print-report

Entanglement entropy and the first law at third order for boosted black branes

Sabyasachi<EMAIL_ADDRESS>and Harvendra
<EMAIL_ADDRESS>

${}^{a}\,$Theory Division, Saha Institute of Nuclear Physics, 1/AF,
Bidhannagar,
West Bengal 700064, India ${}^{b}\,$Homi Bhabha National Institute, Anushakti
Nagar, Mumbai,
Maharashtra 400094, India

###### Abstract

Gauge/gravity duality relates an AdS black hole with uniform boost with a
boosted strongly-coupled CFT at finite temperature. We study the perturbative
change in holographic entanglement entropy for strip sub-region in such
gravity solutions up to third order and try to formulate a first law of
entanglement thermodynamics including higher order corrections. The first law
receives important contribution from an entanglement chemical potential in
presence of boost. We find that suitable modifications to the entanglement
temperature and entanglement chemical potential are required to account for
higher order corrections. The results can be extended to non-conformal cases
and AdS plane wave background.

##  1 Introduction

Of late, studies on quantum information theoretic properties of a strongly
coupled QFT using tools from AdS/CFT correspondence [1, 2, 3] have received
widespread attention. A pioneering work in this course was the holographic
formula for computing entanglement entropy [4, 5] and its covariant
generalization [6]. The holographic entanglement entropy (HEE) formula
inspired the idea that emergence of space-time in the gravity picture is
somehow related to the quantum entanglement degrees of freedom in the dual
field theory [7]. An important observation coming out from the holographic
calculations is that there exists a relationship between entanglement entropy
of an excited state and the corresponding changes in energy and other
macroscopic variables that is reminiscent of the first law of thermodynamics

$T_{E}\Delta S_{E}=\Delta E-\frac{d-1}{d+1}V\Delta P\,,$ (1.1)

where the _entanglement temperature_ , $T_{E}$ is known to obey a universal
behaviour inversely proportional to the size of the entangling region. The
_first law of entanglement thermodynamics_ was initially established by
considering very small excitation over the dual space-time geometry (e.g. pure
AdS) [8, 9, 10] and also from a CFT point of view [11]. In fact, in the later
work it was argued that any conserved charge associated with the excitation
could appear in the law with its conjugate _entanglement chemical potential_ ,
$\mu_{E}$. Over the years, extensive studies have been made on multiple
aspects of the first law of entanglement thermodynamics [12, 13, 14, 15, 16,
17, 18, 19, 20]. The first law like relationship also plays a central role in
reconstructing Einstein’s equations for the dual geometry from boundary CFT
data [21, 22, 23, 24].

In most of the works on the first law of entanglement so far, the bulk
excitation is considered only at leading order and there is no general
consensus on its validity beyond the leading order. We extend upon the work of
[15], where second order corrections to holographic entanglement entropy for a
strip sub-system on the boundary of a boosted AdS black hole were studied.
This gravity solution is dual to a finite temperature CFT with uniform boost
in one of the spatial directions. It was shown in [15] that the first law of
entanglement could incorporate the second order contributions, provided
certain constituents of the law were appropriately modified. In this article
we go one step further and find explicit expressions for third order
perturbative corrections to the HEE in the same setup. We then try to include
these new contributions to the first law by making required redefinition of
our entanglement temperature $(T_{E})$ and chemical potential $(\mu_{E})$.
While doing so, we consider first order variation of all variables instead of
their relative change over the ground state, as is common in bulk of the
literature. Our work is motivated by the idea that the first law of
entanglement is in fact, an exact relationship not limited to first order
fluctuations. Indeed, in the large subsystem limit, the EE is proportional to
the thermal entropy [5] and the entanglement temperature should also, then,
have a flow towards the Hawking temperature of the black hole. This is not,
however, eminent if we restrict the law (1.1) at leading order. Similar line
of thought was also advocated in [25, 26]. In addition, a system with boost
allows us to explicitly check the influence of the Kaluza-Klein momentum on
the first law of entanglement and draw a parallel with the law of
thermodynamics where it is known to contribute [27].

The rest of the paper is organized as follows: in section 2 we introduce the
bulk geometry and write down the area functional for HEE, in section 3 we
consider the narrow strip limit to compute the integrals perturbatively up to
third order. A modified first law involving third order corrections is put
forward in section 4. In section 5 we discuss the HEE of AdS plane wave
geometry by taking a special double limit of the results obtained. In section
6 we extend our calculations to any generic non-conformal Dp-brane geometry
using some simple substitution and argue for a first law in this case as well.
Finally, we conclude in section 7 with a brief discussion.

##  2 Minimal area functional for boosted AdS black hole

For static space-time the HEE can be calculated using the Ryu-Takayanagi
formula [4, 5], which asserts that the entanglement entropy associated with a
region $A$ with boundary $\partial A$ on the asymptotic boundary of the
$(d+1)$-dimensional space-time is dual to the area of a minimal co-dimension
$2$ surface $\gamma_{A}$ in the bulk such that $\partial\gamma_{A}=\partial A$

$S_{A}=\frac{Area(\gamma_{A})}{4G_{N}},$ (2.1)

where $G_{N}$ is the Newton’s constant in $(d+1)$-dimensions. The space-time
of our interest is a boosted AdS black hole described by the line element

$ds^{2}=\frac{R^{2}}{z^{2}}\left(-\frac{f\left(z\right)dt^{2}}{K\left(z\right)}+K\left(z\right)\left(dy-\omega\right)^{2}+dx_{1}^{2}+\cdots+dx_{d-2}^{2}+\frac{dz^{2}}{f\left(z\right)}\right),$
(2.2)

with

$f\left(z\right)=1-\frac{z^{d}}{z_{h}^{d}},~{}~{}~{}~{}K\left(z\right)=1+\beta^{2}\gamma^{2}\frac{z^{d}}{z_{h}^{d}},$
(2.3)

where $z_{h}$ denotes the event horizon of the black hole and $0\leq\beta\leq
1$ is the boost parameter, while $\gamma=\frac{1}{\sqrt{1-\beta^{2}}}$. $R$
sets the overall curvature of the space-time. The boost is taken in
$y$-direction and the same is compactified on a circle of radius $r_{y}$. The
Kaluza-Klein form

$\omega=\beta^{-1}\left(1-\frac{1}{K}\right)dt\,.$ (2.4)

We are interested in the entanglement entropy of a subsystem
$A:x_{1}\in\left[-\frac{\ell}{2},\frac{\ell}{2}\right]$ and
$x_{j}\in\left[0,L\right]~{}\left(j=2,3,\ldots,d-2\right)$ such that
$L\gg\ell$. We will always consider a constant time slice of the metric (2.2).
Then the RT surface is conveniently described by the function
$x_{1}\left(z\right)$ and its area is given by the integral

$\mathcal{A}_{\gamma}=2R^{d-1}V_{d-2}\int_{\epsilon}^{z_{*}}\frac{dz}{z^{d-1}}\sqrt{K}\sqrt{\frac{1}{f}+x_{1}^{\prime}(z)^{2}}\,,$
(2.5)

where $V_{d-2}=2\pi r_{y}L^{d-3}$ is the overall volume of the extended
directions and the y-circle on the boundary and $\epsilon$ is a UV cut-off put
to protect the integral from divergence as $z\to 0$. For ease of calculation
we will often assume $R=1$. Upon extremization this area functional gives a
first integral of motion

$\frac{x_{1}^{\prime}(z)\sqrt{K}}{z^{d-1}\sqrt{\frac{1}{f}+x_{1}^{\prime}(z)^{2}}}=b\,,$
(2.6)

the constant b may be related to the _turning point_ $z_{*}$ by

$K\left(z_{*}\right)-b^{2}z_{*}^{2d-2}=0$

which leads to an integral relation between $z_{*}$ and the strip-width $\ell$

$\frac{\ell}{2}=z_{*}\int_{0}^{1}\frac{dy\,y^{d-1}}{\sqrt{f}\sqrt{\frac{K}{K_{*}}-y^{2d-2}}},$
(2.7)

where $K_{*}=K(z_{*})$ and $y\equiv\frac{z}{z_{*}}$. We obtain by substitution
in equation (2.5) that

$\mathcal{A}_{\gamma}=\frac{2V_{d-2}}{z_{*}^{d-2}}\int_{\frac{\epsilon}{z_{*}}}^{1}\frac{dy}{y^{d-1}}\sqrt{\frac{K}{f\left(1-\frac{K_{*}}{K}y^{2d-2}\right)}}$
(2.8)

##  3 Perturbative calculation of entanglement entropy

Equation (2.8) is unusually hard to solve analytically. In fact, the only
known exact solution except pure AdS was found for the $(2+1)$-dimensional BTZ
black hole [6]. However in certain regime of the solution space the problem
can be addressed.

