First Law and Quantum Correction for Holographic Entanglement Contour

Muxin Han1,2, Qiang Wen3,4*

1 Department of Physics, Florida Atlantic University, 777 Glades Road, Boca
Raton, FL 33431-0991, USA

2 Institut für Quantengravitation, Universität Erlangen-Nürnberg, Staudtstr.
7/B2, 91058 Erlangen, Germany

3 Shing-Tung Yau Center of Southeast University, Nanjing 210096, China

4 School of Mathematics, Southeast University, Nanjing 211189, China

*<EMAIL_ADDRESS>
## Abstract

Entanglement entropy satisfies a first law-like relation, which equates the
first order perturbation of the entanglement entropy for the region $A$ to the
first order perturbation of the expectation value of the modular Hamiltonian,
$\delta S_{A}=\delta\langle K_{A}\rangle$. We propose that this relation has a
finer version which states that, the first order perturbation of the
entanglement contour equals to the first order perturbation of the contour of
the modular Hamiltonian, i.e. $\delta s_{A}(\textbf{x})=\delta\langle
k_{A}(\textbf{x})\rangle$. Here the contour functions $s_{A}(\textbf{x})$ and
$k_{A}(\textbf{x})$ capture the contribution from the degrees of freedom at x
to $S_{A}$ and $K_{A}$ respectively. In some simple cases $k_{A}(\textbf{x})$
is determined by the stress tensor. We also evaluate the quantum correction to
the entanglement contour using the fine structure of the entanglement wedge
and the additive linear combination (ALC) proposal for partial entanglement
entropy (PEE) respectively. The fine structure picture shows that, the quantum
correction to the boundary PEE can be identified as a bulk PEE of certain bulk
region. While the ALC proposal shows that the quantum correction to the
boundary PEE comes from the linear combination of bulk entanglement entropy.
We focus on holographic theories with local modular Hamiltonian and
configurations of quantum field theories where the ALC proposal applies.

###### Contents

  1. 1 Introduction
  2. 2 The first law of entanglement contour and the contour of modular Hamiltonian
  3. 3 Quantum correction to holographic entanglement contour
    1. 3.1 Quantum correction to holographic entanglement entropy
    2. 3.2 Quantum correction to entanglement contour from the fine structure
    3. 3.3 Quantum correction from the additive linear combination proposal for PEE
  4. 4 Discussion

##  1 Introduction

The entanglement entropy captures the quantum entanglement in a pure state
between $A$ and $B$ for a bipartite system $A\cup B$. The study of
entanglement entropy has played an essential role in our understanding of the
emergence of spacetime and holography. These progresses begin with the Ryu-
Takayanagi (RT) proposal [1, 2] that reveals the deep connection between the
spacetime geometry and quantum entanglement. In AdS/CFT [3, 4, 5], consider a
static region $A$ in the boundary field theory and the minimal surface
$\mathcal{E}_{A}$ that is in the dual AdS bulk and anchored to $\partial A$,
the RT formula relates the entanglement entropy of $A$ to the area of
$\mathcal{E}_{A}$ in Planck units, i.e.

$\displaystyle S_{A}=\frac{\text{Area}(\mathcal{E}_{A})}{4G}\,.$ (1)

This relation between the quantum entanglement and geometry has recently been
extended to holographic theories beyond AdS/CFT, for example the (warped)
AdS/(warped) CFT correspondence [6, 7, 8, 9] and 3-dimensional flat holography
[10, 11, 12], whose dual field theory is non-Lorentz invariant. These new
relations are derived firstly via the Rindler method [13], which constructs a
Rindler transformation that maps the entanglement wedge to a Rindler spacetime
with infinitely far away boundaries, then calculates the entanglement entropy
via the thermal entropy in the Rindler spacetime. Later they are also derived
in [14] via the Lewkowycz-Maldacena prescription [15, 16], which directly
applies the replica trick in the bulk to calculate the entanglement entropy.
However, in these cases, there exists a subtle issue about the cut-off in the
bulk causing the RT surfaces not anchored on the boundary111When the boundary
field theory is non-Lorentz invariant, the causal development of an interval
becomes an infinitely long strip instead of a causal diamond. In order to keep
the consistency between the bulk and boundary causal structure, the extremal
surface should not be anchored on the boundary. See [14] for more details..
This issue can be solved at least in 2+1 dimensions by introducing certain
null geodesics emanating from the boundary of $A$. Indeed the analogue of the
RT surface $\mathcal{E}_{A}$ is the extremal geodesic whose length is at the
saddle among all the geodesics that anchored on the null geodesics, then the
holographic entanglement entropy is given by the length of the extremal
geodesic.

These novel null geodesics in holographic theories with non-Lorentz invariant
duals are indeed ingredients of the entanglement wedge’s fine structure based
on the bulk and boundary modular flows [17]. The fine structure also largely
inspires the following study on the entanglement contour or the partial
entanglement entropy [18, 17, 19, 20, 21]. For a given region $A$ and a subset
$A_{i}$ of $A$, the partial entanglement entropy (PEE), denoted by
$s_{A}(A_{i})$, is defined to capture the contribution from $A_{i}$ to the
entanglement entropy $S_{A}$. The key property featured by the PEE is the
additivity, which is not possessed by any other entanglement measures. When
the subsets reduce to single points in $A$, the PEE reduces to a function
$f_{A}(\textbf{x})$ called the entanglement contour [18]. $f_{A}(\textbf{x})$
gives the contribution from the site at the position x in $A$ to $S_{A}$, in
other words, it is the density function of the entanglement entropy $S_{A}$,

$\displaystyle S_{A}=\int_{A}f_{A}(\textbf{x})d^{d}x\,,$ (2)

where $d$ is the dimension of $A$. The PEE $s_{A}(A_{i})$ can also be written
as

$\displaystyle s_{A}(A_{i})=\int_{A_{i}}f_{A}(\textbf{x})d^{d}x\,,$ (3)

hence, only collect the contribution in the subset $A_{i}$.

Though the definition of PEE based on the reduced density matrix is still
missing, the physical meaning as the density function for the entanglement
entropy requires the PEE to satisfy the following physical requirements 222The
requirements 1-6 are firstly given in [18], while the requirement 7 is
recently given in [21]:

  1. 1.

Additivity: If $A_{i}^{a}\cup A_{i}^{b}=A_{i}$ and $A_{i}^{a}\cap
A_{i}^{b}=\emptyset$, by definition we have

$\displaystyle s_{A}(A_{i})$
$\displaystyle=s_{A}(A_{i}^{a})+s_{A}(A_{i}^{b})\,.$ (4)

  2. 2.

Invariance under local unitary transformations: $s_{A}(A_{i})$ is invariant by
any local unitary transformation inside $A_{i}$ or $\bar{A}$.

  3. 3.

Symmetry: For any symmetry transformation $\mathcal{T}$ under which
$\mathcal{T}A=A^{\prime}$ and $\mathcal{T}A_{i}=A^{\prime}_{i}$, we have

$\displaystyle s_{A}(A_{i})=s_{A^{\prime}}(A^{\prime}_{i}).$ (5)

  4. 4.

Normalization: $S_{A}=s_{A}(A_{i})|_{A_{i}\to A}\,.$

  5. 5.

Positivity: $s_{A}(A_{i})\geq 0$.

  6. 6.

Upper bound: $s_{A}(A_{i})\leq S_{A_{i}}\,.$

  7. 7.

Symmetry under the permutation:
$\mathcal{I}(\bar{A},A_{i})=s_{A}(A_{i})=s_{\bar{A}_{i}}(\bar{A})=\mathcal{I}(A_{i},\bar{A})$
.

