.2 PEE threads and bit threads in AdS3

$O$$-\ell$$-\ell$$-R$$R$$y$$Q$$\theta$ Figure 5: Determine the _PEE thread
flow_ $V^{\mu}_{O}$ at the point $Q=(\bar{r},\bar{z})$ by letting the flux of
the PEE threads (the red curves) through the reference RT surface $\Sigma$
(the green semicircle) with the radius $\ell=\sqrt{\bar{r}^{2}+\bar{z}^{2}}$
being equal to the entanglement contour of $r=0$ to the region $[-\ell,\ell]$.

Now we explicitly construct the _PEE thread flow_ $V_{r}^{\mu}$ and the _bit
thread flow_ $V_{A}^{\mu}$ in AdS3. The Poincaré metric on a static time slice
is given by

$\mathrm{d}s^{2}=\frac{1}{z^{2}}(\mathrm{d}r^{2}+\mathrm{d}z^{2}).$ (24)

We first determine $V_{O}^{\mu}$ which describes the PEE threads emanating
from the origin point $O$ settled at $r=0$, then generalize $V_{O}^{\mu}$ to
arbitrary boundary point $V_{r}^{\mu}$ under a translation in the $r$
direction.

For an arbitrary bulk point $Q=(\bar{r},\bar{z})=\ell(\cos\theta,\sin\theta)$
with

$\ell=\sqrt{\bar{r}^{2}+\bar{z}^{2}},\quad\tan\theta=\frac{\bar{z}}{\bar{r}},$
(25)

let us first determine the direction of $V_{O}^{\mu}$, which is the tangent
unit vector of the PEE thread (or geodesic) connecting $O$ and $Q$. Such a
thread will anchor at the boundary on another point $(r,z)=(y,0)$ (see Fig.5),
where $y$ can be determined,

$(\ell\cos\theta-y/2)^{2}+\ell^{2}\sin^{2}\theta=y^{2}/4,\quad\Rightarrow\quad
y=\frac{\ell}{\cos\theta}.$ (26)

Then the tangent unit vector of the PEE thread is given by

$\tau_{O}^{\mu}(Q)=\frac{\ell\sin\theta}{y/2}(\ell\sin\theta,-\ell\cos\theta+y/2)=\frac{2\bar{z}\bar{r}}{\bar{r}^{2}+\bar{z}^{2}}\left(\bar{z},\frac{\bar{z}^{2}-\bar{r}^{2}}{2\bar{r}}\right),$
(27)

and the _PEE thread flow_ can be written as

$V_{O}^{\mu}(Q)=|V_{O}(Q)|\tau_{O}^{\mu}(Q)=\frac{2\bar{z}\bar{r}|V_{O}(Q)|}{\bar{r}^{2}+\bar{z}^{2}}\left(\bar{z},\frac{\bar{z}^{2}-\bar{r}^{2}}{2\bar{r}}\right).$
(28)

To further determine its norm, we should make use of the requirement for the
flow (21). Let us choose a reference surface $\Sigma$ (which is the RT surface
for $[-\ell,\ell]$ and is green in Fig.5) that passes the point $Q$ as

$\Sigma:r^{2}+z^{2}=\ell^{2},$ (29)

with its normal unit vector at the point $Q$ given by

$n_{\Sigma}^{\mu}(Q)=\ell\sin\theta(\cos\theta,\sin\theta).$ (30)

The flux of the vector field $V_{O}^{\mu}$ through the RT surface $\Sigma$ is
given by

$\displaystyle\text{Flux}(V_{O}^{\mu},\Sigma)=$
$\displaystyle\int_{\Sigma}\mathrm{d}\theta\sqrt{h_{\theta\theta}}V_{O}^{\mu}(\theta)n_{\Sigma,\mu}$
(31) $\displaystyle=$
$\displaystyle\int_{\Sigma}\mathrm{d}\theta\sqrt{h_{\theta\theta}}|V_{O}(\theta)|\sin\theta,$

where $h_{\theta\theta}=1/\sin^{2}\theta$ is the $\theta\theta$-component of
the induced metric on $\Sigma$, and $V_{O}^{\mu}(\theta)$ is the value of the
vector field $V_{O}$ on $\Sigma$ parameterized by $\theta$.

According to the requirement (21), the flux equation equals to collecting all
the PEE threads that emanate from $O$ and terminate on the complement
$(-\infty,-\ell)\cup(\ell,\infty)$ region, i.e.

$\int_{0}^{\pi}\mathrm{d}\theta~{}\sqrt{h_{\theta\theta}}|V_{O}(\theta)|\sin\theta=\int_{(-\infty,-\ell)\cup(\ell,\infty)}\mathrm{d}y~{}\mathcal{I}(0,y).$
(32)

We can also parameterize $\Sigma$ using coordinate $r=y$ of the point where
the outer PEE threads terminate, which is related to the parameter $\theta$
via the relation (26), i.e. $y=R/\cos\theta$. This is just the one-to-one
mapping between the points on $\Sigma$ and points on $\bar{A}$ we mentioned
near (22). Then we arrive at the following equation for any $\ell$,

$\int_{(-\infty,-\ell)\cup(\ell,\infty)}\mathrm{d}y~{}\frac{\ell|V_{O}(\theta)|}{y^{2}\sin\theta}=\int_{(-\infty,-\ell)\cup(\ell,\infty)}\mathrm{d}y~{}\mathcal{I}(0,y).$
(33)

where the two-point PEE is given by $\mathcal{I}(0,y)={1}/{(4G_{N}y^{2})}$
following (13). Then the requirement (22) indicates that,

$|V_{O}(Q)|=\frac{\mathcal{I}(0,y)y^{2}\sin\theta}{\ell}=\frac{1}{4G_{N}}\frac{\bar{z}}{\bar{r}^{2}+\bar{z}^{2}},$
(34)

where we have written $\theta$ and $\ell$ in terms of $\bar{z}$ and $\bar{r}$.
With the norm of the _PEE thread flow_ determined, it is now straightforward
to write down the explicit formula for the _PEE thread flow_ ,

$V_{O}^{\mu}(Q)=\frac{1}{4G_{N}}\frac{2\bar{z}^{2}\bar{r}}{(\bar{r}^{2}+\bar{z}^{2})^{2}}\left(\bar{z},\frac{\bar{z}^{2}-\bar{r}^{2}}{2\bar{r}}\right).$
(35)

Due to shift symmetry along $r$-direction, the _PEE thread flow_ with
$r=r_{0}$ can be obtained by replacing $\bar{r}$ with $\bar{r}-r_{0}$ in (35)
and

$V_{r_{0}}^{\mu}(Q)=\frac{1}{4G_{N}}\frac{2\bar{z}^{2}(\bar{r}-r_{0})}{((\bar{r}-r_{0})^{2}+\bar{z}^{2})^{2}}\left(z,\frac{\bar{z}^{2}-(\bar{r}-r_{0})^{2}}{2(\bar{r}-r_{0})}\right).$
(36)

The above expression for the PEE threads from $x$ is of the central importance
of this paper. One can check that the vector field (36) is divergenceless.