Let us assume $z_{*}\ll z_{h}$, in this limit the integrals (2.7) and (2.8)
may be expressed as a power series in $\frac{z_{*}^{d}}{z_{h}^{d}}$. If one
considers $\mathscr{O}\left(\frac{z_{*}^{d}}{z_{h}^{d}}\right)$ deviations
only then the equation (2.7) can be recast as

$\frac{\ell}{2}=z_{*}\int_{0}^{1}\frac{dy\,y^{d-1}}{\sqrt{1-y^{2d-2}}}\left(1+\frac{z_{*}^{d}}{z_{h}^{d}}\left(\frac{y^{d}}{2}+\frac{k}{2}\frac{1-y^{d}}{1-y^{2d-2}}\right)+\mathscr{O}\left(\frac{z_{*}^{2d}}{z_{h}^{2d}}\right)\right),$
(3.1)

where we chose to denote $\beta^{2}\gamma^{2}$ by $k$, the integral can be
solved in terms of beta functions, let us define

$\int_{0}^{1}\frac{dy\,y^{nd-1}}{\sqrt{1-y^{2d-2}}}=\frac{B\left(\frac{nd}{2d-2},\frac{1}{2}\right)}{2d-2}\equiv
b_{n-1}\,.$

We may then express (3.1) as

$\frac{\ell}{2}=z_{*}\left[b_{0}+\frac{z_{*}^{d}}{z_{h}^{d}}\left(z_{10}+k\,z_{11}\right)\right],$
(3.2)

where the coefficients are

$\displaystyle z_{10}$ $\displaystyle=\frac{b_{1}}{2},$ $\displaystyle z_{11}$
$\displaystyle=\left(\frac{d+1}{d-1}\right)\frac{b_{1}}{2}-\frac{1}{d-1}\frac{b_{0}}{2}\,.$
(3.3)

To write the area of the RT surface in terms of the subsystem length $(\ell)$,
we need to invert (3.2). If we denote the turning point for pure AdS by
$\bar{z}_{*}\equiv\frac{\ell}{2b_{0}}$ then an approximate relationship at
leading order is

$z_{*}=\frac{\bar{z}_{*}}{1+\frac{\bar{z}_{*}^{2}}{z_{h}^{d}}\left(\frac{z_{10}+k\,z_{11}}{b_{0}}\right)+\mathscr{O}\left(\frac{\bar{z}_{*}^{2d}}{z_{h}^{2d}}\right)}.$
(3.4)

In a similar vein we can perform a series expansion of the area integral (2.8)

$\mathcal{A}_{\gamma}=\frac{2V_{d-2}}{z_{*}^{d-2}}\int_{\frac{\epsilon}{z_{*}}}^{1}\frac{dy}{y^{d-1}\sqrt{1-y^{2d-2}}}\left(1+\frac{z_{*}^{d}}{z_{h}^{d}}\left(\frac{1+k}{2}y^{d}+\frac{k}{2}\frac{y^{2d-2}\left(1-y^{d}\right)}{1-y^{2d-2}}\right)+\mathscr{O}\left(\frac{z_{*}^{2d}}{z_{h}^{2d}}\right)\right),$
(3.5)

again all integrations here finally boil down to some beta functions. The
ground state (no black hole) result can be very easily read off

$\displaystyle\mathcal{A}_{0}$
$\displaystyle=\frac{2V_{d-2}}{\bar{z}_{*}^{d-2}}\int_{\frac{\epsilon}{z_{*}}}^{1}\frac{dy}{y^{d-1}\sqrt{1-y^{2d-2}}}\,,$
$\displaystyle=\frac{2V_{d-2}}{d-2}\left(\frac{1}{\epsilon^{d-2}}-\frac{b_{0}}{\bar{z}_{*}^{d-2}}\right),$
(3.6)

which exhibits the characteristic UV divergence, contributions at higher
orders are all finite. Making use of the relationship (3.4) they can be
expressed as

$\frac{\mathcal{A}_{\gamma}}{V_{d-2}}=\frac{\mathcal{A}_{0}}{V_{d-2}}+\frac{\bar{z}_{*}^{2}}{z_{h}^{d}}\,\frac{(d-1)+(d+1)k}{2}b_{1}+\mathscr{O}\left(\frac{\bar{z}_{*}^{d}}{z_{h}^{d}}\right)\,.$
(3.7)

The calculation can be continued at higher orders in
$\frac{\bar{z}_{*}^{d}}{z_{h}^{d}}$, albeit with increasing level of
difficulty. The second order results were calculated in [15]. In this work, we
go one step further and seek to evaluate the area up to
$\mathscr{O}\left(\frac{\bar{z}_{*}^{3d}}{z_{h}^{3d}}\right)$, at this order
the approximate relationship between the subsystem length $(\ell)$ and the
turning point $(z_{*})$ is

$z_{*}=\frac{\bar{z}_{*}}{1+\frac{\bar{z}_{*}^{d}}{z_{h}^{d}}\frac{Z_{1}}{b_{0}}+\frac{\bar{z}_{*}^{2d}}{z_{h}^{2d}}\left(\frac{Z_{2}}{b_{0}}-d\,\frac{Z_{1}^{2}}{b_{0}^{2}}\right)+\frac{\bar{z}_{*}^{3d}}{z_{h}^{3d}}\left(\frac{d(3d+1)}{2}\frac{Z_{1}^{3}}{b_{0}^{3}}-3d\,\frac{Z_{1}Z_{2}}{b_{0}^{2}}+\frac{Z_{3}}{b_{0}}\right)+\mathscr{O}\left(\frac{\bar{z}_{*}^{4d}}{z_{h}^{4d}}\right)}\,,$
(3.8)

where $Z_{1}=z_{10}+k\,z_{11}$ and the new coefficient
$Z_{2}~{}\text{and}~{}Z_{3}$ are

$\displaystyle Z_{2}$ $\displaystyle=z_{20}+k\,z_{21}+k^{2}\,z_{22}\,,$
$\displaystyle Z_{3}$
$\displaystyle=z_{30}+k\,z_{31}+k^{2}\,z_{32}+k^{3}\,z_{33}\,,$

such that

$\displaystyle z_{20}$ $\displaystyle=\frac{3}{8}b_{2},$ (3.9) $\displaystyle
z_{21}$
$\displaystyle=\frac{1}{4}\left(-\frac{d+1}{d-1}\,b_{1}+\frac{2d+1}{d-1}\,b_{2}\right),$
(3.10) $\displaystyle z_{22}$
$\displaystyle=\frac{1}{8}\frac{2d-1}{(d-1)^{2}}\,b_{0}-\frac{1}{4}\left(\frac{d+1}{d-1}\right)^{2}b_{1}+\frac{3}{8}\frac{2d+1}{(d-1)^{2}}\,b_{2},$
(3.11) $\displaystyle z_{30}$ $\displaystyle=\frac{5}{16}b_{3},$ (3.12)
$\displaystyle z_{31}$
$\displaystyle=\frac{3}{16}\left(-\frac{2d+1}{d-1}\,b_{2}+\frac{3d+1}{d-1}\,b_{3}\right),$
(3.13) $\displaystyle z_{32}$
$\displaystyle=\frac{1}{16}\frac{(3d-1)(d+1)}{(d-1)^{2}}\,b_{1}-\frac{1}{8}\left(\frac{2d+1}{d-1}\right)^{2}b_{2}+\frac{1}{16}\frac{(d+3)(3d+1)}{(d-1)^{2}}\,b_{3},$
(3.14) $\displaystyle z_{33}$
$\displaystyle=\frac{1}{48}\Bigg{(}\frac{-3-2d(4d-5)}{(d-1)^{3}}\,b_{0}+\frac{3(3d-1)(d+1)^{2}}{(d-1)^{3}}\,b_{1}-9\frac{1+4d(d+1)}{(d-1)^{3}}\,b_{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{15+d(47-d(3d-5))}{(d-1)^{3}}\,b_{3}\Bigg{)}.$
(3.15)