There have been four PEE (or entanglement contour) proposals that satisfies
the above requirements. The first one is the Gaussian formula [22, 18, 23, 24,
25, 26, 27, 28] that applies to the Gaussian states in free theories. The
second proposal is a geometric construction [17, 14, 29] in holographic
theories, based on the fine structure analysis of the entanglement wedge
following the boundary and bulk modular flows. The third one, previously given
by the author in [17, 20], claims that the PEE is given by an additive linear
combination of subset entanglement entropies. Later we will call this proposal
the ALC (additive linear combination) proposal for short333Previously in [20,
21, 30], this proposal was call the “partial entanglement entropy proposal”.
This is a bit misleading since we defined the PEE as (3) rather than the
linear combination (6).. The fourth proposal [21] follows the construction of
the extensive (or additive) mutual information (EMI) [31] (see also [32] for a
similar construction), which tried to solve the above seven requirements in
CFT. The entanglement contour can also be studied under the picture of the bit
threads [33] in holographic theories, see for example [19, 34, 35, 36]. The
PEE calculated by different approaches are highly consistent with one another
[28, 17, 14, 29, 21], suggesting that the PEE should be well-defined and
unique. The uniqueness of the PEE has been confirmed for Poincaré invariant
theories [21], by showing that the above seven requirements in these theories
have unique solution. The PEE is also useful to study the entanglement
structure in condensed matter theories444The entanglement contour gives a
finer description for the entanglement structure. In condense matter theories
it can be used to discriminate between gapped systems and gapless systems with
a finite number of zero modes in $d=3$ [18]. It has been shown to be
particularly useful to characterize the spreading of entanglement when
studying dynamical situations [18, 19, 27]. The entanglement contour is also a
useful probe of slowly scrambling and non-thermalizing dynamics for some
interacting many-body systems [37] and holographic states dual to
Ba$\tilde{\text{n}}$ados geometries, and general excited states in the small
interval limit [38]. Holographically, the correspondence between PEE and bulk
geodesic chords [17, 14] is a finer correspondence between the quantum
entanglement and bulk geometry [14, 39]. Under some balanced condition the PEE
also gives the area of the entanglement wedge cross section [30]. The balanced
PEE can be considered to be an generalization of the reflected entropy [40] to
generic purifications of the bipartite system [30]. . Recently, the
entanglement contour is used to give the entanglement structure of the Hawking
Radiation which shows non-trivial behavior [41, 42] due to an island phase
transition (see [43] for a review on this topic). The above progresses suggest
that the new concept of entanglement contour in quantum information should
play an important role in our understanding of the gauge/gravity duality and
the entanglement structure in quantum field theories (or many-body system).

Figure 1:  A typical region $A$ with a definite order is shown by the red
interval. When an arbitrary subset $\alpha$ is chosen, a natural decomposition
of $A=\alpha_{L}\cup\alpha\cup\alpha_{R}$ is determined. All the degrees of
freedom in $A$ lines in a definite order. When $A$ is a circle, the definition
of $\alpha_{L}$ and $\alpha_{R}$ become ambiguous.

In this paper, we mainly use the fine structure of the entanglement wedge and
the ALC proposal to construct the PEE. The ALC proposal [17, 20] claims that,
the PEE is given by a linear combination of certain subset entanglement
entropies. The ALC proposal is proven to satisfy all the seven requirements
using only the general properties of entanglement entropy. It can be applied
to generic theories, but a definite order is required for all the degrees of
freedom in $A$ for satisfying the additivity.

  * •

The ALC proposal: Given a region $A$ and an arbitrary subset $\alpha$, when
there is a definite order inside $A$, it can be unambiguously partitioned into
three non-overlapping subregions $A=\alpha_{L}\cup\alpha\cup\alpha_{R}$ (see
for example Fig.1), where $\alpha_{L}$ ($\alpha_{R}$) denotes the subset on
the left (right) hand side of $\alpha$. In this configuration, the ALC
proposal claims that

$\displaystyle
s_{A}(\alpha)=\frac{1}{2}\left(S_{\alpha_{L}\cup\alpha}+S_{\alpha\cup\alpha_{R}}-S_{\alpha_{L}}-S_{\alpha_{R}}\right)\,.$
(6)

The ALC proposal can be used to calculate the entanglement contour for one
dimensional regions in general theories with a definite order[17, 20]. It also
works for highly symmetric regions in higher dimensions, which can be
characterized by a single coordinate [29]. Furthermore, this linear
combination can be understood as a conditional mutual information [42]

$\displaystyle
s_{A}(\alpha)=\frac{1}{2}I(\alpha:\bar{A}|\alpha_{L})=\frac{1}{2}I(\alpha:\bar{A}|\alpha_{R})\,.$
(7)

We will briefly introduce the fine structure approach in section 3 later.

By an infinitesimal variation of the state, the perturbation of entanglement
entropy $S_{A}$ satisfies a first law-like relation $\delta
S_{A}=\delta\langle K_{A}\rangle$, where $\langle K_{A}\rangle$ is the
expectation value of the modular Hamiltonian $K_{A}$ [44, 45]. In holographic
theories where the RT formula applies, $\delta S_{A}$ affects the dynamics of
the bulk geometry: The first law of entanglement entropy has been used to
derive the linearized Einstein’s equations in the bulk spacetime [46, 45]. The
first law and linearized Einstein’s equations have also been discussed in
holographies beyond AdS/CFT [12, 47, 48]. In this paper we propose that the
first law of entanglement entropy has a finer description: in a given region
$A$, the first order perturbation of the entanglement contour at each site
equals the first order variation of the expectation value of the modular
Hamiltonian’s contour, which is a similar density function for the modular
Hamiltonian.

Another important topic of holographic entanglement entropy is the quantum
correction. The RT formula only concerns the leading order contribution to the
entanglement entropy $S_{A}$. It is shown in [49] that the quantum correction
to $S_{A}$ is just the bulk entanglement entropy of the homology surface
$\Sigma_{A}$ in the entanglement wedge. The evaluation of the quantum
correction plays an essential role in our understanding of holography and
spacetime beyond the classical level, see for example [50, 51, 52, 53, 54,
55]. For holographic theories, the entanglement contour derived [14, 17, 29]
via the fine structure of the entanglement wedge is also only at the leading
order. Then it is very interesting to explore the finer description of the
quantum correction. More explicitly, for a give subset $A_{i}$ of $A$, we want
to evaluate the quantum correction to the PEE $s_{A}(A_{i})$. Furthermore, for
the cases that the modular Hamiltonian is local, we identify a bulk sub-region
$a_{i}$ of the homology surface $\Sigma_{A}$, such that the contribution from
$a_{i}$ to the bulk entanglement entropy $S_{\Sigma_{A}}$ gives the quantum
correction to $s_{A}(A_{i})$.

##  2 The first law of entanglement contour and the contour of modular
Hamiltonian

The state of a generic quantum system can be described by the density matrix
$\rho_{total}$. Let us consider an arbitrary subsystem $A$ and its complement
$\bar{A}$, the state of $A$ is then described by the reduced density matrix
$\rho_{A}=\text{Tr}_{\bar{A}}\rho_{total}$. If the total system is in a pure
state, the entanglement between $A$ and $\bar{A}$ is captured by the
entanglement entropy that is the von Neumann entropy $S_{A}$ of $\rho_{A}$

$\displaystyle S_{A}=-\text{Tr}\rho_{A}\log\rho_{A}\,.$ (8)

The modular Hamiltonian $K_{A}$ is a state-dependent operator defined by

$\displaystyle\rho_{A}\equiv e^{-K_{A}}\,.$ (9)

One may multiply a constant to the right hand side of the above equation to
ensure $\text{Tr}\rho_{A}=1$. Usually the modular Hamiltonian is non-local.
For the cases where $K_{A}$ is local, usually it can be written as
$K_{A}=-H/T$, where $H$ is the ordinary Hamiltonian measured by the local
observer (or Rindler observer) confined in the causal development of $A$.

Let us consider any infinitesimal perturbation to the density matrix
$\rho_{total}$, the first order perturbation of the entanglement entropy is
given by

$\displaystyle\delta S_{A}=$
$\displaystyle-\text{Tr}(\delta\rho_{A}\log\rho_{A})-\text{Tr}(\rho_{A}\rho_{A}^{-1}\delta\rho_{A})$
(10) $\displaystyle=$
$\displaystyle\text{Tr}(\delta\rho_{A}K_{A})-\text{Tr}\delta\rho_{A}\,.$ (11)
$\displaystyle=$ $\displaystyle\delta\langle K_{A}\rangle$ (12)

Here we have used the fact the $\text{Tr}\delta\rho_{A}=0$, since
$\text{Tr}\rho_{A}=1$ always holds and $K_{A}$ is defined by the unperturbed
state. The above equality between variations of the entanglement entropy and
the modular Hamiltonian’s expectation value is called the first law of
entanglement entropy. For thermal states where $K_{A}=-H/T$, this relation
becomes the quantum version of the first law of thermodynamics, $\delta\langle
H\rangle=T\delta S_{A}$.