Then we turn to the _bit thread flow_ $V_{A}$, which is the summation (or
superposition) of all the _PEE thread flow_ $V_{r_{0}}^{\mu}$ emanating from
the points inside an interval $A=[-R,R]$. The computation is just a simple
integration of $V_{r_{0}}$ over the interval $[-R,R]$,

$\displaystyle V_{A}^{\mu}(Q)$
$\displaystyle=\int^{R}_{-R}\mathrm{d}r_{0}V_{r_{0}}^{\mu}=\frac{\bar{z}^{2}}{4G_{N}}\frac{2R}{((R-\bar{r})^{2}+\bar{z}^{2})((R+\bar{r})^{2}+\bar{z}^{2})}\left(2\bar{z}\bar{r},R^{2}-\bar{r}^{2}+\bar{z}^{2}\right).$
(37)

$O$$-R$$R$ Figure 6: Bit threads configuration (the red curve) in pure AdS3
constructed from PEE threads. This bit threads configuration is the integral
curves of the _bit thread flow_ $V^{\mu}_{A}$. They are the geodesics normal
to the RT surface of $A$.

With the explicit formula for the _bit thread flow_ $V_{A}$ given, it is
straightforward to check that $V_{A}$ satisfies all the three requirements for
it to describe bit threads. Firstly, since the _PEE thread flow_ $V_{r_{0}}$
is divergenceless, $V_{A}$ should also be divergenceless. Secondly, if we set
$(\bar{r},\bar{z})=(R\cos\theta,R\sin\theta)$ hence $Q$ is on the RT surface,
we find that

$\displaystyle
V_{A}^{\mu}(Q)=\frac{R\sin\theta}{4G_{N}}(\cos\theta,\sin\theta)\,,$ (38)

where we can read $|V_{A}(Q)|=1/(4G_{N})$ and $V_{A}(Q)$ is normal to the RT
surface. At last, we can calculate $|V_{A}|$ anywhere in the bulk which is
given by

$\displaystyle|V_{A}|=\frac{2R\bar{z}}{\sqrt{\left(R^{2}+\bar{r}^{2}+\bar{z}^{2}\right)^{2}-4R^{2}\bar{r}^{2}}}.$
(39)

For any bulk point $(\bar{r},\bar{z})=l(\cos\theta,\sin\theta)$ away from the
RT surface, i.e. $l\neq R$, we have
$\left(R^{2}+\bar{r}^{2}+\bar{z}^{2}\right)^{2}-4R^{2}\bar{r}^{2}>4R^{2}l^{2}\sin^{2}\theta$,
which indicates that $|V_{A}|<1/(4G_{N})$ at the points away from the RT
surface.

In fact, according to the PEE threads picture, that $|V_{A}|<1/4G_{N}$ away
from RT surface is manifest, and we have the following statement:

  * •

Provided that, for any region $A$ the _bit thread flow_ $V_{A}^{\mu}$ is
normal to the RT surface and satisfies $|V_{A}|=1/4G$ on the RT surface, then
we have $|V_{A}|<1/4G$ away from RT surface.

We leave the proof of this statement in Appendix A. With these requirements
satisfied, it is sufficient to say that, the _bit thread flow_ gives an
optimal bit thread configuration.

In Fig.6, we plot this bit threads configuration using the integral curves of
the _bit thread flow_ $V^{\mu}_{A}$, and find that they are just the bulk
geodesics normal to the RT surface $\mathcal{E}_{A}$. This is not surprising
as we can check that, the _bit thread flow_ $V_{A}^{\mu}$ (37) we constructed
coincides exactly with the one found in [42] in $d=2$. As we have mentioned,
this is perhaps the most natural bit thread configuration that respect the
symmetries of the configuration. In the following subsection, we will show
that the coincidence also happens in higher dimensions.

###  3.3 PEE threads and bit threads in AdSd+1

For the vacuum CFTd on a static time slice, the two-point PEE is given by

$\mathcal{I}(\textbf{x}_{1},\textbf{x}_{2})=\frac{c}{6}\frac{2^{d-1}(d-1)}{\Omega_{d-2}|\textbf{x}_{2}-\textbf{x}_{1}|^{2(d-1)}}.$
(40)

The metric of the dual Poincaré AdSd+1 at a time slice is given by

$\mathrm{d}s^{2}=\frac{1}{z^{2}}\left(\mathrm{d}\textbf{x}^{2}+\mathrm{d}z^{2}\right)=\frac{1}{z^{2}}\left(\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{d-2}^{2}+\mathrm{d}z^{2}\right),$
(41)

where $\textbf{x}=\left(x_{1},x_{2},\cdots,x_{d-1}\right)$ and

$\displaystyle\mathrm{d}\Omega_{d-2}=\sin^{d-3}\phi_{1}\sin^{d-4}\phi_{2}...\sin\phi_{d-3}\mathrm{d}\phi_{1}\mathrm{d}\phi_{2}...\mathrm{d}\phi_{d-2}.$
(42)

Again, we first consider the _PEE thread flow_ $V_{O}$ for the origin $O$ at
$r=0$. Due to the rotational symmetry of $V_{O}$, we will restrict to the
2-dimensional slice with $\phi_{i}=0$ and employ the similar strategy as in
case of AdS3. This simplifies the analysis since the bulk geodesics have the
same function as those in AdS3 at this slice. For a bulk point
$Q=(\bar{r},\bar{z})=(\ell\cos\theta,\ell\sin\theta)$ on the $\phi_{i}=0$
slice, the _PEE thread flow_ is tangent to the PEE threads, hence

$V_{O}^{\mu}(Q)=\frac{2\bar{z}\bar{r}|V_{O}(Q)|}{\bar{r}^{2}+\bar{z}^{2}}\left(\bar{z},\frac{\bar{z}^{2}-\bar{r}^{2}}{2\bar{r}}\right).$
(43)

The flux equation (21) becomes

$\int_{\Sigma}\mathrm{d}\theta~{}\frac{\bar{r}^{d-2}}{\bar{z}^{d-2}}|V_{O}|=\int\mathrm{d}y~{}y^{d-2}\mathcal{I}(0,y),$
(44)

where $\Sigma$ is a reference surface $r^{2}+z^{2}=\ell^{2}$ that passes
through $Q$. Since the PEE threads have the same function as those in AdS3,
the one-to-one mapping between the points on $\Sigma$ and the boundary points
outside the region associated to $\Sigma$ on the $\phi_{i}=0$ slice is again
given by the relation (26), i.e. $y=\ell/\cos\theta$. Then the norm of the
_PEE thread flow_ is given by