The explicit integrals that produce these coefficients are listed in appendix
A. The minimal area at this order takes the following form

$\begin{split}\frac{\mathcal{A}_{\gamma}^{(3)}}{V_{d-2}}=\frac{\mathcal{A}_{0}}{V_{d-2}}\,+\,&\frac{\bar{z}_{*}^{2}}{z_{h}^{d}}\frac{I_{1}b_{1}}{2}-\frac{\bar{z}_{*}^{d+2}}{z_{h}^{2d}}\frac{\left(d+2\right)I_{1}^{2}b_{1}^{2}-I_{2}b_{0}b_{2}}{4\left(d-1\right)\left(d+2\right)b_{0}}\,+\\\
&\frac{\bar{z}_{*}^{2d+2}}{z_{h}^{3d}}\,\frac{2(d+1)(2d+3)I_{1}^{3}b_{1}^{3}-6(d+1)I_{1}I_{2}b_{0}b_{1}b_{2}+I_{3}b_{0}^{2}b_{3}}{48(d-1)^{2}(d+1)b_{0}^{2}},\end{split}$
(3.16)

where,

$\displaystyle I_{1}$ $\displaystyle=(d-1)+(d+1)k\,,$ (3.17) $\displaystyle
I_{2}$ $\displaystyle=3(d-1)^{2}+2(d-1)(2d+1)k+3(2d+1)k^{2}\,,$ (3.18)
$\displaystyle I_{3}$
$\displaystyle=15(d-1)^{3}+9(d-1)^{2}(3d+1)k+3(d-1)(3d+1)(d+3)k^{2}$
$\displaystyle~{}~{}~{}~{}-(d-5)(d+3)(3d+1)k^{3}\,,$ (3.19)

all of which are positive quantities. The expressions correctly reproduce the
first and second order results of [15], while the third order result is new.
Let us point out that the absolute sign of the change in area of the RT
surface alternates at each order; being positive at leading order, negative at
second order and so on. This is consistent with earlier observations and hints
towards the existence of a complete, non-perturbative expression.

From equation (3.16) The change in HEE over pure AdS could be written as

$S^{(3)}=\frac{\mathcal{A}_{\gamma}^{(3)}-\mathcal{A}_{0}}{4G_{N}^{(d+1)}}$
(3.20)

### Numerical evaluation of HEE

The perturbation method illustrated above should work well if
$\frac{\bar{z}_{*}^{d}}{z_{h}^{d}}\ll 1$ or in other words $\ell\ll z_{h}$
(very narrow strip width). It is instructive to do a numerical evaluation of
the minimal area and compare how close our perturbation series analysis can
mimic the behaviour.

To perform the numerics we restrict ourselves to $\left(4+1\right)$
dimensions. We first regularize the area integral (2.8) by separating out the
divergent piece and write the finite part as

$\mathcal{A}_{\gamma}=\frac{2}{z_{*}^{d-2}}\left[\int_{0}^{1}\frac{dy}{y^{d-1}}\left(\frac{K(y)}{\sqrt{f(y)}\sqrt{K(y)-K_{*}y^{2d-2}}}-1\right)-\frac{1}{d-2}\right].$
(3.21)

Similarly we also write down the area integral for ordinary AdS space-time

$\mathcal{A}_{0}=\frac{2}{\bar{z}_{*}^{d-2}}\left[\int_{0}^{1}\frac{dy}{y^{d-1}}\left(\frac{1}{\sqrt{1-y^{2d-2}}}-1\right)-\frac{1}{d-2}\right].$
(3.22)

We choose a few values for the turning point $z_{*}$ and obtain the
corresponding subsystem lengths $\ell$ from equation (2.7), for the same
values we also integrate equations (3.21) and (3.22) to obtain the area
difference $\Delta A=\mathcal{A}_{\gamma}-\mathcal{A}_{0}$. We then plot
$\Delta A$ on the y-axis and $\ell$ on the x-axis.

Figure 1: Area difference of the minimal surface from ground state at
different orders for AdS5 and their comparison with numerical result, plot
drawn by choosing $\beta=0.25~{}\text{and}~{}z_{h}=10$.

The results are succinctly expressed in figure 1, we observe that perturbative
results agree to a good extent with numerical values in the region
$\ell<z_{h}$ and the accuracy increases with inclusion of higher order terms.

##  4 First law of entanglement thermodynamics

The HEE is found to depend on two parameters, viz. the horizon distance
$z_{h}$ and the boost parameter $\beta$ (or equivalently $k$); hence, any
infinitesimal change in these two parameters is bound to affect the HEE. Our
next task is to express first order variation of $S^{(3)}$ with respect to
these parameters as a ‘first law’ like relationship.

To this end, we first require the components of the boundary stress-energy
tensor which can be found from a Fefferman-Graham expansion [28, 29, 30] of
the metric (2.2), the expansion leads us to [15]

$\langle
T_{\mu\nu}\rangle\sim\frac{d}{z_{h}^{d}}\begin{pmatrix}\frac{d-1}{d}+k&\beta\gamma^{2}&0&\cdots\\\
\beta\gamma^{2}&\frac{1}{d}+k&0&\cdots\\\ 0&0&\frac{1}{d}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}$ (4.1)

The $T_{01}$ component quantifies the charge due to momentum in the
y-direction, we choose an appropriate normalization factor and write the
following conserved quantities

$\displaystyle E$ $\displaystyle=\frac{V_{d-2}}{16\pi
G_{N}^{d+1}}\left(\frac{d-1}{d}+k\right)\frac{d\,\ell}{z_{h}^{d}}\,,$ (4.2)
$\displaystyle P$
$\displaystyle=\frac{r_{y}}{8G_{N}^{d+1}}\frac{1}{z_{h}^{d}}\,,$ (4.3)
$\displaystyle N$ $\displaystyle=\frac{r_{y}V_{d-2}}{16\pi
G_{N}^{d+1}}\beta\gamma^{2}\frac{d\,\ell}{z_{h}^{d}}\,,$ (4.4)

In the above expressions, $E,P\text{~{}and~{}}N$ stand for the changes in
energy, pressure transverse to the boost and the momentum charge due to boost
in $y$-direction, respectively. We also define an _entanglement chemical
potential_ $\mu_{E}$ as the value of the Kaluza-Klein gauge field at the
turning point [15]

$\mu_{E}\equiv\frac{1}{r_{y}\beta}\left(1-\frac{1}{K_{*}}\right),$ (4.5)

it can be expressed order by order using the perturbative expression (3.8)

$\displaystyle\mu_{E}^{(1)}$
$\displaystyle=\frac{\beta\gamma^{2}}{r_{y}}\frac{\bar{z}_{*}^{d}}{z_{h}^{d}}\,,$
$\displaystyle\mu_{E}^{(2)}$
$\displaystyle=\frac{\beta\gamma^{2}}{r_{y}}\left(\frac{\bar{z}_{*}}{z_{h}}\right)^{d}\left(1-\left(\frac{\bar{z}_{*}}{z_{h}}\right)^{d}\left(k+d\frac{Z_{1}}{b_{0}}\right)\right).$
(4.6)

It is straightforward to write the first law at leading order, the
entanglement entropy is

$S^{(1)}=\frac{V_{d-2}}{8G_{N}^{d+1}}\frac{\bar{z}_{*}^{2}}{z_{h}^{d}}\,\left((d-1)+(d+1)k\right)b_{1}$

from the expressions of $\mu_{E}$ we make out that any contribution from a
$\mu_{E}\delta N$ like term should not occur before second order. One can
easily check that at first order [8, 9, 15]