It is very interesting to explore a finer version of the above first law, i.e.
the relation between variations of the entanglement contour and certain local
properties of modular Hamiltonian. Here we focus on the configurations where
the ALC proposal applies. This includes single intervals in 2-dimensional
theories and spherical (or strip) regions in higher dimensions with a definite
order. Again, let us consider a region $A$ and its non-overlapping connected
subsets $\\{A_{1},A_{2},A_{3}\\}$, the PEE is given by the ALC proposal

$\displaystyle
s_{A}(A_{2})=\frac{1}{2}\left(S_{12}+S_{23}-S_{1}-S_{3}\right).$ (13)

Here we write, for example, $S_{A_{1}\cup A_{2}}=S_{12}$. Similarly the
modular Hamiltonian of $A_{1}\cup A_{2}$ is denoted by $K_{A_{1}\cup
A_{2}}=K_{12}$.

Let us perform an infinitesimal perturbation on both sides of (13), then apply
the first law to all the subset entanglement entropies on the right hand side,
we get

$\displaystyle\delta s_{A}({A_{2}})=$
$\displaystyle\frac{1}{2}\left(\delta\langle K_{12}\rangle+\delta\langle
K_{23}\rangle-\delta\langle K_{1}\rangle-\delta\langle K_{3}\rangle\right)$
(14) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\text{Tr}(\delta\rho_{12}K_{12})+\text{Tr}(\delta\rho_{23}K_{23})-\text{Tr}(\delta\rho_{1}K_{1})-\text{Tr}(\delta\rho_{3}K_{3})\right)$
(15)

We assume that the Hilbert space $\mathcal{H}_{A}$ of $A$ factorizes
$\mathcal{H}_{A}=\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\mathcal{H}_{A_{3}}$.
The modular Hamiltonian acts trivially outside the region where it is defined.
So it is convenient to extend it to an operator acting on the whole region
$A$, for example

$\displaystyle K_{12}\equiv K_{12}\otimes I_{3},$ (16)

where $I_{3}$ is the identity operator on $\mathcal{H}_{A_{3}}$. This is
crucial to write

$\displaystyle\text{Tr}(\delta\rho_{12}K_{12})=\text{Tr}(\delta\rho_{A}K_{12})\,,$
(17)

where the trace on the left hand side is over
$\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}$, while on the right hand side
the trace is over $\mathcal{H}_{A}$. We rewrite other terms similarly to
obtain,

$\displaystyle\delta s_{A}({A_{2}})=$
$\displaystyle\frac{1}{2}\text{Tr}[\delta\rho_{A}\left(K_{12}+K_{23}-K_{1}-K_{3}\right)]\,.$
(18)

Similarly we may express the PEE in terms of the modular Hamiltonians,

$\displaystyle
s_{A}(A_{2})=\frac{1}{2}\text{Tr}\rho_{A}\left(K_{12}+K_{23}-K_{1}-K_{3}\right)\,.$
(19)

It is easy to see that, the linear combination of the modular Hamiltonians in
the above equation is exactly the same as the subset entanglement entropies in
the ALC proposal. It has been proven that this linear combination was
additive. More explicitly, let us define a new non-local operator on $A$,

$\displaystyle
k_{A}(A_{2})\equiv\frac{1}{2}\left(K_{12}+K_{23}-K_{1}-K_{3}\right)\,.$ (20)

If $A_{2}$ is divided into two non-overlapping connected subregions
$A=A_{2}^{a}\cup A_{2}^{b}$, we have

$\displaystyle k_{A}(A_{2})=k_{A}(A_{2}^{a})+k_{A}(A_{2}^{b})\,.$ (21)

It is natural to take $K_{A}\to 0$ when $A$ vanishes, hence when we take the
limit $A_{2}\to A,\,A_{1}\to\emptyset,\,A_{3}\to\emptyset$, we get the
normalization property,

$\displaystyle k_{A}(A_{2})|_{A_{2}\to A}=K_{A}.$ (22)

Due to the additivity and normalization of the operator $k_{A}(A_{2})$, we
call $k_{A}(A_{2})$ the partial modular Hamiltonian. Furthermore, if we know
the modular Hamiltonian for all the subregions inside $A$, we can determine
the contour function $k_{A}(\textbf{x})$555Note that, one should not take the
contour function $k_{A}(\textbf{x})$ as a local function of x since it also
depend on the region $A$. Also it is an operator in the sense of (20) rather
than a number. for $K_{A}$ by taking $A_{2}$ to be a single site at the
position $\textbf{x}=\\{t,\vec{x}\\}$, hence

$\displaystyle K_{A}=\int_{A}k_{A}(\textbf{x})d\vec{x}\,.$ (23)

Similar to the entanglement contour, $k_{A}(\textbf{x})$ is understood as a
density function for the modular Hamiltonian $K_{A}$. The partial modular
Hamiltonian $k_{A}(A_{2})$ gives the contribution from the subregion $A_{2}$,
i.e.

$\displaystyle k_{A}(A_{2})=\int_{A_{2}}k_{A}(\textbf{x})d\vec{x}\,,$ (24)

where the domain of the integration is confined in $A_{2}$. Note that both the
contour function $k_{A}(\textbf{x})$ and the partial modular Hamiltonian
$k_{A}(A_{2})$ are operators defined on $A$ rather than the point x or the
subregion $A_{2}$.

As a result, the equation (14) can be written as

$\displaystyle\delta s_{A}(A_{2})=\delta\langle k_{A}(A_{2})\rangle\,,$ (25)

which we call the first law of partial entanglement entropy. If we know all
the partial modular Hamiltonians, we can determine the contour function hence
get a finer version of the above relation

$\displaystyle\delta s_{A}(\textbf{x})=\delta\langle
k_{A}(\textbf{x})\rangle\,,$ (26)

which we call the first law of entanglement contour. For any site x in $A$,
the first law of entanglement contour states that the perturbation of the
contribution to $S_{A}$ at x equals the perturbation of the expectation value
of $k_{A}(\textbf{x})$, which is the contribution to $K_{A}$ at x. Though it
is derived for the special cases where the ALC proposal applies, we conjecture
it to be valid for more general configurations. We hope this can be confirmed
in the future.

This finer version of the first law is useful, because the modular Hamiltonian
has been extensively explored in many configurations, especially when the
modular Hamiltonian is local. More importantly, the modular Hamiltonian
$K_{A}$ is usually written as an integration over the region $A$, hence
perfectly match with our introduction of the contour of the modular
Hamiltonian. One simple and renowned case is the modular Hamiltonians for
ball-shaped regions $A$ in $d$-dimensional CFTs. If we consider the vacuum
state of the CFT and a static and ball-shaped region with radius $R$ and
center position $\textbf{x}_{0}=\\{t_{0},\vec{x}_{0}\\}$, then the modular
Hamiltonian takes the simple form [13, 56],

$\displaystyle
K_{A}=2\pi\int_{A}\frac{R^{2}-|\vec{x}-\vec{x}_{0}|^{2}}{2R}T_{tt}(t_{0},\vec{x})d\vec{x}\,,$
(27)

where $T_{\mu\nu}$ is the stress tensor and $\vec{x}$ is the coordinates on
$A$. More generally, the modular Hamiltonian can be written in a covariant
way,

$\displaystyle
K_{A}=\int_{A}d\Sigma~{}\eta^{\mu}(\textbf{x})~{}T_{\mu\nu}(\textbf{x})~{}\xi^{\nu}(\textbf{x}),$
(28)

where $d\Sigma$ is an infinitesimal volume on the spacelike co-dimension-one
region $A$ with the normal vector $\eta^{\mu}$, while the vector field
$\xi^{\mu}$ describes the modular flow (or geometric flow) which is generated
by the modular Hamiltonian.

There are two ways to derive the modular flow. The first one relies on the
construction of the Rindler transformation $R$, which is a symmetry
transformation that maps the causal development $\mathcal{D}_{A}$ of $A$ to a
Rindler spacetime with infinitely far away boundaries. The normal Hamiltonian,
which generates the Rindler time translation $\partial_{\tau}$ in the Rindler
spacetime, is mapped to the modular Hamiltonian of $A$. In other words,
$\partial_{\tau}$ maps to the modular flow $\xi^{u}$ in $\mathcal{D}_{A}$ by
the inverse Rindler transformation. However the construction of the Rindler
mapping is highly non-trivial. The Rindler transformation for static balls in
CFTs are constructed in [13]. While the Rindler transformation for covariant
intervals in warped CFTs and BMSFTs (theories with BMS3 symmetries) are
constructed in [6, 8] and [10]. See also [48] for a related construction of
the modular Hamiltonian. The Rindler transformation can be extended into the
bulk in the context of holography, hence plays an essential role to derive the
geometric picture of the entanglement entropy [8, 10].