$|V_{O}(Q)|=\frac{1}{4G_{N}}\frac{2^{d-1}(d-1)}{\Omega_{d-2}}\frac{\bar{z}^{d-1}}{(\bar{r}^{2}+\bar{z}^{2})^{d-1}}.$
(45)

And then the _PEE thread flow_ is given by

$V_{O}^{\mu}(Q)=\frac{2^{d}\bar{z}^{d}}{4G_{N}}\frac{(d-1)}{\Omega_{d-2}}\frac{\bar{r}}{(\bar{r}^{2}+\bar{z}^{2})^{d}}\left(\bar{z},\frac{\bar{z}^{2}-\bar{r}^{2}}{2\bar{r}}\right).$
(46)

Note that, on the $\phi_{i}=0$ slice the $V_{O}^{\phi_{i}}$ components are
zero.

When $d\geq 3$, the explicit formula of the vector field $V_{P}$ for an
arbitrary boundary point $P$ is much more complicated. Nevertheless, due to
the symmetry of the configuration, we will see that solving $V_{O}^{\mu}$ is
enough to determine the _bit thread flow_ $V_{A}^{\mu}$ for any static
spherical region $A$. For any point $P:(\textbf{x},0)$ on the boundary and
$Q:(\textbf{y},\bar{z})$ in the bulk, one can consider the projection point
$Q_{b}:(\textbf{y},0)$ of $Q$ on the boundary. Due to the translational and
rotational symmetries of $V_{P}$, it will be useful to note that, $V_{P}(Q)$
takes the same formula as (46), with $\bar{r}$ replaced by
$|\textbf{x}-\textbf{y}|$ and the $r$ coordinate parameterizing the direction
from $P$ to $Q$.

Now we turn to the _bit thread flow_ $V_{A}$. For example, given a spherical
region $A=\\{\textbf{x}||\textbf{x}|<R\\}$, let us again confine the analysis
on the $\phi_{i}=0$ slice and first consider the cases with $\bar{r}>R$. In
other words, the projection point $Q_{b}=(\bar{r},0)$ of the bulk point
$Q=(\bar{r},\bar{z})$ on the boundary is outside $A$. Our strategy is that, we
first decompose the region $A$ into layers of partial spherical shells
$\leavevmode\hbox to9.3pt{\vbox to9.3pt{\pgfpicture\makeatletter\hbox{\hskip
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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}$ centered at
the projection point $Q_{b}$, with radius $r_{0}>\bar{r}-R$ (see Fig.7). The
angular coordinate that parameterize the shell is $\phi_{1}$. Then we use (46)
to calculate the contribution to $V_{A}(Q)$ from each shell. And finally, we
integrate the contribution from all these shells to get $V_{A}(Q)$.

$\bar{\phi}_{1}$$r_{0}$$\bar{r}$$Q_{b}$$R$$Q$$\bar{z}$$A$ Figure 7: The grey
area represents a spherical region $A$ with radius $R$, centered at the
origin. The region $A$ is layered with partial spherical shells. These shells
are centered at the boundary-projected point $(\bar{r},z=0,\phi_{i}=0)$ with a
radius $r_{0}>\bar{r}-R$.

The area of a partial spherical shells $\leavevmode\hbox to9.3pt{\vbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}$ is given by,

$\displaystyle\int_{0}^{\bar{\phi}_{1}}\mathrm{d}\phi_{1}\Omega_{d-3}r_{0}^{d-2}\sin^{d-3}\phi_{1},$
(47)

where the upper limit of the integration $\bar{\phi}_{1}$ is given by,

$\cos(\bar{\phi}_{1})=\frac{r_{0}^{2}+\bar{r}^{2}-R^{2}}{2r_{0}\bar{r}}.$ (48)

Since $V_{A}$ is $\phi_{i}$-rotational symmetrical, the direction of
$V_{A}(Q)$ lies exactly at the $\phi_{i}=0$ slice, which means we only need to
compute $V^{r}_{A}(Q)$ and $V^{z}_{A}(Q)$. The reason we classify the points
using the such spherical shells is that, points in the same shell has the same
distance from the projection point $Q_{b}$, hence their contribution to
$V_{A}(Q)$ takes the same formula as (46), with $\bar{r}$ replaced by their
distance to $Q_{b}$ and $r$ representing the direction pointing to $Q_{b}$.
Then it is straightforward to write down the contribution to $V^{z}_{A}(Q)$
from $\leavevmode\hbox to9.3pt{\vbox
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}$, which is
just the summation from all the points on $\leavevmode\hbox to9.3pt{\vbox
to9.3pt{\pgfpicture\makeatletter\hbox{\hskip 4.65001pt\lower-4.65001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}$,

$\begin{split}V^{z}_{\text{partial}\leavevmode\hbox to9.3pt{\vbox
to9.3pt{\pgfpicture\makeatletter\hbox{\hskip 4.65001pt\lower-4.65001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}}(Q)=&\int_{0}^{\bar{\phi}_{1}}\mathrm{d}\phi_{1}V_{\textbf{x}=0}^{z}(r_{0})\Omega_{d-3}r_{0}^{d-2}\sin^{d-3}\phi_{1}\\\
=&V_{\textbf{x}=0}^{z}(r_{0})\Omega_{d-3}r_{0}^{d-2}\left(-_{2}F_{1}\left(\frac{1}{2},\frac{4-d}{2},\frac{3}{2},\cos^{2}\bar{\phi}_{1}\right)\cos\bar{\phi}_{1}+\frac{\pi^{3/2}\cos^{-1}(\frac{(d-3)\pi}{2})}{(d-3)\Gamma(\frac{4-d}{2})\Gamma(\frac{d-3}{2})}\right),\end{split}$
(49)

while the contribution to $V^{r}_{A}(Q)$ should have an additional multiplier
$\cos\phi_{1}$ to get the projection on the $r$ direction,

$\begin{split}V^{r}_{\text{partial}\leavevmode\hbox to9.3pt{\vbox
to9.3pt{\pgfpicture\makeatletter\hbox{\hskip 4.65001pt\lower-4.65001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}}(Q)=&\int_{0}^{\bar{\phi}_{1}}\mathrm{d}\phi_{1}V_{\textbf{x}=0}^{r}(r_{0})\Omega_{d-3}r_{0}^{d-2}\sin^{d-3}\phi_{1}\cos\phi_{1}\\\
=&V_{\textbf{x}=0}^{r}(r_{0})\Omega_{d-3}r_{0}^{d-2}\frac{\sin^{d-2}\bar{\phi}_{1}}{d-2}.\end{split}$
(50)