$\delta S^{(1)}=\beta_{E}^{(1)}\left(\delta E-\frac{d-1}{d+1}V\delta
P\right),$ (4.7)

where $V=\ell V_{d-2}$ is the total volume of the subsystem on the boundary.
The inverse _entanglement temperature_ (at first order) $\beta_{E}^{(1)}$ is
given by

$\beta_{E}^{(1)}=\left(\frac{d+1}{d}\right)\frac{\pi
b_{1}}{2b_{0}^{2}}\ell\,.$ (4.8)

Non-triviality enters at sub-leading orders as it is not transparent how one
should take care of subsequent corrections. The authors of [15] argued that
higher order corrections could be incorporated in the law through appropriate
re-definition of the entanglement temperature, chemical potential and
subsystem volume. We take a related but slightly different approach. Let us
illustrate with the change in HEE up-to second order; we claim

$\delta S^{(2)}=\boldsymbol{\alpha}\left(\delta E-\frac{d-1}{d+1}V\delta
P\right)-\boldsymbol{\zeta}\,\mu_{E}^{(1)}\delta N\,,$ (4.9)

which, after performing the variation, leads to

$\begin{split}\frac{V_{d-2}}{8G_{N}^{(d+1)}}\left[\delta
z_{h}^{-d}\frac{\partial S^{(2)}}{\partial z_{h}^{d}}+\delta k\frac{\partial
S^{(2)}}{\partial k}\right]=\delta
z_{h}^{-d}\left(\boldsymbol{\alpha}\left(\frac{\partial E}{\partial
z_{h}^{d}}-\frac{d-1}{d+1}V\frac{\partial P}{\partial
z_{h}^{d}}\right)-\boldsymbol{\zeta}\mu_{E}^{(1)}\frac{\partial N}{\partial
z_{h}^{d}}\right)+\\\ \delta k\left(\boldsymbol{\alpha}\frac{\partial
E}{\partial k}-\boldsymbol{\zeta}\mu_{E}^{(1)}\frac{\partial N}{\partial
k}\right).\end{split}$

We refrain from writing full expressions to avoid clutter. To determine the
coefficients $\boldsymbol{\alpha}$ and $\boldsymbol{\zeta}$ we may equate the
coefficients of $\delta z_{h}^{-2}$ and $\delta k$ from both sides, this
yields the unique solution

$\displaystyle\boldsymbol{\alpha}$
$\displaystyle\simeq\beta_{E}^{(1)}\left(1+\left(\frac{\bar{z}_{*}}{z_{h}}\right)^{d}\frac{\alpha_{1}}{\beta_{E}^{(1)}}+\mathscr{O}\left(\frac{\bar{z}_{*}^{2d}}{z_{h}^{2d}}\right)\right),$
(4.10) $\displaystyle\boldsymbol{\zeta}$ $\displaystyle\simeq\frac{\pi
b_{2}}{b_{0}^{2}}\frac{d^{2}+d-2-k\left(2+3d-2d^{2}\right)}{d(d+2)(d-1+k(d-3))}\ell+\mathscr{O}\left(\frac{\bar{z}_{*}^{d}}{z_{h}^{d}}\right)\,,$
(4.11)

where,

$\alpha_{1}=-\frac{(d+1)\left((d+2)b_{1}^{2}-3b_{0}b_{2}\right)}{d(d+2)b_{0}}+k\,\frac{(d+1)\left((2d-5)(2d+1)b_{0}b_{2}-\left(d^{3}-7d-6\right)b_{1}^{2}\right)}{d(d-1)(d+2)((d-3)k+d-1)b_{0}}\,.\\\
$ (4.12)

It is sufficient to determine $\boldsymbol{\zeta}$ at leading order since
$\mu_{E}^{(1)}\delta N$ is itself a second order quantity, we also note that
at leading order $\boldsymbol{\alpha}=\beta_{E}^{(1)}$. It is then appropriate
to define $\boldsymbol{\alpha}$ as the inverse entanglement temperature at
second order and redefine our entanglement chemical potential as

$\displaystyle\beta_{E}^{(2)}$ $\displaystyle\equiv\boldsymbol{\alpha}\,,$
(4.13) $\displaystyle\bar{\mu}_{E}^{(1)}$
$\displaystyle\equiv\frac{\boldsymbol{\zeta}}{\boldsymbol{\alpha}}\mu_{E}^{(1)}\simeq\frac{2\left(d^{2}+d-2-k\left(2+3d-2d^{2}\right)\right)b_{2}}{(d+1)(d+2)(d-1+k(d-3))b_{1}}\,\mu_{E}^{(1)}\,.$
(4.14)

The above redefinition allows us to express the variation of HEE in the
desired form

$\delta S_{E}^{(2)}=\beta_{E}^{(2)}\left(\delta E-\frac{d-1}{d+1}V\delta
P-\bar{\mu}_{E}^{(1)}\delta N\right).$ (4.15)

In a similar way we can extend the first law up-to third order, the modified
entanglement temperature and chemical potential would be

$\displaystyle\beta_{E}^{(3)}$
$\displaystyle=\beta_{E}^{(1)}\left(1+\left(\frac{\bar{z}_{*}}{z_{h}}\right)^{d}\frac{\alpha_{1}}{\beta_{E}^{(1)}}+\left(\frac{\bar{z}_{*}}{z_{h}}\right)^{2d}\frac{\alpha_{2}}{\beta_{E}^{(1)}}+\mathscr{O}\left(\frac{\bar{z}_{*}^{3d}}{z_{h}^{3d}}\right)\right),$
(4.16) $\displaystyle\bar{\mu}_{E}^{(2)}$
$\displaystyle=\frac{\boldsymbol{\zeta}}{\boldsymbol{\alpha}}\mu_{E}^{(2)}\,,$
(4.17)

the new coefficient $\alpha_{2}$ is given by

$\alpha_{2}=\frac{\alpha_{20}+k\,\alpha_{21}+k^{2}\,\alpha_{22}}{8d(d-1)^{2}\left((d-3)k+d-1\right)b_{0}^{2}}$
(4.18)

where,

$\begin{split}\alpha_{20}&=\left(15b_{0}^{2}b_{3}-18(d+1)b_{0}b_{1}b_{2}+2(d+1)(2d+3)b_{1}^{3}\right)\left((d-1)^{2}\left((d-3)k+d-1\right)\right),\\\
\alpha_{21}&=(d-1)\left((3d+1)(11d-15)b_{0}^{2}b_{3}+2(d(45-4d(6d+1))+25)b_{0}b_{1}b_{2}\right.\\\
&\left.~{}~{}~{}~{}+2(d+1)^{2}(2d+3)(3d-5)b_{1}^{3}\right)\,,\\\
\alpha_{22}&=(d+3)(3d-7)(3d+1)b_{0}^{2}b_{3}-2(2d+1)(d+1)((4d-3)d-19)b_{0}b_{1}b_{2}\\\
&~{}~{}~{}~{}+2(2d+3)(d-3)(d+1)^{3}b_{1}^{3}\,.\end{split}$

The correct dressing factor for the chemical potential at this order involves
first order terms in both $\boldsymbol{\zeta}$ and $\boldsymbol{\alpha}$; it
is expressed as

$\begin{split}\frac{\boldsymbol{\zeta}}{\boldsymbol{\alpha}}=&\frac{2\left(d^{2}+d-2-k\left(2+3d-2d^{2}\right)\right)b_{2}}{(d+1)(d+2)(d-1+k(d-3))b_{1}}\\\
&-\left(\frac{\bar{z}_{*}^{d}}{z_{h}^{d}}\right)\Big{[}2(d-2)(d+1)(d+2)k((d-3)k+d-1)\left(d^{2}+(d-2)(2d+1)k+d-2\right)b_{1}b_{2}\\\
&+(d+2)^{2}((d-3)k+d-1)\left(3d\left(d^{2}-3\right)+(d-2)(d+3)(3d+1)k^{2}\right.\\\
&\left.+2d(d-1)(3d+1)k+6\right)b_{1}b_{3}-4(d+1)\left(d^{2}+(d-2)(2d+1)k+d-2\right)\\\
&\left((2d-5)(2d+1)k^{2}+6(d-1)^{2}k+3(d-1)^{2}\right)b_{2}^{2}\Big{]}\\\
&\left(2b_{1}^{2}(d-1)(d+1)^{2}(d+2)^{2}((d-3)k+d-1)^{2}\right)^{-1}.\end{split}$
(4.19)