Recently another way to generate the modular flow is proposed in [20] base on
the properties of the PEE. The key of this approach is that the PEE should be
invariant under the modular flow. The property is observed in the entanglement
wedge’s fine structure, which we will introduce later. Using the ALC proposal,
it is easy to derive the orbit of the modular flow in $\mathcal{D}_{A}$, if we
know all the entanglement entropies for sub-intervals inside $D_{A}$. This
approach reproduces the previous results quite easily. More importantly, it
does not rely on the Rindler transformations.

In the Rindler spacetime, the modular Hamiltonian is just the energy. Since
the Rindler spacetime is invariant under the translation along the spacial
directions, the contour function for the modular Hamiltonian should respect
this symmetry, thus is a constant. Applying the inverse Rindler
transformation, this flat contour maps to the contour function of the modular
Hamiltonian $K_{A}$ in $A$. This contour function is nothing but the integrand
of (27) and (28), i.e.

$\displaystyle
k_{A}(\textbf{x})=\eta^{\mu}(\textbf{x})~{}T_{\mu\nu}(\textbf{x})~{}\xi^{\nu}(\textbf{x})\,.$
(29)

According to the first law of the entanglement contour, we have

$\displaystyle\delta
s_{A}(x)=\eta^{\mu}(\textbf{x})~{}\xi^{\nu}(\textbf{x})~{}\delta\langle
T_{\mu\nu}(\textbf{x})\rangle\,,$ (30)

which states that, the first order variation of the entanglement contour
relates to the first order variation of the stress tensor. Note that in the
above equation we used the relation
$\delta\left(\eta^{\mu}(\textbf{x})\xi^{\nu}(\textbf{x})\right)=0$ because the
geometry is at the saddle hence the first order perturbation of geometry
vanishes. The above relation is also in some sense applied in [42] to evaluate
the entanglement contour for low energy excited states of CFT near the vacuum.

However, in more generic configurations the modular Hamiltonian cannot be
written as an integration like (28), hence one may worry about the validity of
(23). We stress that, the reason we can write $K_{A}$ as an integral of
$k_{A}(\textbf{x})$ is the additivity of the partial modular Hamiltonian,
which is always true when the ALC proposal applies. The ALC proposal does not
select theories. For example, let us consider an interval $A$ in a 2-dimension
theory which is not conformal invariant, where $K_{A}$ can not be written as
(28). Since in this case the ALC proposal applies thus the partial modular
Hamiltonian is additive, $K_{A}$ can still be written as (23) with
$k_{A}(\textbf{x})$ not directly related to the stress tensor. So (28) is not
necessary for the validity of (23).

##  3 Quantum correction to holographic entanglement contour

###  3.1 Quantum correction to holographic entanglement entropy

The RT formula only gives the leading order contribution for the holographic
entanglement entropy, i.e. at the order
$\mathcal{O}(1/G_{N})\sim\mathcal{O}(N^{2})$. The Faulkner-Lewkowycz-Maldacena
(FLM) formula [45] corrects the RT formula to the next order $\mathcal{O}(1)$,

$\displaystyle
S_{A}=\frac{\text{Area}(\mathcal{E}_{A})}{4G_{N}}+S_{bulk}(\Sigma_{A})\,,$
(31)

where $S_{bulk}(\Sigma_{A})$ is the bulk entanglement entropy for the homology
surface $\Sigma_{A}$, which is any Cauchy surface with its boundary satisfying
$\partial\Sigma_{A}=A\cup\mathcal{E}_{A}$. Later in this paper we use the
short-hand notation $\Sigma_{A}\equiv a$. The entanglement wedge
$\mathcal{W}_{A}$ is the causal development of $\Sigma_{A}$. Note that,
compared with the full expression for the quantum correction to the
holographic entanglement entropy [45], Eq. (31) omitted the terms that are
given by local integrals on the original minimal surface, including the terms
that cancel the UV divergences of the bulk entanglement entropy.

Since the quantum correction is taken into account, the minimization on the
area of the RT surface should be adjusted to the minimization of the quantum
extremal surface [51],

$\displaystyle
S_{A}=\text{min}\left(\frac{\text{Area}(\tilde{\mathcal{E}}_{A})}{4G_{N}}+S_{bulk}(\tilde{\Sigma}_{A})\right)\,.$
(32)

Accordingly the surface satisfying the minimization changes from
$\mathcal{E}_{A}$ to $\tilde{\mathcal{E}}_{A}$, and $\Sigma_{A}$ changes to
$\tilde{\Sigma}_{A}$. However, this difference usually only affect $S_{A}$ at
the order $\mathcal{O}({G_{N}})$ 666The difference only gives significant
corrections when we approach a phase transition, where the RT surface jumps
discontinuously or when the bulk entanglement entropy is comparable to the
area term., hence we will directly apply (31) instead of (32) to avoid
unnecessary complications. Our discussion focuses on the configurations where
the quantum correction is much smaller than the leading contribution from the
RT formula.

The relation (31) implies an important relation between the bulk and boundary
modular Hamiltonian [53],

$\displaystyle K_{A}=\frac{\hat{\mathcal{E}}_{A}}{4G_{N}}+K_{a}\,,$ (33)

where $K_{a}$ is the modular Hamiltonian of the bulk region $\Sigma_{A}$, and
$\hat{\mathcal{E}}_{A}$ is the bulk area operator whose expectation value
gives the area of the RT surface.

Then it is quite interesting to discuss the quantum corrections to the
entanglement contour. Firstly we will explore the spatial distribution of the
bulk entanglement entropy on the homology surface $\Sigma_{A}$, i.e. the
entanglement contour or PEE of the bulk degrees of freedom. The essential
entanglement contour that we study is the contour on the boundary region $A$,
so the PEE from any bulk degrees of freedom will be assigned to the PEE of a
boundary degrees of freedom as the quantum correction to the entanglement
contour of $A$. Secondly, we will explore how to assign the bulk PEE to the
boundary PEE. We will study the quantum correction of the contour using both
the fine structure analysis with the modular slices and the ALC proposal.

###  3.2 Quantum correction to entanglement contour from the fine structure

#### Fine structure of the entanglement wedge

In holography, when the modular Hamiltonian is local, the entanglement contour
can be described by a geometric picture 777This picture works also for
holographic theories beyond AdS/CFT [14], for example, the (warped) AdS/
(warped) CFT correspondence and the flat holography. However, similar
construction cannot be straitforwardly generalized to the cases of multi-
intervals and a large enough boundary interval in the BTZ background with
disconnected RT surface, since the modular flow becomes nonlocal. This is an
important problem we hope to understand further in the future., which is
constructed in a series of papers [17, 14, 20, 29]. Since the modular
Hamiltonian is local thus generates a geometric modular flow, there exists a
natural slicing of the entanglement wedge. More explicitly, from any point $P$
in $A$, the boundary modular flow generates an orbit, which we call the
boundary modular flow curve. Then we let the points on the boundary modular
flow curve flow under the bulk modular flow. Trajectories of this flow form a
two dimensional bulk surface, which we call a modular slice. The causal
development $\mathcal{D}_{A}$ is a slicing of the boundary modular flow
curves. Similarly the entanglement wedge $\mathcal{W}_{A}$ is a slicing of the
modular slices. Note that, since the boundary modular flow is also a bulk
modular flow, when the boundary modular flow curve is settled exactly at the
boundary, i.e. $z=0$, its trajectory under the bulk modular flow is just
itself. Here by the trajectory of the boundary modular flow curve, we mean the
trajectory of the curve settled at the limit $z\to 0$ but $z\neq 0$, hence
points on the curve can flow into the bulk. More explicitly, points on the
curve flow into the bulk, then get to a turning point, and eventually flow
back to some points on exactly the same boundary modular flow curve.

See Fig.2 for an explicit example in AdS3/CFT2. In the right figure the black
curve is the boundary modular flow curve that passes the point $P$, the orange
curves are orbits of points on the black curve under the bulk modular flow.
The modular slice intersects with the RT surface $\mathcal{E}_{A}$ at the
partner point $\tilde{P}$ of $P$. The outermost straight orange lines are the
normal null geodesics emanated from $\mathcal{E}_{A}$, and are also bulk
modular flow curves that end on the future and past tips of the causal
development $\mathcal{D}_{A}$. $\tau_{m}$ denotes the Rindler time in
different causal wedges, which can be covered by a single complex “time”
coordinate,

$\displaystyle\tau_{m}=\tau+\frac{m-1}{2}\pi i\,.$ (34)

In this coordinate, the thermal circle in the entanglement wedge
$\mathcal{W}_{A}$ is just the imaginary circle $\tau\sim\tau+2\pi i$ of the
Rindler time. The dashed line $\gamma_{P}$ is where the modular slice
intersect with the homology surface $\Sigma_{A}$.