At last, we integrate over all the partial spherical shells to get the _bit
thread flow_ ,

$\displaystyle V_{A}^{r}(Q)=$
$\displaystyle\int_{\bar{r}-R}^{\bar{r}+R}\mathrm{d}r_{0}~{}V^{r}_{\text{partial}\leavevmode\hbox
to9.3pt{\vbox to9.3pt{\pgfpicture\makeatletter\hbox{\hskip
4.65001pt\lower-4.65001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}}=\frac{1}{4G_{N}}\frac{\bar{r}\bar{z}}{R}\left(\frac{2R\bar{z}}{\sqrt{(R^{2}+\bar{r}^{2}+\bar{z}^{2})^{2}-4R^{2}\bar{r}^{2}}}\right)^{d},$
(51) $\displaystyle V_{A}^{z}(Q)=$
$\displaystyle\int_{\bar{r}-R}^{\bar{r}+R}\mathrm{d}r_{0}~{}V^{z}_{\text{partial}\leavevmode\hbox
to9.3pt{\vbox to9.3pt{\pgfpicture\makeatletter\hbox{\hskip
4.65001pt\lower-4.65001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}}=\frac{1}{4G_{N}}\frac{R^{2}-\bar{r}^{2}+\bar{z}^{2}}{2R}\left(\frac{2R\bar{z}}{\sqrt{(R^{2}+\bar{r}^{2}+\bar{z}^{2})^{2}-4R^{2}\bar{r}^{2}}}\right)^{d}.$
(52)

$Q_{b}$$Q$$\bar{z}$$A$ Figure 8:  The bulk point is projected inside the
region $A$. The shaded area has no contribution to $V_{A}^{r}$.

Then we consider the case with $\bar{r}<R$, where the projection point $Q_{b}$
of $Q$ lies inside $A$. Following the same strategy as the case with
$\bar{r}>R$, the region $A$ is also foliated by layers of partial spherical
shells centered at $\bar{r}$ and one may obtain $V^{\mu}_{A}$ by integrating
over all the shells. The only difference is that, in the spherical region
centered at $Q_{b}$ with radius $R-\bar{r}$, the whole spherical shell
contribute to $V_{A}$ (see Fig.8). For symmetry reason, this spherical region
has no contribution to $V^{r}_{A}$, and its contribution to $V^{z}_{A}$ is
computed by,

$\begin{split}\int_{0}^{R-\bar{r}}\mathrm{d}r_{0}V^{z}_{\text{whole}\leavevmode\hbox
to9.3pt{\vbox to9.3pt{\pgfpicture\makeatletter\hbox{\hskip
4.65001pt\lower-4.65001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}}=&\int_{0}^{R-\bar{r}}\mathrm{d}r_{0}\int_{0}^{\pi}\mathrm{d}\phi_{1}V_{\textbf{x}=0}^{z}(r_{0})\Omega_{d-3}r_{0}^{d-2}\sin^{d-3}\phi_{1}\end{split}.$
(53)

The contribution from the other points on the partial spherical shell can be
carried out following our discussion for the case with $\bar{r}>R$. Then we
get the _bit thread flow_ $V_{A}(Q)$

$\displaystyle V_{A}^{r}(Q)=$
$\displaystyle\int_{R-\bar{r}}^{\bar{r}+R}\mathrm{d}r_{0}~{}V^{r}_{\text{partial}\leavevmode\hbox
to9.3pt{\vbox to9.3pt{\pgfpicture\makeatletter\hbox{\hskip
4.65001pt\lower-4.65001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}}=\frac{1}{4G_{N}}\frac{\bar{r}\bar{z}}{R}\left(\frac{2R\bar{z}}{\sqrt{(R^{2}+\bar{r}^{2}+\bar{z}^{2})^{2}-4R^{2}\bar{r}^{2}}}\right)^{d},$
(54)

and

$\displaystyle V_{A}^{z}(Q)=$
$\displaystyle\int_{0}^{R-\bar{r}}\mathrm{d}r_{0}V^{z}_{\text{whole}\leavevmode\hbox
to9.3pt{\vbox to9.3pt{\pgfpicture\makeatletter\hbox{\hskip
4.65001pt\lower-4.65001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}}+\int_{R-\bar{r}}^{\bar{r}+R}\mathrm{d}r_{0}~{}V^{z}_{\text{partial}\leavevmode\hbox
to9.3pt{\vbox to9.3pt{\pgfpicture\makeatletter\hbox{\hskip
4.65001pt\lower-4.65001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\pgfsys@roundcap\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{ }
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.90001pt}{0.0pt}\pgfsys@curveto{3.90001pt}{2.15393pt}{2.15393pt}{3.90001pt}{0.0pt}{3.90001pt}\pgfsys@curveto{-2.15393pt}{3.90001pt}{-3.90001pt}{2.15393pt}{-3.90001pt}{0.0pt}\pgfsys@curveto{-3.90001pt}{-2.15393pt}{-2.15393pt}{-3.90001pt}{0.0pt}{-3.90001pt}\pgfsys@curveto{2.15393pt}{-3.90001pt}{3.90001pt}{-2.15393pt}{3.90001pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.3pt}{0.0pt}\pgfsys@curveto{1.3pt}{0.71797pt}{0.71797pt}{1.3pt}{0.0pt}{1.3pt}\pgfsys@curveto{-0.71797pt}{1.3pt}{-1.3pt}{0.71797pt}{-1.3pt}{0.0pt}\pgfsys@curveto{-1.3pt}{-0.71797pt}{-0.71797pt}{-1.3pt}{0.0pt}{-1.3pt}\pgfsys@curveto{0.71797pt}{-1.3pt}{1.3pt}{-0.71797pt}{1.3pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r_{0}}}$ (55)
$\displaystyle=$
$\displaystyle\frac{1}{4G_{N}}\frac{R^{2}-\bar{r}^{2}+\bar{z}^{2}}{2R}\left(\frac{2R\bar{z}}{\sqrt{(R^{2}+\bar{r}^{2}+\bar{z}^{2})^{2}-4R^{2}\bar{r}^{2}}}\right)^{d}.$

The above results share the same expressions as (51) and (52), which means we
do not need to distinguish between the case with $\bar{r}>R$ and the case with
$\bar{r}>R$.

Remarkably, in general dimensions the resulting $V^{\mu}_{A}$ also coincides
with the bit threads configuration (16) constructed in [42].