The first law of entanglement including third order corrections is, therefore,

$\delta S^{(3)}=\beta_{E}^{(3)}\left(\delta E-\frac{d-1}{d+1}V\delta
P-\bar{\mu}_{E}^{(2)}\delta N\right),$ (4.20)

In figure 2 we have shown the dependence of entanglement temperature on
subsystem width ($\ell$) with and without higher order corrections. At the
leading order $T_{E}\sim\ell^{-1}$ and it simply decays to zero as
$\ell\to\infty$. We note that the decay is somewhat dampened by corrections,
the leading order of which is positive. We also note an alternating property
of signs as we go on including higher order terms to $T_{E}$, with the first
order being positive, the second order negative and so on. This strongly
suggests that the entanglement temperature finally converges to a unique value
once all possible corrections are taken care for and matches with the black
hole temperature in the large system size limit. However, it is not clear if
such a conclusion can be drawn from a simple perturbation series analysis as
ours.

(a) AdS4

(b) AdS5

Figure 2: The entanglement temperature with and without higher order
corrections for $d=3~{}\text{and}~{}4$, plot drawn for $z_{h}=100$ and
$\beta=0.25$.

##  5 Large boost limit and AdS plane wave

It is possible to consider a simultaneous limit where the black hole horizon
is allowed to shrink while the boost parameter is taken to be very large, we
let

$z_{h}\to\infty,~{}~{}~{}~{}\beta\to
1,~{}~{}~{}~{}\frac{\beta^{2}\gamma^{2}}{z_{h}^{d}}=\frac{1}{z_{I}^{d}}=\text{fixed}.$
(5.1)

Such a double limit has been explored before in [31, 32] in the context of
non-relativistic holography. In this limit the background geometry (2.2)
reduces to an AdS plane wave

$ds^{2}=\frac{R^{2}}{z^{2}}\left(-\frac{dt^{2}}{K}+K\left(dy-\left(1-K^{-1}\right)dt\right)^{2}+dx_{1}^{2}+\cdots+dx_{d-2}^{2}+dz^{2}\right),$
(5.2)

where $K=1+\frac{z^{d}}{z_{I}^{d}}$. $z_{I}$ is representative of the energy
or momentum of the wave travelling in $y$-direction. The entanglement entropy
of AdS plane wave has been studied previously in [33, 34, 35]. We can use our
results (3.16) and (3.17) - (3.19) and apply the above double limit to obtain
the HEE for a strip system in the plane wave background, given that we
maintain $\frac{\bar{z}_{*}^{d}}{z_{I}^{d}}\ll 1$. Up to third order, then,
the change in HEE over ground state can be expressed as

$\begin{split}S^{(3)}_{\mathrm{plane}}=\frac{V_{d-2}}{8G_{N}^{(d+1)}}\frac{\bar{z}_{*}^{2}}{z_{I}^{d}}&\left(I_{1}b_{1}-\frac{\bar{z}_{*}^{d}}{z_{I}^{d}}\frac{\left(d+2\right)I_{1}^{2}b_{1}^{2}-I_{2}b_{0}b_{2}}{2\left(d-1\right)\left(d+2\right)b_{0}}\,+\right.\\\
&\left.\frac{\bar{z}_{*}^{2d}}{z_{I}^{2d}}\,\frac{2(d+1)(2d+3)I_{1}^{3}b_{1}^{3}-6(d+1)I_{1}I_{2}b_{0}b_{1}b_{2}+I_{3}b_{0}^{2}b_{3}}{24(d-1)^{2}(d+1)b_{0}^{2}}\right),\end{split}$
(5.3)

which is similar in appearance to that for a black hole, except that

$\displaystyle I_{1}$ $\displaystyle=(d+1)\,,$ $\displaystyle I_{2}$
$\displaystyle=3(2d+1)\,,$ $\displaystyle I_{3}$
$\displaystyle=-(d-5)(d+3)(3d+1)\,,$

are the new coefficients obtained by taking the limit (5.1) in eqs.
(3.17)-(3.19).

Let us try to figure out the form of the entanglement first law for AdS plane
wave. The energy and momentum charge of the CFT excitation dual to this
geometry become

$\displaystyle E_{\mathrm{plane}}$ $\displaystyle=\frac{V_{d-2}}{16\pi
G_{N}^{d+1}}\frac{d\,\ell}{z_{I}^{d}}\,,$ (5.4) $\displaystyle
N_{\mathrm{plane}}$ $\displaystyle=\frac{r_{y}V_{d-2}}{16\pi
G_{N}^{d+1}}\frac{d\,\ell}{z_{I}^{d}}\,,$ (5.5)

while the pressure along all $x_{i}$ directions vanish identically. Similarly,
the entanglement chemical potential for the plane wave geometry at different
orders can be written by applying the limit (5.1) in eqs. (4)

$\displaystyle\mu_{E}^{(1)}$
$\displaystyle=\frac{1}{r_{y}}\frac{\bar{z}_{*}^{d}}{z_{I}^{d}}\,,$
$\displaystyle\mu_{E}^{(2)}$
$\displaystyle=\frac{1}{r_{y}}\left(\frac{\bar{z}_{*}}{z_{I}}\right)^{d}\left(1-\left(\frac{\bar{z}_{*}}{z_{I}}\right)^{d}\left(1+d\frac{z_{11}}{b_{0}}\right)\right),$
(5.6)

with $z_{11}$ being the same as in eqn. (3.3). Again we see that any
contribution to the law from a $\mu_{E}\delta N_{\mathrm{plane}}$ like term
cannot occur at first order where the HEE simplifies to

$S_{\mathrm{plane}}^{(1)}=\frac{V_{d-2}}{8G_{N}^{(d+1)}}\frac{\bar{z}_{*}^{2}}{z_{I}^{d}}(d+1)b_{1}.$

If we consider first order variation of this expression w.r.t the new scale
$z_{I}$ we can easily establish that

$\delta S_{\mathrm{plane}}^{(1)}=\left(\frac{d+1}{d}\right)\frac{\pi
b_{1}}{2b_{0}^{2}}\ell\,\delta E_{\mathrm{plane}}.$ (5.7)

Thus the entanglement temperature for AdS plane wave is the same as that for
AdS black hole at first order: $\beta_{E}^{(1)}=\frac{\pi
b_{1}}{2b_{0}^{2}}\left(\frac{d+1}{d}\right)\ell$.