Figure 2: A typical example of the modular slice in AdS3/CFT2. In the left
figure, the modular slice at the point $P$ is embedded in the entanglement
wedge. The right figure shows how the boundary modular flow curve flows in the
bulk under the bulk modular flow. The dashed curve $\gamma_{P}$ is where is
modular slice intersect with the homology surface. $\tau_{m}$ denotes the
modular flow in different causal wedges in the bulk. The modular flow curves
become null at the boundary of the entanglement wedge, and are just the null
geodesic congruence emanating from the RT surface $\mathcal{E}_{A}$
vertically.

#### The holographic entanglement contour from the fine structure

The relation between the fine structure and the entanglement contour appears
as we consider the replica story of a single point $P$ in $A$. When applying
the replica trick, we prepare $n$ copies of the system and cut the region $A$
open for all the copies, then we glue them cyclically to form a $n$-manifold.
Correspondingly, in the gravity side we cut the entanglement wedge open along
any homology surface $\Sigma_{A}$, then glue all the copies of the bulk
spacetime cyclically [15, 16].

While applying replica trick on $A$, let us focus on the replica story of a
single point $P$, and see how it affects the boundary and bulk modular flow.
Firstly we cut $P$ open for each copy. This cuts the modular flow curve open
at $P$. Then we glue all the open curves cyclically at $P$ in each copy, hence
the modular flow in the $i$th copy will flow into the $(i+1)$th copy of the
curve through $P$. See Fig.3 for a simple example with $n=2$. Here the
boundary modular flow curve along $\tau_{1}$ contains the lower half line in
the first copy and the upper half line in the second copy. Then we prepare two
copies of the modular slices and see how the bulk modular flow is affected.
Note that, the bulk modular flow lines emanating from the $\tau_{1}$ boundary
modular flow curve should return to the same boundary modular flow curve. The
fact that the $\tau_{1}$ boundary curve now contains two parts in different
copies implies that, the $\tau_{1}$ bulk modular flow curves should also be
cut open and glued cyclically, thus can flow back to the second part of the
$\tau_{1}$ boundary curve in the second copy. The place where we cut the bulk
modular flow curves open is just the curves $\gamma_{P}$, which are the purple
lines in Fig.3.

Figure 3:  The replica trick applied to the modular slice when $n=2$. Each
slice is cut open at $\gamma_{P}$ then glued cyclically. The dashed lines show
the gluing boundary conditions.

In summary, the replica story of a single point in $A$ induces the replica
story of the corresponding modular slice. Since the modular flows are local,
it will not affect the modular slices in the neighborhood. The replica story
on all the modular slices are relatively independent and together form the
replica story of the entanglement wedge. Following the calculation of [15,
16], if we evaluate the partition functions at the classical level, the cyclic
gluing at the region $A$ turns on the extensive contribution at the bulk fixed
points of the replica symmetry, i.e. the RT surface. Accordingly, the cyclic
gluing of any point $P$ exactly turns on the contribution to the entanglement
entropy at the partner point $\tilde{P}$. This is the original statement of
[17]. In the same sense this relation implies a correspondence between the
geodesic chords $\mathcal{E}_{i}$ on $\mathcal{E}_{A}$ and the PEE of certain
subset $A_{i}$ in $A$,

$\displaystyle s_{A}(A_{i})=\frac{Length\left(\mathcal{E}_{i}\right)}{4G}\,,$
(35)

where $\mathcal{E}_{i}$ is the set of partner points of $A_{i}$. The one-to-
one correspondence between all the points on $A$ and $\mathcal{E}_{A}$ gives
the entanglement contour of $A$.

In the context of AdS/CFT, given a static region $A$ (spheres or intervals)
and a static homology surface $\Sigma_{A}$, $\gamma_{P}$ for any point $P$ is
just a static geodesic normal to $\mathcal{E}_{A}$ [29] 888The curves
$\gamma_{P}$ coincide with a special bit-thread configuration constructed in
[57] following the bulk geodesics.. See the purple dashed lines in the left
figure of Fig.4. The correspondence between the PEE of the subsets $A_{i}$ and
geodesic chords $\mathcal{E}_{i}$ in the sense of (35) is shown in the right
figure of Fig.4. Accordingly the homology surface is also decomposed by two
$\gamma_{P}$ curves for two points that decomposes $A$,

$\displaystyle\Sigma_{A}\equiv a=a_{1}\cup a_{2}\cup a_{3}\,.$ (36)

Figure 4: The above two figures show a time slice of the entanglement wedge.
The purple dashed lines are the $\gamma_{P}$ curves, which are static
geodesics normal to $\mathcal{E}_{A}$. In the right figure, the decomposition
of $A$ induces a decomposition of the homology surface $\Sigma_{A}$ and the RT
surface $\mathcal{E}_{A}$.

#### Quantum correction to the holographic entanglement contour

In the above discussion the partition functions are only evaluated at the
classical level, hence the entanglement contour from the slicing of the
entanglement wedge by the modular slices is only at the leading order. When
including the quantum corrections to the partition functions, i.e. computing
the partition function of all bulk quantum fluctuations around the classical
geometry, the entanglement entropy and entanglement contour should receive
quantum corrections. As we previously pointed out in section 3.1, the first
order quantum correction to the entanglement entropy comes from the
entanglement entropy of the bulk region in the entanglement wedge $S_{a}$.
Studying the correction to the entanglement contour relates to the following
question: how do we distribute the bulk entanglement entropy to the degrees of
freedom in $A$?

The answer is indeed hidden in the fine structure of the entanglement wedge.
Now we introduce the entanglement contour $s_{a}(\textbf{x})$, which represent
the contribution from the site x to the bulk entanglement entropy $S_{a}$. The
cyclic gluing of the single point $P$ not only turns on the leading
contribution to $S_{A}$ at its partner point $\tilde{P}$, but also induces the
cyclic gluing of the bulk points on the curve $\gamma_{P}$. Note that the
cyclic gluing of all the points in the homology surface $\Sigma_{A}$ coincides
with the replica trick in the bulk for the bulk entanglement entropy $S_{a}$.
This indicates that, the quantum correction to the PEE of the point $P$ comes
from the bulk PEE of the curve $\gamma_{P}$, which we denote as
$s_{a}(\gamma_{P})$.

It is more convenient to consider the quantum correction to the PEE
$s_{A}(A_{2})$ of a subregion $A_{2}$. For example, see the left figure in
Fig.5, where $A$ is a static interval which is divided into three non-
overlapping parts $A=A_{1}\cup A_{2}\cup A_{3}$. According to the fine
correspondence, the geodesic chord $\mathcal{E}_{2}$ gives the PEE
$s_{A}(A_{2})$ at the leading order. The curves $\gamma_{P}$ for all the
points in $A_{2}$ form the bulk region enclosed by $A_{2}$, $\mathcal{E}_{2}$
and the $\gamma_{P}$ curves for the two endpoints of $A_{2}$. This region is
denoted as $a_{2}$, which is the yellow region in Fig.5. In other words, we
have

$\displaystyle s_{A}(A_{2})=\frac{Area(\mathcal{E}_{2})}{4G}+s_{a}(a_{2})\,,$
(37)

where the bulk PEE $s_{a}(a_{2})$ is the quantum correction to the PEE
$s_{A}(A_{2})$.

The above statement can be easily understood in the Rindler bulk spacetime.
The Rindler transformations map $\mathcal{W}_{A}$ to the Rindler bulk
spacetime, which is an AdS black brane with translation symmetries along the
directions, say $\vec{x}=\\{x_{i}\\}$, that are extensive on the horizon or
boundary. The regions $A_{i}$, $a_{i}$ and $\mathcal{E}_{i}$ are mapped to
$A^{\prime}_{i}$, $a^{\prime}_{i}$ and $\mathcal{E}^{\prime}_{i}$
respectively. See the right figure in Fig.5 for a time slice of the Rindler
bulk. Note that, due to the translation symmetry $a^{\prime}_{i}$ and
$\mathcal{E}^{\prime}_{i}$ are the regions projected to $A^{\prime}_{i}$ along
the $r$ direction.