##  4 PEE threads for multi-intervals

$B_{1}$$A_{2}$$A_{1}$$B_{2}$$B_{2}$$\omega=2$$\omega=0$$\omega=1$$\alpha>1/2$

$B_{1}$$A_{2}$$A_{1}$$B_{2}$$B_{2}$$\omega=2$$\omega=0$$\omega=1$$0<\alpha<1/2$

Figure 9: Connected phase (the upper figure) and disconnected phase (the lower
figure) for $A=A_{1}\cup A_{2}$. The threads connecting $A_{1}$ and $A_{2}$ in
disconnected phase are _outer threads_ as they pass through the boundary of
$\mathcal{W}_{A}$ (blue shaded regions), while those in connected phase are
_inner threads_ as they are confined in $\mathcal{W}_{A}$.

Previously, we mainly focus on the PEE thread configurations for intervals or
spherical boundary regions, which are connected regions. We have shown that
the entanglement entropy is given by the flux associated with all the _outer
threads_ of the connected region. In this section, we turn to the cases of
disconnected intervals and show how the holographic entanglement entropy can
be reproduced by counting certain classes of PEE threads in the bulk. This is
a non-trivial problem as the entanglement entropy for disconnected regions
undergoes a phase transition where the RT surfaces in the bulk changes
discontinuously [77]. Upon such phase transitions, the bit thread
configurations should also change while the PEE thread configurations should
not change. A naive generalization of the scheme in the previous section
cannot reproduce the holographic entanglement entropy for disconnected regions
[59]. In the following, we propose that due to the phase transition of the
entanglement wedge, new rules to define the _inner_ and _outer PEE threads_
should be taken into account to reproduce the holographic entanglement
entropy.

In this section we focus on the case of Poincaré AdS3. Consider a disconnected
interval $A=A_{1}\cup A_{2}$ and its complement $B=B_{1}\cup B_{2}$ on the
boundary, and $A\cup B$ makes up the vacuum state of the boundary holographic
CFT2. Let us mark $A_{1}=[a_{1},a_{2}]$ and $A_{2}=[a_{3},a_{4}]$, and mark
the entanglement wedge $\mathcal{W}_{A}$ of $A$ in blue (see Fig.9). According
to the RT formula [2, 3, 77] when the cross ratio

$\displaystyle\alpha=\frac{(a_{2}-a_{1})(a_{4}-a_{3})}{(a_{3}-a_{1})(a_{4}-a_{2})}>\frac{1}{2},$
(56)

$\mathcal{W}_{A}$ is connected and we have
$\mathcal{W}_{A}\supset\mathcal{W}_{A_{1}}\cup\mathcal{W}_{A_{2}}$. In this
connected phase the entanglement entropy is calculated by

$\textit{Connected phase}:\quad S_{A}=S_{B_{1}}+S_{A_{1}B_{1}A_{2}},$ (57)

where $B_{1}$ is the sandwiched interval between $A_{1}$ and $A_{2}$. As
$A_{1}$ and $A_{2}$ get far apart $0<\alpha<\frac{1}{2}$, the
$\mathcal{W}_{A}$ becomes disconnected and
$\mathcal{W}_{A}=\mathcal{W}_{A_{1}}\cup\mathcal{W}_{A_{2}}$. In this
disconnected phase, the entanglement entropy becomes

$\textit{Disconnected phase}:\quad S_{A}=S_{A_{1}}+S_{A_{2}}.$ (58)

Previously we define the PEE threads that connecting one point inside $A$ and
another point outside $A$ as the _outer threads_ , which are the threads that
contribute non-trivially to the entanglement entropy. A simple example to
illustrate this problem is the two interval in disconnected phase. If we stick
to this definition for the inner and _outer threads_ , then the PEE threads
connecting points in $A_{1}$ and points in $A_{2}$ should be _inner threads_
hence do not contribute to $S_{A}$, and we get

$\displaystyle
S_{A}=\mathcal{I}(A,B)=\mathcal{I}(A_{1},B)+\mathcal{I}(A_{2},B)\,.$ (59)

This is not consistent with the RT formula (58). Also, it was claimed in [59]
that, the normalization property $S_{A}=\mathcal{I}(A,B)|_{B\to\bar{A}}$ of
the PEE does not hold for the disjoint intervals. On the right hand side of
(58), $S_{A_{1}}$ contains all the _outer threads_ of $A_{1}$, which can be
decomposed into $\mathcal{I}(A_{1},B)\cup\mathcal{I}(A_{1},A_{2})$, and a
similar decomposition applies to $S_{A_{2}}$ hence we should have

$\displaystyle
S_{A}=\mathcal{I}(A_{1},B)+\mathcal{I}(A_{2},B)+2\mathcal{I}(A_{1},A_{2})\,.$
(60)

The above equation indicates that, the PEE threads connecting $A_{1}$ and
$A_{2}$ not only contribute non-trivially to $S_{A}$, but also to $S_{B}$ as
$S_{A}=S_{B}$. More interestingly, the contribution from such PEE threads is
doubly counted.

For the purpose to reproduce the holographic entanglement entropy from the PEE
threads, we should reconsider the classification of _inner_ and _outer
threads_. Also, note that the PEE threads connecting $A_{1}$ and $A_{2}$ in
the disconnected phase pass through the entanglement wedge $\mathcal{W}_{B}$
twice in the bulk. This is a new phenomenon compared with the configurations
where $A$ is a single interval and the _inner threads_ are all confined in
$\mathcal{W}_{A}$. The holographic entanglement entropy between $A$ and $B$ is
indeed the generalized gravitational entanglement entropy between
$\mathcal{W}_{A}$ and $\mathcal{W}_{B}$ [5]. If the PEE threads are some
physical objects in the bulk that representing the entanglement flow in the
dual gravity, then it is natural to count the threads that pass through the
boundaries between $\mathcal{W}_{A}$ and $\mathcal{W}_{B}$.

Inspired by the above discussions, we give a new definition for the _inner_
and _outer PEE threads_ based on the entanglement wedge configuration, which
solves all the problems one has in AdS3. Given two (connected or disconnected)
regions $A$ and $B$, which together make up the whole boundary, and their
entanglement wedges in the bulk, the _inner_ and _outer threads_ are defined
in the following:

  * •

outer threads: PEE threads that pass through the boundary between
$\mathcal{W}_{A}$ and $\mathcal{W}_{B}$.

  * •

inner threads: PEE threads that are confined inside $\mathcal{W}_{A}$ or
$\mathcal{W}_{B}$.

In addition we also define a new parameter for the _outer threads_ ,

  * •

weight of an outer thread $\omega$: the number of the times that the outer
thread passes through the boundary between $\mathcal{W}_{A}$ and
$\mathcal{W}_{B}$. The _inner threads_ are just threads with $\omega=0$.

For any connected regions $A$ and $B$, we also refer to
$\omega_{AB}=\omega_{BA}$ as the weight of the threads connecting $A$ and $B$.
The threads with $\omega=0$ are just _inner threads_. Obviously, the above
definition reduces to the definition in section 3.1 for the cases of $A$ being
single intervals.