We should, nevertheless, include the chemical potential when we consider
second order or higher corrections because $\mu_{E}$ is significant from
second order onwards. Hence, at third order we are led to adding the following
contribution to the first law

$\mu_{E}^{(2)}\delta N_{\mathrm{plane}}=\frac{V_{d-2}}{16\pi
G_{N}^{d+1}}d\,\ell\left(\frac{\bar{z}_{*}}{z_{I}}\right)^{d}\left(1-\left(\frac{\bar{z}_{*}}{z_{I}}\right)^{d}\left(1+d\frac{z_{11}}{b_{0}}\right)\right)\delta
z_{I}^{-d}\,.$

So that the law at third order takes the canonical form

$\delta S_{\mathrm{plane}}^{(3)}=\beta_{E}^{(3)}\left(\delta
E_{\mathrm{plane}}-\mu_{E}^{(2)}\delta N_{\mathrm{plane}}\right),$ (5.8)

with the inverse entanglement temperature at third order, $\beta_{E}^{(3)}$
being given by

$\begin{split}\frac{\beta_{E}^{(3)}}{\beta_{E}^{(1)}}=~{}&\frac{\delta
E_{plane}-\mu_{E}^{(2)}\delta N_{plane}}{\beta_{E}^{(1)}\delta
S_{plane}^{(3)}}\\\
\approx~{}&1-\left(\frac{\bar{z}_{*}}{z_{I}}\right)^{d}\left(1+\frac{b_{1}^{2}(d+1)^{2}(d+2)-3b_{0}b_{2}(2d+1)}{b_{0}b_{1}(d-1)(d+1)(d+2)}\right)\\\
&+\left(\frac{\bar{z}_{*}}{z_{I}}\right)^{2d}\left(\frac{b_{0}d-2b_{0}+b_{1}d^{2}+b_{1}d}{2b_{0}(d-1)}+\frac{\left(b_{1}^{2}(d+1)^{2}(d+2)-3b_{0}b_{2}(2d+1)\right)^{2}}{b_{0}^{2}b_{1}^{2}(d-1)^{2}(d+1)^{2}(d+2)^{2}}\right.\\\
&\left.+\frac{b_{1}^{2}(d+1)^{2}(d+2)-3b_{0}b_{2}(2d+1)}{b_{0}b_{1}(d-1)(d+1)(d+2)}\right.\\\
&\left.-\frac{b_{0}^{2}b_{3}(5-d)(d+3)(3d+1)-18b_{0}b_{1}b_{2}(2d+1)(d+1)^{2}+2b_{1}^{3}(2d+3)(d+1)^{4}}{8b_{0}^{2}b_{1}(d-1)^{2}(d+1)^{2}}\right)\end{split}$
(5.9)

(a) AdS5

(b) AdS7

Figure 3: The entanglement temperature with and without higher order
corrections of AdS plane wave for $d=4~{}\text{and}~{}6$, plot drawn for
$z_{I}=100$.

The entanglement temperatures for AdS plane wave in $5$ and $7$ dimensions is
shown in figure 3, we note that in contrast with the AdS black hole, the first
order correction to the temperature is negative and second order is positive.
The entanglement temperature flows towards zero with increase in system size,
which is expected since the background is dual to zero temperature CFT
excitation. We learn that the higher order corrections are significant for any
finite value of the system size $\ell$; for $\ell\to 0$, $T_{E}\to\infty$ and
that is trivial indication of having no subsystem at all.

##  6 Extension to non-conformal Dp branes

The gauge/gravity duality dictionary can also be extended to non-conformal
field theories on the boundary. The supergravity description is obtained from
type II solution describing $N$ coincident Dp-branes in the field theory limit
[36]. A non-conformal black p-brane background in the Einstein frame dual to a
boosted field theory at finite temperature can be written as

$ds^{2}=\frac{\left(2\pi\right)^{\frac{p-2}{2}}}{g_{YM}}\frac{z^{\frac{\theta}{4}}}{r_{p}^{\frac{(7-p)^{2}}{8}}}\left[-\frac{f\left(z\right)}{K\left(z\right)}\frac{dt^{2}}{z^{2}}+K\left(z\right)\frac{\left(dy-\omega\right)^{2}}{z^{2}}+\frac{dx_{2}^{2}+\cdots+dx_{p}^{2}}{z^{2}}\right.\\\
\left.+\frac{4}{(5-p)^{2}}r_{p}^{7-p}\frac{dz^{2}}{f\left(z\right)\,z^{2}}+r_{p}^{7-p}d\Omega_{8-p}^{2}\right],$
(6.1)

with the dilaton being given by
$e^{\phi}=\left(2\pi\right)^{2-p}g^{2}_{YM}\left(r_{p}^{7-p}z^{\frac{2(7-p)}{(5-p)}}\right)^{\frac{3-p}{4}}$
and the functions

$f(z)=1-\frac{z^{\tilde{p}}}{z_{h}^{\tilde{p}}}\,,~{}~{}~{}K(z)=1+\beta^{2}\gamma^{2}\frac{z^{\tilde{p}}}{z_{h}^{\tilde{p}}}\,,~{}~{}~{}\omega(z)=\beta^{-1}\left(1-\frac{1}{K}\right)dt,$
(6.2)

where
$\tilde{p}=\frac{14-2p}{5-p}~{}\text{and}~{}\theta=-\frac{(3-p)^{2}}{5-p}$,
the parameter $r_{p}$ is related to the gauge coupling by
$r_{p}=g^{2}_{YM}d_{p}N\,.$

The space-time (6.1) is only conformally $AdS_{p+2}\times S^{8-p}$, for $p=3$
the conformal factor vanishes and one obtains the well known $AdS_{5}\times
S^{5}$ geometry which corresponds to a stack of $N$ D3-branes in the
decoupling limit and is dual to an $\mathcal{N}=4$ SYM on the AdS boundary
[1].

We can use the Ryu-Takayanagi formula and find the HEE for non-conformal
geometries using the Einstein frame metric (6.1), see [5, 12, 35, 37]. For a
strip of width $\ell$ as considered before: $\\{-\frac{\ell}{2}\leq
x_{2}\leq\frac{\ell}{2};~{}x_{3}=x_{4}=\cdots=x_{p}=L\\}$, we obtain the
following area integral

$\displaystyle\mathcal{A}_{\gamma}$
$\displaystyle=\frac{(2\pi)^{2p-4}V_{p-1}\Omega_{8-p}}{g_{YM}^{4}r_{p}^{(7-p)}}\int_{\epsilon}^{z_{*}}\frac{dz}{z^{p-\theta}}\sqrt{K(z)}\sqrt{x_{2}^{\prime
2}+\frac{4}{(5-p)^{2}}r_{p}^{7-p}f^{-1}(z)}\,,$
$\displaystyle=\frac{2(2\pi)^{2p-4}}{(5-p)g_{YM}^{4}r_{p}^{\frac{7-p}{2}}}V_{p-1}\Omega_{8-p}\int_{\epsilon}^{z_{*}}\frac{dz}{z^{\tilde{p}-1}}\sqrt{K(z)}\sqrt{\bar{x}_{2}^{\prime
2}+f^{-1}(z)}\,,$ (6.3)

where we did a rescaling
$\bar{x}_{2}(z)=\frac{(5-p)}{2}r_{p}^{-(7-p)}\,x_{2}(z)$ to reach the last
line. Here, $V_{p-1}=2\pi r_{y}L^{p-2}$ is the volume of the unconstrained
space directions on the brane $\left(x_{3},x_{4},\ldots,x_{p}\right)$ and the
y-circle, and $\Omega_{8-p}$ denotes the volume of the $(8-p)$-sphere.

The area integral is identical to (2.5) for the conformal case, except for the
replacement $d\to\tilde{p}$. Consequently, we can use all results from
previous sections by making the replacement in $d$. Instead of repeating them,
we go straight into writing a first law for non-conformal cases; the
components of the conserved boundary stress-energy tensor for this geometry
are [35]

$\displaystyle E$ $\displaystyle=\frac{Q_{p}V_{p-1}\Omega_{8-p}}{16\pi
G_{N}^{10}}\left(\frac{\tilde{p}-1}{\tilde{p}}+k\right)\frac{(7-p)\,\ell}{z_{h}^{\tilde{p}}}\,,$
(6.4) $\displaystyle P$
$\displaystyle=\frac{r_{y}Q_{p}\Omega_{8-p}}{8G_{N}^{10}}\frac{(7-p)}{z_{h}^{\tilde{p}}}\,,$
(6.5) $\displaystyle N$
$\displaystyle=\frac{r_{y}Q_{p}V_{p-1}\Omega_{8-p}}{16\pi
G_{N}^{10}}\beta\gamma^{2}\frac{(7-p)\,\ell}{z_{h}^{\tilde{p}}}\,,$ (6.6)

where $Q_{p}$ stands for the numerical pre-factor in (6),
$Q_{p}=\frac{2(2\pi)^{2p-4}}{(5-p)g_{YM}^{4}r_{p}^{\frac{7-p}{2}}}$. We shall
denote total volume of the boundary subsystem by $V=\ell V_{p-1}$. The
entanglement chemical potential $\left(\mu_{E}\right)$ can again be derived
perturbatively from equation (4.5), with $d$ replaced by $\tilde{p}$ in all
expressions.