The quantum correction to the entanglement entropy (thermal entropy in this
case) of $A^{\prime}=A^{\prime}_{1}\cup A^{\prime}_{2}\cup A^{\prime}_{3}$ is
just the bulk entanglement entropy for the region $a^{\prime}$, which is the
exterior region of the Rindler horizon. Since the entanglement contour
respects the symmetries, it only depends on the radial coordinate and is flat
along the $x_{i}$ directions, i.e.

$\displaystyle s_{a^{\prime}}(\vec{x},r)=s_{a^{\prime}}(r)\,.$ (38)

It is convenient to define the constant

$\displaystyle\mathcal{C}=\int s_{a^{\prime}}(r)dr\,,$ (39)

where the domain of the integration is from the horizon to the boundary. Thus
$\mathcal{C}$ is the density function for bulk entanglement entropy after
integration over the radius direction.

On the boundary $A^{\prime}$, let us denote the leading order and quantum
correction of $S_{A^{\prime}}$ by $S^{(0)}_{A^{\prime}}$ and
$S^{(1)}_{A^{\prime}}$ respectively. Due to translation symmetries,
$S^{(1)}_{A^{\prime}}$ and $S^{(0)}_{A^{\prime}}$ should be equally
distributed to all degrees of freedom on $A^{\prime}$ hence present a volume
law. Accordingly, the contour function is a constant given by,

$\displaystyle s_{A^{\prime}}(\vec{x})=\frac{1}{4G_{N}}+\mathcal{C}\,,$ (40)

where the first term come from the RT (or Bekenstein-Hawking) formula while
the second term come from quantum correction. For a subregion $A^{\prime}_{2}$
with length $l^{\prime}_{2}$, the PEE $s_{A^{\prime}}(A^{\prime}_{2})$ is just
given by

$\displaystyle
s_{A^{\prime}}(A^{\prime}_{2})=l^{\prime}_{2}\left(\frac{1}{4G_{N}}+\mathcal{C}\right)\,.$
(41)

Figure 5: The homology surface $a$ at a time slice is mapped to the time slice
of the Rindler bulk under the Rindler transformation. $\gamma_{P}$ curves are
mapped to the curves along the $r$ direction. The bulk region $a_{2}$ is just
mapped to the bulk region $a^{\prime}_{2}$, which is the projection region of
$A^{\prime}_{2}$ along the $r$ direction.

It is easy to see the length of $\mathcal{E}^{\prime}_{2}$ equals to
$l^{\prime}_{2}$, and
$s_{a^{\prime}}(a^{\prime}_{2})=l_{2}^{\prime}\,\mathcal{C}$. Then we can
write the PEE (41) in the following way,

$\displaystyle
s_{A^{\prime}}(A^{\prime}_{2})=\frac{Area(\mathcal{E}^{\prime}_{2})}{4G}+s_{a^{\prime}}(a^{\prime}_{2})\,,$
(42)

The modular slices in $\mathcal{W}_{A}$ are just mapped to the AdS2 slices
with fixed $\vec{x}$ in the Rindler bulk. Note that the Rindler transformation
is also a symmetry of the theory, according to the symmetry property of the
PEE we have

$\displaystyle s_{a}(a_{i})=s_{a^{\prime}}(a^{\prime}_{i})\,,\qquad
s_{A}(A_{i})=s_{A^{\prime}}(A^{\prime}_{i})\,.$ (43)

The length of the geodesic chords is also invariant under the Rindler
transformation,

$\displaystyle Area(\mathcal{E}_{i})=Area(\mathcal{E}^{\prime}_{i})\,.$ (44)

Following (42),(43) and (44), we immediately recover (37).

Our discussion shows that, the quantum correction to the entanglement
entropies, PEE or entanglement contour in holographic CFTs are indeed
proportional the leading contribution. This is consistent with the quantum
result of entanglement contour given in [21].

###  3.3 Quantum correction from the additive linear combination proposal for
PEE

Unlike the geometric construction, the ALC proposal is not limited to the
leading order. In holographic field theories, we can expand the entanglement
entropy with respect to $G_{N}$, i.e.

$\displaystyle S_{A}=S^{(0)}_{A}+S^{(1)}_{A}+S^{(2)}_{A}+\cdots\,,$ (45)

where $S^{(i)}_{A}$ is of order $\mathcal{O}(G_{N}^{i-1})$. Similarly we can
expand the PEE in the same way and the ALC proposal should hold at all orders,
i.e.

$\displaystyle
s^{(i)}_{A}(A_{2})=\frac{1}{2}\left(S^{(i)}_{12}+S^{(i)}_{23}-S^{(i)}_{1}-S^{(i)}_{3}\right).$
(46)

All properties of the PEE should be satisfied respectively at all orders. In
the following, we only consider the first order correction, which are the bulk
entanglement entropies of $\Sigma_{A_{i}}$,

$\displaystyle S^{(1)}_{A_{i}}=S_{\Sigma_{A_{i}}}\,.$ (47)

So we get another formula for the quantum correction to the PEE

$\displaystyle
s^{(1)}_{A}(A_{2})=\frac{1}{2}\left(S_{\Sigma_{12}}+S_{\Sigma_{23}}-S_{\Sigma_{1}}-S_{\Sigma_{3}}\right)\,,$
(48)

where $\Sigma_{i}$ means $\Sigma_{A_{i}}$ and $\Sigma_{ij}$ means
$\Sigma_{A_{i}}\cup\Sigma_{A_{j}}$.

This looks quite confusing. On the one hand, the linear combination (48) are
exactly the same as the ALC proposal, hence looks like a bulk PEE
$s_{\Sigma_{A}}(\Sigma_{2})$. On the other hand the ALC proposal requires
$\Sigma_{A}=\Sigma_{1}\cup\Sigma_{2}\cup\Sigma_{3}$ where $\Sigma_{i}$ are
non-overlapping, and furthermore

$\displaystyle\Sigma_{12}\cap\Sigma_{23}=\Sigma_{2}\,,\quad\Sigma_{12}=\Sigma_{1}\cup\Sigma_{2}\,,\quad\Sigma_{23}=\Sigma_{2}\cup\Sigma_{3}\,.$
(49)

Obviously, these requirements are satisfied by the regions $a_{i}$ rather than
the regions $\Sigma_{i}$ or $\Sigma_{ij}$ in the bulk. So the linear
combination in (48) is not a PEE defined by the ALC proposal and is not
guaranteed to be additive. However, previously we get the result
$s^{(1)}_{A}(A_{2})=s_{a}(a_{2})$ using the fine structure analysis of the
entanglement wedge. This implies that the left hand side of (48) is a PEE in
the bulk thus should be additive.

The confusion can be resolved if one associate the bulk entanglement entropies
to their corresponding boundary regions $A_{i}$ and $A_{ij}$ rather than the
bulk regions $\Sigma_{i}$ and $\Sigma_{ij}$. In other words, the left hand
side of (48) can be understand as a PEE on the boundary following the ALC
proposal. Thus, the additivity immediately follows.

Now we show the additivity of (48) from a more intuitive perspective in the
Rindler spacetime. Again we consider the simple case of AdS3/CFT2 which is
shown in Fig.6. Though the RT surfaces for $A_{1},A_{3},A_{1}\cup A_{2}$ and
$A_{2}\cup A_{3}$ look quite different from one another (see the blue solid
lines in the upper figure of Fig.6), their images in the Rindler bulk are
indeed the same curve up to a translation or reflection (see the solid blue
lines in the lower figure of Fig.6), because
$A^{\prime}_{1},A^{\prime}_{3},A^{\prime}_{1}\cup A^{\prime}_{2}$ and
$A^{\prime}_{2}\cup A^{\prime}_{3}$ are all infinitely long half lines. For
example, consider the RT surface emanating from $x=x_{0}$ and moving along the
$+x$ direction, it approaches the horizon in the following way
$r(x)=r_{h}(1-e^{-(x-x_{0})})^{-1}$, where $r=r_{h}$ is the horizon. In the
large $|x|$ region, the RT surfaces just move along the horizon, thus the
translation symmetry emerges and the entanglement contour is flat at the large
$|x|$ limit. It is obvious that if we translate $\mathcal{E}^{\prime}_{12}$ by
$l^{\prime}_{2}$, it exactly matches with $\mathcal{E}^{\prime}_{1}$. The only
difference is that $\mathcal{E}^{\prime}_{12}$ is longer by $l^{\prime}_{2}$
near the cut off region, where the volume law applies. Also the bulk region
$\Sigma^{\prime}_{12}$ is only larger than $\Sigma^{\prime}_{1}$ by a region
that exactly matches with $a^{\prime}_{2}$ under a translation. Then we have