Let us use the above definitions to explain the entanglement entropy $S_{A}$
in the connected phase (see the upper figure in Fig.9). In this case, given
$i,j=1,2$ we have $\omega_{A_{i}A_{j}}=0$, $\omega_{A_{i}B_{j}}=1$ and
$\omega_{B_{i}B_{j}}=2$, thus, by counting the PEE threads with different
weights, the entanglement entropy is given by

$\displaystyle S_{A}=$ $\displaystyle\mathcal{I}(A_{1},B_{1}\cup
B_{2})+\mathcal{I}(A_{2},B_{1}\cup B_{2})+2\mathcal{I}(B_{1},B_{2})$ (61)
$\displaystyle=$ $\displaystyle\mathcal{I}(A_{1}\cup A_{2}\cup
B_{2},B_{1})+\mathcal{I}(A_{1}\cup A_{2}\cup B_{1},B_{2})$ (62)
$\displaystyle=$ $\displaystyle S_{B_{1}}+S_{B_{2}}\,,$ (63)

which coincides with the RT formula.

$B_{2}$$A_{3}$$A_{2}$$B_{1}$$B_{3}$$B_{3}$$A_{1}$$\omega=1$$\omega=2$$\omega=0$$\omega=3$
Figure 10: The entanglement wedge and PEE threads with different weights for
$A=A_{1}\cup A_{2}\cup A_{3}$ with disconnected $\mathcal{W}_{A_{1}\cup
A_{2}\cup A_{3}}$ and connected $\mathcal{W}_{A_{2}\cup A_{3}}$. Here the
direction of the PEE flow is from the inside to the outside of the
entanglement wedge.

We can further explore more complicated scenarios, such as a multi-interval
configuration $A=A_{1}\cup A_{2}\cup A_{3}$ depicted in Fig.10. In this setup,
$A_{1}$ is situated sufficiently far from $A_{2}$ and $A_{3}$ so that the
entanglement wedge $\mathcal{W}_{A_{1}\cup A_{2}\cup A_{3}}$ becomes
disconnected. Moreover, $A_{2}$ and $A_{3}$ maintain close proximity,
resulting in a connected $\mathcal{W}_{A_{2}\cup A_{3}}$. The complementary
region to $A$ is identified as $B=B_{1}\cup B_{2}\cup B_{3}$. Here, $B_{2}$ is
settled between $A_{2}$ and $A_{3}$, while $B_{1}$ is flanked by $A_{1}$ and
$A_{2}$. According to the RT formula, the entanglement entropy is given by:

$\displaystyle S_{A}=$ $\displaystyle S_{A_{1}}+S_{B_{2}}+S_{A_{2}\cup
B_{2}\cup A_{3}}\,.$ (64)

To recover the RT result, we read out the weights of _outer threads_ from the
entanglement wedge configuration Fig.10. The weights of the _outer threads_
are presented in Table 1. By counting the _outer threads_ , we obtain the
entanglement entropy:

$\displaystyle S_{A}=$
$\displaystyle\mathcal{I}(A_{1},B_{1}B_{3})+2\mathcal{I}(A_{1},A_{2}A_{3})+3\mathcal{I}(A_{1},B_{2})+\mathcal{I}(A_{2}A_{3},B_{1}B_{2}B_{3})+2\mathcal{I}(B_{1}B_{3},B_{2})$
$\displaystyle=$
$\displaystyle\mathcal{I}(A_{1},A_{2}A_{3}B_{1}B_{2}B_{3})+\mathcal{I}(B_{2},A_{1}A_{2}A_{3}B_{1}B_{3})+\mathcal{I}(A_{2}B_{2}A_{3},A_{1}B_{1}B_{3})$
$\displaystyle=$ $\displaystyle S_{A_{1}}+S_{B_{2}}+S_{A_{2}\cup B_{2}\cup
A_{3}},$ (65)

This result aligns with the RT formula.

$\omega$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $B_{1}$ | $B_{2}$ | $B_{3}$
---|---|---|---|---|---|---
$A_{1}$ | 0 | 2 | 2 | 1 | 3 | 1
$A_{2}$ | 2 | 0 | 0 | 1 | 1 | 1
$A_{3}$ | 2 | 0 | 0 | 1 | 1 | 1
$B_{1}$ | 1 | 1 | 1 | 0 | 2 | 0
$B_{2}$ | 3 | 1 | 1 | 2 | 0 | 2
$B_{3}$ | 1 | 1 | 1 | 0 | 2 | 0
Table 1: The weight of PEE threads for $A=A_{1}\cup A_{2}\cup A_{3}$ with
disconnected $\mathcal{W}_{A_{1}\cup A_{2}\cup A_{3}}$ and connected
$\mathcal{W}_{A_{2}\cup A_{3}}$.

Note that the above definition for the _inner_ and _outer threads_ and the
weight of the threads depends on an explicit configuration for the
entanglement wedge. One may conclude that, the PEE threads may not be a
complete reformulation of the RT formula in AdS3. Nevertheless, we can get rid
of this dependence by considering all the possible assignments for the weights
and choose the one that gives the minimal value for the entanglement entropy
$S_{A}$.

More explicitly, for a region $A$ with multi sub-intervals $A=\cup A_{i}$ and
its complement $B=\cup B_{j}$, let us consider an arbitrary (extremal) surface
in the bulk that is homologous to $A$, thus we can get a corresponding
assignment for the weights $\omega_{A_{i}A_{j}}$, $\omega_{A_{i}B_{j}}$ and
$\omega_{B_{i}B_{j}}$ by counting the number of times the thread intersect
with the surface. Then the entanglement entropy is given by minimizing the
following summation,

$\displaystyle
S_{A}=S_{B}=\text{min}\left[\sum_{i,j}\omega_{A_{i}A_{j}}\mathcal{I}(A_{i},A_{j})+\sum_{i,j}\omega_{B_{i}B_{j}}\mathcal{I}(B_{i},B_{j})+\sum_{i,j}\omega_{A_{i}B_{j}}\mathcal{I}(A_{i},B_{j})\right]$
(66)

There will be a class of homologous surfaces that minimize the above
summation, and the extremal one will coincide with the RT surface. Since $A$
and $B$ are separated by a homologous surface, we should in general have that:

  * •

$\omega_{A_{i}B_{j}}$ should be odd numbers, i.e.
$\omega_{A_{i}B_{j}}=1,3,5\cdots$,

  * •

$\omega_{A_{i}A_{j}}$ and $\omega_{B_{i}B_{j}}$ can be zero or non-zero even
numbers, i.e. $\omega_{A_{i}A_{j}},\omega_{B_{i}B_{j}}=0,2,4\cdots$.