One can write down a first law of entanglement involving third order
corrections by following the procedure of section 4. The entanglement
temperature at first order is

$T_{E}^{(1)}=\left(\frac{7-p}{\tilde{p}+1}\right)\frac{2b_{0}^{2}}{\pi
b_{1}}\ell\,,$ (6.7)

which for $p=3$ matches with the result (4.8) of AdS5; we note, however, that
$\ell$ in this expression is associated with the rescaled dimension
$\bar{x}_{2}$ from equation (6). The higher order modifications can be
determined but we do not write them explicitly, instead we refer to figure 4
for their effect on $T_{E}$.

(a) $p=2$

(b) $p=4$

Figure 4: The entanglement temperature with and without higher order
corrections of D2 and D4 branes, plot drawn for $z_{h}=100$ and $\beta=0.25$.

##  7 Conclusion

In this work, we studied the holographic entanglement entropy between a strip
sub-region and its complement on the boundary of a boosted AdS black hole. We
assumed the strip-width $(\ell)$ to be very narrow compared with the black
hole parameter $(z_{h})$ and performed a perturbation series analysis of the
HEE up to third order in the dimensionless parameter $\frac{\ell}{z_{h}}$.
Further, we tried to establish a first law of entanglement thermodynamics
including corrections at this order. Due to boost in the direction $(y)$ being
compact, the momentum in $y$-direction is quantized and appears as a conserved
$U(1)$ charge in a lower dimensional theory. We find that it lends important
contribution to the first law of entanglement. We have shown that changes in
the HEE beyond leading order can be absorbed in a redefined entanglement
temperature $\left(T_{E}\right)$ and chemical potential
$\left(\mu_{E}\right)$, thus allowing the canonical first law relationship

$\delta S_{E}^{(3)}=\beta_{E}^{(3)}\left(\delta E-\frac{d-1}{d+1}V\delta
P-\bar{\mu}_{E}^{(2)}\delta N\right),$ (7.1)

to remain true irrespective of the order of perturbation series. The same
trick is shown to work for other holographic gravity duals e.g. AdS plane wave
and non-conformal black Dp-branes as well. While we have only considered strip
entangling region and third order corrections, the algorithm should work for
all entangling regions and can incorporate corrections at any arbitrary order.
At the same time, this way of rewriting the first law of entanglement avoids
inclusion of any foreign quantity other than the conserved charges of the
theory in the law.

The entanglement entropy in a general state is also known to obey a first law
of entanglement with respect to the modular Hamiltonian [10]

$\Delta S\leq\langle\Delta H\rangle.$ (7.2)

In general, $H$ is a non-local operator; so it is curious that $H$ can somehow
be expressed in terms of the subregion thermodynamic variables so that the
more conventional looking eq. (7.1) is satisfied. The resolution could be that
for the special case of interest as ours, the modular Hamiltonian is local;
unfortunately this suspicion cannot be confirmed unless a closed form
expression for $H$ is found for a strip entangling surface in a CFT111There
indeed are cases where $H$ is local and can be expressed as integral over
components of the stress-energy tensor, e.g. the vacuum state of a CFT and
spherical entangling surface [10].. Another possibility is that the modular
Hamiltonian effectively looks local in the small subsystem size limit
$\left(\frac{\ell}{z_{h}}\ll 1\right)$ that we assume in this work and can be
expressed in terms of subregion thermodynamic variables. In fact, it was shown
in [38] that the reduced density matrix always thermalizes as long as the
subsystem size is much less than the total system size; this could as well be
the reason that $H$ becomes proportional to subregion energy, pressure etc.
and a local first law (7.1) is satisfied222We thank the anonymous referee for
their comments in this regard..

While certainly not conclusive, we hope our work helps in deciding whether a
first law like relationship is a generic feature of entanglement in
holographic field theories.

## Acknowledgements

SM acknowledgess the Yukawa Institute for Theoretical Physics at Kyoto
University, where a version of this work was presented during the YITP
workshop YITP-W-20-03 on ‘Strings and Fields 2020’.

##  Appendix A List of integrals

In the following, we list the integrals encountered while solving the
equations (2.7) and (2.8) and write their simplified form in terms of Beta
functions, first let us recall the definition

$\int_{0}^{1}\frac{dy\,y^{nd-1}}{\sqrt{R}}=\frac{B\left(\frac{nd}{2d-2},\frac{1}{2}\right)}{2d-2}\equiv
b_{n-1}\,,$ (A.1)

where $R\equiv 1-y^{2d-2}$.

We start with the integrals occurring while solving for the turning point
$\left(z_{*}\right)$ from equation (2.7)

$\displaystyle\int_{0}^{1}\frac{y^{d-1}\left(1-y^{d}\right)dy}{\sqrt{R}}$
$\displaystyle=\frac{d+1}{d-1}b_{1}-\frac{1}{d-1}b_{0}\,,$ (A.2)
$\displaystyle\int_{0}^{1}\frac{y^{2d-1}\left(1-y^{d}\right)dy}{R^{3/2}}$
$\displaystyle=-\frac{2d+1}{d-1}b_{2}-\frac{d+1}{d-1}b_{1}\,,$ (A.3)
$\displaystyle\int_{0}^{1}\frac{y^{d-1}\left(1-y^{d}\right)^{2}dy}{R^{5/2}}$
$\displaystyle=\frac{2d+1}{(d-1)^{2}}b_{2}+\frac{8}{3}\frac{(d+1)(d-3)}{(2d-2)^{2}}b_{1}-\frac{4}{3}\frac{2d-3}{(2d-2)^{2}}b_{0}\,,$
(A.4)
$\displaystyle\int_{0}^{1}\frac{y^{3d-1}\left(1-y^{d}\right)dy}{R^{3/2}}$
$\displaystyle=\frac{3d+1}{d-1}b_{3}-\frac{2d+1}{d-1}b_{2}\,,$ (A.5)
$\displaystyle\int_{0}^{1}\frac{y^{2d-1}\left(1-y^{d}\right)^{2}dy}{R^{5/2}}$
$\displaystyle=\frac{4}{3}\frac{(3d+1)(d+3)}{(2d-2)^{2}}b_{3}-\frac{2(2d+1)}{(d-1)^{2}}b_{2}-\frac{4}{3}\frac{(d+1)(d-3)}{(2d-2)^{2}}b_{1}\,,$
(A.6)
$\displaystyle\begin{split}\int_{0}^{1}\frac{y^{d-1}\left(1-y^{d}\right)^{3}dy}{R^{7/2}}&=-\frac{8}{15}\frac{(d-5)(d+3)(3d+1)}{(2d-2)^{3}}b_{3}+\frac{24}{5}\frac{(2d-5)(2d+1)}{(2d-2)^{3}}b_{2}\\\
&~{}~{}~{}~{}+\frac{8}{5}\frac{(3d-5)(d-3)(d+1)}{(2d-2)^{3}}b_{1}-\frac{8}{15}\frac{(4d-5)(2d-3)}{(2d-2)^{3}}b_{0}\,.\end{split}$
(A.7)