$\displaystyle S^{(0)}_{A^{\prime}_{1}\cup
A^{\prime}_{2}}-S^{(0)}_{A^{\prime}_{1}}=\frac{l^{\prime}_{2}}{4G_{N}}\,,\quad
S^{(0)}_{A^{\prime}_{2}\cup
A^{\prime}_{3}}-S^{(0)}_{A^{\prime}_{3}}=\frac{l^{\prime}_{2}}{4G_{N}}\,.$
(50)

and

$\displaystyle
S_{\Sigma^{\prime}_{12}}-S_{\Sigma^{\prime}_{1}}=l^{\prime}_{2}\mathcal{C}\,,\qquad
S_{\Sigma^{\prime}_{23}}-S_{\Sigma^{\prime}_{3}}=l^{\prime}_{2}\mathcal{C}\,.$
(51)

Plugging the above equations to the ALC proposal, as expected at the leading
order, we find the PEE is just given by,

$\displaystyle
s^{(0)}_{A^{\prime}}(A^{\prime}_{2})=\frac{l^{\prime}_{2}}{4G_{N}}\,,$ (52)

While the quantum correction (48) can also be calculated by

$\displaystyle s^{(1)}_{A}(A_{2})=$
$\displaystyle\frac{1}{2}\left(S_{\Sigma_{12}}+S_{\Sigma_{23}}-S_{\Sigma_{1}}-S_{\Sigma_{3}}\right)$
(53) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(S_{\Sigma^{\prime}_{12}}+S_{\Sigma^{\prime}_{23}}-S_{\Sigma^{\prime}_{1}}-S_{\Sigma^{\prime}_{3}}\right)$
(54) $\displaystyle=$ $\displaystyle l^{\prime}_{2}\mathcal{C}$ (55)

The result recovers the previous result of $s_{a^{\prime}}(a^{\prime}_{2})$ or
$s_{a}(a_{2})$ using the fine structure of the entanglement wedge, so the
additivity of the right hand side of (48) is justified in this case. This is
also a consistency check between the two approaches to evaluate the quantum
corrections.

Figure 6: The upper figure shows the RT curves (solid blue curves) for the
subregions in the linear combination in the ALC proposal. The lower figure
shows the images of the above RT curves in the Rindler spacetime. Since
$A^{\prime}_{1},A^{\prime}_{3},A^{\prime}_{1}\cup A^{\prime}_{2}$ and
$A^{\prime}_{2}\cup A^{\prime}_{3}$ are all infinite half lines, their RT
surfaces are the same up to a translation or reflection.

##  4 Discussion

In this paper, we explore two important aspects about entanglement contour.
Firstly we explore the “first law” of entanglement contour. For a given
region, the first law tells us that first order variation of the contour (or
density) function of the entanglement entropy equals the first order variation
of the expectation value of the contour (or density) function of the modular
Hamiltonian, i.e. $\delta s_{A}(\textbf{x})=\delta\langle
k_{A}(\textbf{x})\rangle$. This gives a much stronger and finer description of
the variation of the entanglement structure under the variation of the state.
Note that, this relation is only derived for the configurations where the ALC
proposal applies. However this relation seems to be a quite natural extension
of the first law of entanglement entropy $\delta S_{A}=\delta\langle
K_{A}\rangle$. We conjecture it to be true for more generic configurations. It
may be quite useful to calculate the entanglement contour for low energy
exited states (see for example [42]).

The second aspect is the quantum correction to the holographic entanglement
contour. Firstly, using the fine structure picture, we find that the quantum
correction to the PEE of a subset is captured by the bulk PEE of a certain
bulk subregion inside the homology surface, i.e.
$s^{(1)}_{A}(A_{2})=s_{a}(a_{2})$. This observation gives a fine relation
between the PEE of the bulk degrees of freedom and the PEE of the boundary
degrees of freedom. Secondly, for the configurations where the ALC proposal
applies, the quantum corrections to the PEE computed by the ALC proposal is a
linear combination of the bulk entanglement entropies of certain bulk regions.
For example, see the right hand side of (48). The results of the two
approaches are confirmed to be consistent in the Rindler bulk spacetime. Note
also that, the additivity of the linear combination (48) is not manifest. It
comes from the additivity of the boundary PEE and the fact that the bulk
entanglement entropies are quantum corrections to the entanglement entropies
of certain boundary regions.

One can test the first law of the entanglement contour at the leading order
using a perturbed geometry around the pure AdS space. On one hand, the
perturbation of the geometry perturbs the stress tensor of the boundary CFT,
which furthermore perturbs the entanglement contour according to the first
law. On the other hand, the perturbation of the geometry perturbs the fine
correspondence between points in $A$ and $\mathcal{E}_{A}$ which also gives a
perturbation of the entanglement contour. The first law can be confirmed if
the two perturbations of the entanglement contour coincide with each other.

Explicit configurations of bit threads is a good way to describe the
entanglement contour. However, the entanglement contour is assumed to be
unique while the bit thread configuration is highly non-unique even when the
state and region are determined. So far, it is not well undertood how we can
impose physical requirements to determine the bit thread configuration for a
given entanglment wedge. We propose that, reproducing the right entanglement
contour should be a reasonable physical requirement. This is recently explored
in [36] by applying the locking theorems [58, 59] of bit threads to construct
a concrete locking scheme for the RT surfaces in the entanglement wedge. In
[60] two perturbations of the bit threads configurations are explicitly
considered. One of them is for the geodesic bit threads normal to the RT
surface [57], consistent with our fine structure analysis [29]999See also [61,
62] for another related flow picture based on a fracton model which satisfies
several major properties of AdS/CFT.. The other is the canonical perturbation
of the bit threads configuration following the Iyer-Wald formalism [63]. These
perturbations of bit threads give perturbations of the entanglement contour,
hence is useful to test the first law.

We do not explicitly discuss the dependence of the coordinates of the bulk
entanglement contour $s_{a}(\textbf{x})$. It is interesting since it gives a
fine description of the entanglement structure in the bulk and affects the
boundary entanglement contour at the quantum level. The bulk entanglement
contour is also mentioned recently in [34, 35], which extend the concept of
bit threads to the quantum bit threads by allowing the bit threads to start
and terminate in the bulk. In such a way they can use the quantum bit threads
to describe the quantum correction of the holographic entanglement entropy.
However an explicate configuration of the quantum bit threads is necessary to
give a contour function.

We propose that the entanglement contour $s_{a}(\textbf{x})$ should be
evaluated by applying the first law of entanglement contour in the bulk. More
explicitly let us consider the low energy excitations (for example the Hawking
Radiation) in the bulk which induce a perturbation of the stress tensor, while
the backreaction to the geometry can be omitted. According to the first law of
entanglement contour, the perturbation of the contour function is proportional
to the perturbation of the stress tensor. This approach may be valid at the
early age of a black hole. Together with the picture we give in section 3.2,
the time evolution or perturbation of the bulk entanglement contour is
furthermore related to the evolution or perturbation of the boundary
entanglement contour at the quantum level using the relation between the bulk
PEE and boundary PEE. See Fig.7 for example.

Another important relevant question that we come up with is what the first law
of the entanglement contour at the quantum level can tell us about the
dynamics in the bulk. The linearized Einstein’s equations in the bulk have
already been derived by the first law of the entanglement entropy at the
classical level. According to our discussion on both of the first law and
quantum correction of the entanglement contour, the perturbation of the
boundary entanglement contour at the quantum level should relate to the
perturbation of the energy-momentum tensor in the bulk, and further relate to
the perturbation of the bulk geometry. We hope this can give us further
understanding about the dynamics of geometry beyond the linearized Einstein’s
equations or quantum excitations of gravity.

Figure 7: Here the homology surface $\Sigma_{A}$ is divided into $a_{1}\cup
a_{2}\cup a_{3}$ as in Fig.4. The background state is the vacuum of the
boundary CFT, and the stars are low energy excitations of the stress tensor
inside $a_{2}$. The red curve on the boundary is the perturbation of the
entanglement contour at the quantum level caused by the bulk excitations,
which is only non-zero on $A_{2}$.

## Acknowledgments

We would like to thank Andrew Rolph and Juan Pedraza for a careful reading of
this manuscript and helpful comments. QW is supported by the “Zhishan”
Scholars Programs of Southeast University. MH receives support from the
National Science Foundation through grant PHY-1912278.