Now we revisit the two interval case shown in Fig.9. In this case the minimal
weight of $\omega_{A_{i}B_{j}}$ is 1. While the minimal weight for
$\omega_{A_{1}A_{2}}$ and $\omega_{B_{1}B_{2}}$ is zero. Nevertheless, they
can not be zero simultaneously, since there is no homologous surface that is
consistent with the assignment $\omega_{A_{1}A_{2}}=\omega_{B_{1}B_{2}}=0$. So
we conclude that, the possible assignments for the weights that may give the
minimal $S_{A}$ are listed in the following:

  1. 1.

$\omega_{\omega_{A_{i}B_{j}}}=1$, $\omega_{A_{1}A_{2}}=0$,
$\omega_{B_{1}B_{2}}=2$,

  2. 2.

$\omega_{\omega_{A_{i}B_{j}}}=1$, $\omega_{A_{1}A_{2}}=2$,
$\omega_{B_{1}B_{2}}=0$.

Then the corresponding entanglement entropy $S_{A}$ is calculated by (66),
which is the minimal weighted summation between the following two results:

  1. 1.

$\mathcal{I}(A,B)+2\mathcal{I}(B_{1},B_{2})=S_{A_{1}B_{2}A_{2}}+S_{B_{2}}$.

  2. 2.

$\mathcal{I}(A,B)+2\mathcal{I}(A_{1},A_{2})=S_{A_{1}}+S_{A_{2}}$.

When $A_{1}$ and $A_{2}$ is far enough such that $0<\alpha<1/2$, the first
result will give the minimal value for $S_{A}$. While the second result
minimizes $S_{A}$ when $A_{1}$ and $A_{2}$ get closer such that $\alpha>1/2$.
This gives a complete reformulation for the phase transition of the RT surface
in terms of the PEE threads.

##  5 Discussions

In this paper, we propose a natural scheme to geometrize the PEE using bulk
geodesics, which we refer to as PEE threads. The configuration of these PEE
threads is solely determined by the state and represents an intrinsic
structure that is independent of the specific region under consideration. In
the context of Poincaré AdS spacetime, we demonstrate that for any static
interval or spherical region, a unique configuration of bit threads can emerge
from the PEE thread configuration by superimposing all the _PEE thread flows_
emanating from that region. These bit thread configurations are regarded as
the most natural ones as they respect the symmetries of the system and possess
a clear physical interpretation inherited from PEE.

In AdS3/CFT2, we further study the reformulation of the RT formula for multi-
interval regions based on the PEE thread configuration. We find that, the PEE
threads should be weighted by the number of times that they intersect with the
RT surface. Remarkably, we can get rid of the predetermined RT surface by
considering all the possible assignments that are consistent with any
homologous surface of the region, and choosing the assignment that gives the
minimal summation (66). The assignment that minimizes the summation will
correspond to certain type of homologous surface, which is exactly the RT
surface if we require it to be extremal. This reformulation also works for
single intervals.

For the cases of multi intervals, since the PEE threads can intersect with the
RT surface multiple times, the claim that there is a definite direction for
any PEE threads does not make sense. This means our strategy to generate bit
thread configurations from PEE threads breakdown. Nevertheless, we think the
formulation of weighted PEE threads makes more sense. The reason is that, the
holographic entanglement entropy is the generalized gravitational
(entanglement) entropy [5] between entanglement wedges $\mathcal{W}_{A}$ and
$\mathcal{W}_{B}$, which includes different bulk regions at the critical
point. It is reasonable that, the parts of the PEE threads in different
entanglement wedges play different roles. One can also define an in-definite
flow direction which is always locally from one side of the homologous surface
to the other side (see the arrows in Fig. 10). Then the norm of the PEE flow
on homologous surface can recover $1/(4G)$ only if this homologous surface
coincide with the RT surface.

The success of our reformulation indicates that the PEE threads may be a
certain physical quantity that lies in the bulk gravity theory. It seems that,
the part of the PEE thread that lies in $\mathcal{W}_{A}$ should be considered
as degrees of freedom in $\mathcal{W}_{A}$ while the other part belongs to
$\mathcal{W}_{B}$. The entanglement entropy between the two parts at leading
order is just proportional to the number of partition points, i.e. the weight
$\omega$. It will be very interesting to explore a deeper physical
interpretation of the PEE threads in the bulk (quantum) gravity or its toy
models. For example, the PEE threads is very reminiscent of the so-called
pentagon-edge geodesics defined in the toy model of AdS3/CFT2 based on fracton
models with subsystem symmetry [78, 79]. Also the PEE threads may be closed
related to the minimal path along the tensor network as toy models of quantum
gravity (see [80, 81, 82] for examples).

One may wonder if we can generate a bit thread configuration for non-spherical
but connected regions in higher dimensions. We give a simple exploration for
the _bit thread flow_ $V_{A}$ for a strip in Appendix.B. Unfortunately, as we
can see $|V_{A}|<1/4G_{N}$ on the RT surface of strip and thus does not
satisfy the requirements for bit threads. Moreover, a naive integration of the
PEE flow does not reproduce the entanglement entropy calculated by the RT
formula. One important reason for this failure could be the fact that, unlike
the spherical regions, the modular Hamiltonian is non-local for strip. Another
possible reason is that, unlike the spherical regions, the PEE threads with
both its endpoints inside or outside the region can also intersect with the RT
surface twice. This indicates that we should classify the PEE threads and
weight them differently as in the multi-interval cases, hence the entanglement
entropy may coincide with the RT formula. Indeed this is a non-trivial task
even for strips. We leave this for further investigations.

There are absolutely many other interesting future directions relevant to the
PEE threads. For example, we can explore the problems of how to generalize the
PEE threads to its covariant version, how to extend our discussion from
Poincaré AdS to AdS black holes and how to define a quantum version of PEE
threads to include the quantum correction [6] in the bulk gravity as was done
in [46, 45] for a quantum version of bit threads. The PEE has been explored in
holographic theories beyond AdS/CFT [83, 60, 65, 84, 69]. Generalizing the PEE
threads to more generic holographic theories is also very interesting.

## Acknowledgment

J. Lin is supported by the National Natural Science Foundation of China under
Grant No.12247117, No.12247103 and No.12047502. Y. Lu received supported from
the China Postdoctoral Science Foundation under Grant No.2022TQ0140, the
National Natural Science Foundation of China under Grant No.12247161, and the
NSFC Research Fund for International Scientists (Grant No. 12250410250). Q.
Wen would like to thank the “Zhishan Scholars” program of Southeast
University.

##  Appendix A Proof of a statement

Let us prove the statement for static intervals in the following:

  * •

Provided that, for any region $A$ the _bit thread flow_ $V_{A}^{\mu}$ is
normal to the RT surface and satisfy $|V_{A}|=1/4G$ on the RT surface, then we
have $|V_{A}|<1/4G$ away from RT surface.

The RT surfaces are static semicircles in pure AdS3 spacetime. As is shown in
Fig.11, for a bulk point $Q$ inside the entanglement wedge $\mathcal{W}_{A}$,
only the PEE threads emanating from the red region $[r_{1},r_{2}]$ flows
outside the $\mathcal{W}_{A}$ and contribute non-trivially to
$V_{A}^{\mu}(Q)$. The vector $V_{A}^{\mu}(Q)$ determines a RT surface
$\mathcal{E}_{R_{Q}}$ of a region $R_{Q}$, which passes through $Q$ and normal
to $V_{A}^{\mu}(Q)$. Generally, there are following two possibilities:

  1. 1.

The whole red region lies inside $R_{Q}$, i.e. $[r_{1},r_{2}]\subset R_{Q}$.

  2. 2.

Only a part of the red region lies inside $R_{Q}$.

The PEE threads from the red region $[r_{1},r_{2}]$ and inside $R_{Q}$ will
contribute positively to $V_{A}^{\mu}(Q)$ while those from the red region but
outside $R_{Q}$ contribute negatively to $V_{A}^{\mu}(Q)$.

On the other hand, let us consider the region $R_{Q}$ where the _bit thread
flow_ $V_{R_{Q}}(Q)$ has the same direction as $V_{A}(Q)$ and
$|V_{R_{Q}}(Q)|=\frac{1}{4G}$. All the _PEE thread flow_ $V_{\textbf{x}}$ for
$R_{Q}$ contribute positively along the direction of $V_{A}(Q)$. This means
the contribution from the red region $[r_{1},r_{2}]$ that lies inside $R_{Q}$
to $V_{A}(Q)$ is less than $\frac{1}{4G}$. As we have shown that the red
region that lies outside $R_{Q}$ contribute negatively to $V_{A}(Q)$, we can
conclude that $|V_{A}(Q)|<\frac{1}{4G}$. A similar argument also applies to
the case where the field point lies outside $\mathcal{E}_{A}$, hence the
statement is proved.

$V_{A}^{\mu}$$r_{1}$$r_{2}$$Q$$\mathcal{E}_{R_{Q}}$$\mathcal{E}_{A}$

$V_{A}^{\mu}$$r_{1}$$r_{2}$$Q$$\mathcal{E}_{R_{Q}}$$\mathcal{E}_{A}$

Figure 11:  The total vector field $V^{\mu}_{A}$ at $Q$ receives the net
contributions from the PEE threads of the red boundary region. For $Q$ inside
the $\mathcal{W}_{A}$, this red region $[r_{1},r_{2}]$ is determined by two
intersecting geodesics (red dashed curves). The green dashed semicircle
denotes the RT surface $\mathcal{E}_{R_{Q}}$ passing through the field point
$Q$ and normal to $V^{\mu}_{A}$. $R_{Q}$ is the associated boundary region of
$\mathcal{E}_{R_{Q}}$.

##  Appendix B PEE threads for a strip region

Now let us construct the _bit thread flow_ $V_{A}^{\mu}$ for
$(d-1)$-dimensional strip region with

$x_{1}\in[-R,R],\quad x_{i}\in(-\infty,\infty),\text{for}\ i=2,...,d-1.$ (67)

And the corresponding RT surface is given by [85]

$\pm
r(z)=\frac{z_{*}\sqrt{\pi}}{2(d-1)}\frac{\Gamma(\frac{1}{2}+\frac{1}{2(d-1)})}{\Gamma(1+\frac{1}{2(d-1)})}-\frac{z}{d}\left(\frac{z}{z_{*}}\right)^{d-1}\
_{2}F_{1}\left(\frac{1}{2},\frac{1}{2}+\frac{1}{2(d-1)},\frac{3}{2}+\frac{1}{2(d-1)},\frac{z^{2(d-1)}}{z_{*}^{2(d-1)}}\right),$
(68)

where $z_{*}$ is the deepest point of minimal surface in the bulk, which reads

$z_{*}=\frac{2R(d-1)}{\sqrt{\pi}}\frac{\Gamma(1+\frac{1}{2(d-1)})}{\Gamma(\frac{1}{2}+\frac{1}{2(d-1)})}.$
(69)

Due to the translational symmetry along $x_{i}$-direction, these
$x_{i}$-components of $V_{A}^{\mu}$ vanishes and $V_{A}^{\mu}$ also enjoy this
translational symmetry along $x_{i}$-direction. Thus it is enough to evaluate
the field point at $(\bar{x}_{1},\bar{z})$ with $\bar{x}_{i\geq 2}=0$.

$x_{0}$$\Pi_{x_{0}}$$x_{1}$$x_{2}$$\bar{x}_{1}$ Figure 12: The blue-shaded
region is a strip with the width $2R$ centered at the origin. The strip is
foliated by layers of thin plates $\Pi_{x_{0}}$ along the $x_{1}$-direction
and $x_{0}$ is distance between the plate and the field point.

To get the _bit thread flow_ $V^{\mu}_{A}$, let us follow the similar trick as
the sphere case. We first foliate the strip into layers of thin plates
$\Pi_{x_{0}}$ along $x_{1}$-direction (see Fig. 12 for an illustration), where
$x_{0}$ is the distance between the field point and the plate. Then we use
(46) to determine the contribution from each thin plate to $V_{A}^{\mu}$ and
integrate over all the plates to get the total $V_{A}^{\mu}$. The results are

$V^{x_{1}}_{A}(\bar{x}_{1},\bar{z})=\frac{\bar{z}^{d+1}}{4G_{N}}\left(\frac{1}{\left((R-{\bar{x}_{1}})^{2}+\bar{z}^{2}\right)^{d/2}}-\frac{1}{\left((R+{\bar{x}_{1}})^{2}+\bar{z}^{2}\right)^{d/2}}\right),$
(70)
$V_{A}^{z}(\bar{x}_{1},\bar{z})=\frac{\bar{z}^{d}}{4G_{N}}\left(\frac{\bar{x}_{1}+R}{((R+\bar{x}_{1})^{2}+\bar{z}^{2})^{d/2}}-\frac{\bar{x}_{1}-R}{((\bar{x}_{1}-R)^{2}+\bar{z}^{2})^{d/2}}\right).$
(71)

One could check that this extrapolates to $d=2$ to recover (37).

Unlike the spherical regions, however, the _bit thread flow_ for strip
$|V_{A}|<1/4G_{N}$ on the RT surface (68) and thus is not a bit thread
configuration.

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