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11institutetext: Department of Physics, University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA 22institutetext: Department of Physics,
University of Houston, Houston, TX 77204, USA 33institutetext: Department of
Physics, University of Wuppertal, Wuppertal D-42119, Germany

# Off-of-equilibrium effects on Kurtosis Along Strangeness-Neutral
Trajectories

Travis Dore 11 Jamie Karthein 22 Debora Mroczek 11 Paolo Parotto 33 Jacquelyn
Noronha-Hostler 11 Claudia Ratti 22

###### Abstract

The Beam Energy Scan program at the Relativistic Heavy Ion Collider (RHIC) is
searching for the QCD critical point. The main signal for the critical point
is the kurtosis of the distribution of proton yields obtained on an event-by-
event basis where one expects a peak at the critical point. However, its exact
behavior is still an open question due to out-of-equilibrium effects and
uncertainty in the equation of state. Here we use a simplistic hydrodynamic
model that enforces strangeness-neutrality, selecting trajectories that pass
close to the critical point. We vary the initial conditions to estimate the
effect of out-of-equilibrium hydrodynamics on the kurtosis signal.

##  1 Introduction

As the search for the QCD critical point Aoki:2006br ; Borsanyi:2010bp ;
Endrodi:2011gv ; Bellwied:2015rza  continues, it becomes increasingly
important to have a thorough understanding of its signals, including what
factors influence them and how strong those influences are. One such signal is
non-monotonic behavior of the kurtosis of the equilibrium net baryon
distribution Mroczek:2020rpm ; Stephanov:2011pb , specifically, a large peak
in the vicinity of the critical region signifying large fluctuations of net
baryon number. Since the QCD equation of state (EoS) at finite baryon
densities is not known from first-principles due to the Fermi sign problem,
one must rely on phenomenological methods and models.

|
---|---
Figure 1: Left: A comparison of isentropes from an EoS realization which
enforces $\mu_{B}=\mu_{S}=0$ vs. one that enforces heavy-ion trajectories.
Right: Demonstration of the ability of the system to pass through a large
portion of the phase diagram depending on what its initial conditions are for
its out-of-equilibrium components.

One such implementation is done in Ref. Karthein:2021nxe  in which a critical
region with the expected 3D Ising universality class is smoothly matched to
the Lattice QCD EoS at zero net baryon density and uses a Taylor expansion for
the non-critical contribution to the pressure at finite net baryon densities.
The size, shape, and strength of the critical region can be selected through
the use of multiple numerical parameters, each of which produce a realization
of a QCD EoS with a critical point that can be constrained against
experimental results and theoretical consistency checks (e.g. $0\leq
c_{s}^{2}\leq 1$). The aforementioned work is an extension of a previous one
Parotto:2018pwx , which now has the ability to capture ‘heavy-ion’
trajectories through the phase diagram by enforcing both net-strangeness
neutrality as well as finite electric charge, instead of $\mu_{B}=\mu_{S}=0$.
The difference in the isentropes produced (which are trajectories in the phase
diagram of constant entropy to baryon number ratio, $s/\rho_{B}$) by the two
EoS’s can be seen in Fig. 1.

The EoS developed in that previous work is used here, with the
parameterization being the same as that of the paper Karthein:2021nxe . Here,
we use this EoS in conjunction with a simplified hydrodynamic model to begin
exploring out of equilibrium effects on the kurtosis at freeze-out. The value
of the kurtosis at freeze-out is intimately related to the fluctuations in
baryon number from event to event when the system freezes out Stephanov:1998dy
; Stephanov:1999zu ; Stephanov:2008qz ; Stephanov:2011pb , thus directly tying
it to an observable. The work here does not use a realistic hydrodynamic model
and thus cannot make quantitative predictions. Instead, these results are
qualitative idea of the kinds of effects one might expect to encounter in
realistic model-to-data comparisons.

##  2 Out-of-Equilibrium Effects

It has been shown in previous works Dore:2020jye ; Du:2021zqz  that out of
equilibrium effects can have an impact on the $(T,\mu_{B})$ trajectories
explored by the system throughout its evolution. This can have a direct
influence on searches for the critical point by possibly pushing the system
towards or away from the critical point on an event-by-event basis, and even
affecting the path the system takes through the critical region. In Fig. 1, we
demonstrate the ability of a system to pass through a large portion of the
phase diagram, using the same energy density, three different baryon
densities, and many different initial values for the shear-stress tensor and
the bulk pressure.

It is also expected that in the vicinity of the critical region the bulk
viscosity, $\zeta/s$, should scale with the correlation length to the third
power, i.e. $\zeta/s\sim(\xi/\xi_{0})^{3}$ where $\xi_{0}$ is a scaling
constant picked to scale the correlation length appropriately in the critical
region Monnai:2016kud . In order to incorporate as many critical effects as
possible, we implement a bulk viscosity that has this critical scaling, the
exact form of which, can be seen in the next section.

##  3 Hydrodynamic Model

As a starting point for future work, we use a simplified Bjorken expanding
model Bjorken:1982qr  to obtain qualitative results and a basic understanding
of the sought after effects. Here we use the DNMR equations of motion detailed
in Dore:2020jye . These equations of motion require initial conditions for
$\pi_{\eta}^{\eta}$ \- the rapidity eigenvalue of the shear stress tensor,
$\Pi$ \- the bulk pressure, the energy density $\epsilon$, and $\rho$ \- the
baryon density. Second order transport coefficients are the same as in
Denicol:2018wdp ; Bazow:2016yra . We use a temperature dependent shear
viscosity ($\eta T/w$ where $w=e+p$ is the enthalpy), given by an excluded
volume calculation below the transition temperature NoronhaHostler:2012ug ;
McLaughlin:2021dph  (using the PDG16+ Alba:2017mqu ) that is matched onto a
QCD motivated parameterization in the deconfined phase Christiansen:2014ypa ;
Dubla:2018czx . The form of the non-critical and critical bulk viscosities are

$\displaystyle\frac{\zeta T}{w}$
$\displaystyle=36\times\frac{1/3-c_{s}^{2}}{8\pi}$ (1)
$\displaystyle\left(\frac{\zeta T}{w}\right)_{CS}$ $\displaystyle=\frac{\zeta
T}{w}\left[1+\left(\frac{\xi}{\xi_{0}}\right)^{3}\right]$ (2)

##  4 Effects On Probed Kurtosis

Figure 2: Trajectories of different calculations that pass through the
critical region.

In Fig. 2, one can see a variety of different hydrodynamic $(T,\mu_{B})$
trajectories that the system can pass through and still go through the
critical region. Here, we study which value of the kurtosis is actually probed
by these runs, and check this at three different values of temperature after
the critical point, specifically, 1, 3, and 5 MeV below $T_{c}$. In Fig. 3 we
show which different values of $\chi_{4}$ are probed in our simulations. In
this work, $\chi_{4}$ is used as a substitute for the kurtosis since the two
are closely related.

|  |
---|---|---
Figure 3: The fourth moment of the equilibrium net baryon distribution at
fixed values of temperature. The vertical lines are bands around possible
values of the kurtosis seen including out of equilibrium effects. The dots are
where the isentropes intersect the parabolas of constant temperature.

It can be seen in Fig. 3 that out of equilibrium effects lead to a spread in
$\chi_{4}$ probed on an event-by-event basis. The vertical lines for each
figure represent the bounds given by out-of-equilibrium effects in our work,
the dots show where the isentrope probes $\chi_{4}$ at a given $(T,\mu_{B})$.
Interestingly, in the bottom right panel, the isentrope probes a region
outside the bounds probed by out-of-equilibrium effects.

##  5 Conclusion

In this work we have explored some consequences that out-of-equilibrium
effects could have on the search for the critical pointby looking at the
fourth moment of the equilibrium net-baryon distribution. We found that, for a
given EoS, the value of kurtosis you probe at freeze-out is heavily dependent
on the out-of-equilibrium initial conditions of the system. This can lead to
different values of the kurtosis being probed on an event-by-event basis.

This material is based upon work supported by the the US-DOE Nuclear Science
Grant No. DE-SC0019175; the National Science Foundation under grants no.
PHY-1654219, PHY-1560077, and DGE â 1746047; and the DFG grant SFB/TR55.
J.N.H. and T.D. are supported by the US-DOE Nuclear Science Grant No. DE-
SC0020633. DM is partially supported by the National Science Foundation
Graduate Research Fellowship Program under Grant No. DGE â 1746047

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