Next, The integrals that arise when solving for the extremized area are

$\displaystyle\int_{0}^{1}\frac{dy}{y^{d-1}\sqrt{R}}=-\frac{b_{0}}{d-2}\,,$
(A.8)
$\displaystyle\int_{0}^{1}\frac{y^{d}dy}{y^{d-1}\sqrt{R}}=\frac{d+1}{2}b_{1}\,,$
(A.9)
$\displaystyle\int_{0}^{1}\frac{y^{2d-2}(1-y^{d})dy}{y^{d-1}R^{3/2}}=\frac{d+1}{d-1}b_{1}-\frac{1}{d-1}b_{0}\,,$
(A.10)
$\displaystyle\int_{0}^{1}\frac{y^{2d}dy}{y^{d-1}\sqrt{R}}=\frac{2d+1}{d+2}b_{2}\,,$
(A.11)
$\displaystyle\int_{0}^{1}\frac{y^{3d-2}(1-y^{d})dy}{y^{d-1}R^{3/2}}=\frac{2d+1}{d-1}b_{2}-\frac{d+1}{d-1}b_{1}\,,$
(A.12)
$\displaystyle\int_{0}^{1}\frac{y^{4d-4}(1-y^{d})^{2}dy}{y^{d-1}R^{5/2}}=\frac{4}{3}\frac{3d(2d+1)}{(2d-2)^{2}}b_{2}-\frac{8}{3}\frac{2d(d+1)}{(2d-2)^{2}}b_{1}+\frac{4}{3}\frac{d}{(2d-2)^{2}}b_{0}\,,$
(A.13)
$\displaystyle\int_{0}^{1}\frac{dyy^{3d}}{y^{d-1}\sqrt{R}}=\frac{3d+1}{2d+2}b_{3}\,,$
(A.14)
$\displaystyle\int_{0}^{1}\frac{y^{4d-2}(1-y^{d})dy}{y^{d-1}R^{3/2}}=\frac{3d+1}{d-1}b_{3}-\frac{2d+1}{d-1}b_{2}\,,$
(A.15)
$\displaystyle\int_{0}^{1}\frac{y^{5d-4}(1-y^{d})^{2}dy}{y^{d-1}R^{5/2}}=\frac{4}{3}\frac{4d(3d+1)}{(2d-2)^{2}}b_{3}-\frac{8}{3}\frac{3d(2d+1)}{(2d-2)^{2}}b_{2}+\frac{4}{3}\frac{2d(d+1)}{(2d-2)^{2}}b_{1}\,,$
(A.16)
$\begin{split}\int_{0}^{1}\frac{y^{6d-6}(1-y^{d})^{3}dy}{y^{d-1}R^{7/2}}&=-\frac{8}{15}\frac{4d(6d-2)(3d+1)}{(2d-2)^{3}}b_{3}-\frac{8}{5}\frac{3d(5d-2)(2d+1)}{(2d-2)^{3}}b_{2}\\\
&~{}~{}~{}~{}+\frac{8}{5}\frac{2d(4d-2)(d+1)}{(2d-2)^{3}}b_{1}-\frac{8}{15}\frac{d(3d-2)}{(2d-2)^{3}}b_{0}\,.\end{split}$
(A.17)

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# A Forecaster’s Review of Judea Pearl’s Causality: Models, Reasoning and
Inference, Second Edition, 2009

Feng Li<EMAIL_ADDRESS>School of Statistics and Mathematics, Central
University of Finance and Economics, Beijing 102206, China

###### Abstract

With the big popularity and success of Judea Pearl’s original causality book,
this review covers the main topics updated in the second edition in 2009 and
illustrates an easy-to-follow causal inference strategy in a forecast
scenario. It further discusses some potential benefits and challenges for
causal inference with time series forecasting when modeling the
counterfactuals, estimating the uncertainty and incorporating prior knowledge
to estimate causal effects in different forecasting scenarios.

###### keywords:

Causality; Forecasting; Bayesian Network; Machine Learning

††journal: arXiv

##  1 Introduction and contents

With the big popularity and success of Judea Pearl’s original causality book
in 2000, the second edition published in 2009 further updates and clarifies
many details in all ten chapters of the original book. The book has a mixed
targeting audience. Readers with basic probability and statistical knowledge
would be easy to follow the contents. Readers with nonmathematical backgrounds
could skip most of the formulas and still find the concepts and discussion
useful. Moreover, each chapter of the second edition elaborates summaries of
new developments, discussions and annotated bibliographies for in-depth
reading.

The second edition of the book starts with the basics of probabilities and
graphs. The causal model is designed with causal networks as oracles. Then the
second chapter describes the possibility of learning causal relationships from
raw data with a causal discovery framework. The next chapter explores the ways
of inferring such relationships by computing the effect of interventions and
controlling confounding bias. The inference rule is represented by the _do-
calculus_ which gives these causal notions a clear empirical interpretation.
Chapter 4 extends the effect of a fixed constant $x$ in $do(x)$ with a
probabilistic setting of $x$. This is to facilitate the evaluation of the
effect of novel actions and policies. Sequences of time-varying actions could
also be designed to evaluate the effects with a graphical method. Chapter 5
demonstrates graphical models and the logic of intervention can alleviate the
current difficulties in structural equation modeling for causal analysis.
Furthermore, Chapter 6 addresses the difficulties encountered when we attempt
to define and control confounding by using statistical criteria. And Chapter 7
provides a formal analysis of structure-based counterfactuals. Chapter 8
describes how graphical and counterfactual models can combine to elicit causal
information from imperfect experiments. Chapter 9 provides formal semantics
for the probability that one event was a necessary or sufficient cause (or
both) of another event using counterfactual interpretations. Chapter 10 shows
the actual causation that can be formulated in structural model semantics. The
last chapter reflects, elaborates, and discusses many philosophical and
empirical cases with readers. The epilogue is an introduction to the non-
mathematical aspects of causation.

Causality inference has achieved significant advances in the past two decades.
With the increase in available data, causal inference with graphic models has
been adapted to handle large-scale and high-dimensional datasets. It provides
flexible approaches to handling complex data structures and incorporates prior
knowledge for causal inference. Various causal discovery and inference
algorithms have been developed. These algorithms provide automated methods to
infer causal relationships from observational and experimental data. New
methods have been developed to address challenges in inferring causality from
observational data, such as propensity score matching, and regression
discontinuity designs. These techniques aim to overcome confounding biases and
establish causal relationships.

The integration of causality and machine learning has received significant
attention. Methods that combine causal inference with machine learning
techniques, such as causal forest, causal boosting, and causal generative
models, have been proposed to leverage the strengths of both fields and enable
causal reasoning in predictive models. Interestingly, although many machine
learning and deep learning methods have been successfully adopted in the
forecast community, there is still relatively little focus on causality for
time series forecasting. A shred of evidence is that Pearl nicely collected
journal reviews for the causality book in various disciplines 111Available at
http://bayes.cs.ucla.edu/BOOK-2K/book_review.html. which is a nice addition to
the book. It is not surprising that I did not find any related review from a
forecasting perspective.

To review the concepts of causality from a forecasting perspective, we
consider a typical forecasting scenario where we want to forecast the sales
performance of different retail stores over time such as the M5 competition
[5]. A few questions immediately interest us.

  * •

Why should forecasters know about causality?

  * •

How do I know if I have enough data for causal inference?

  * •

What tools can be directly picked up by forecasters?

  * •

Are we still suffering from computational challenges?

##  2 Causality in a forecast scenario

Business planners may not be satisfied by merely tackling forecast accuracy. A
forecaster (or more generally, a prophet) would be expected to provide more
insights on top of forecasts. Although not explicitly declared in Pearl’s
causality book, it is assumed the data are always available. Our
aforementioned scenario required the dataset with information on various
factors such as store size, location, promotional activities, and historical
sales data. Unfortunately, not every time series dataset contained such
detailed information. It is not uncommon that only historical sales data are
available at hand. Historically, forecasters were in difficulty involving
causality analysis. Nowadays, the situation is much better because the
external variables are easily collected. For example, the M5 competition [5]
dataset includes many external variables as features. Machine learning
algorithms could work on such features to improve forecasting performance.

Now we could construct a causal model that represents the relationships among
these variables. We can use directed acyclic graphs (DAGs) to visually
represent the causal structure. For example, we may have arrows from store
size, location, and promotional activities to sales performance, indicating
their potential causal influence.

With the causal model and data in place, we can use the _do-calculus_ to
specify interventions. When we examine the causal effect of store size on
sales performance, we can use the _do-operator_ and write it as “ _do_(store
size)”. This signifies that we are considering a hypothetical scenario where
we actively intervene and change the store size, regardless of its original
causal mechanisms. By applying the _do-calculus_ , we can evaluate the causal
effect of the intervened variable (in this case, store size) on the outcome
variable (sales performance). We can compare the outcomes under the
intervention “ _do_(store size)” to the outcomes without the intervention to
estimate the causal effect. This allows us to isolate the impact of store size
on sales performance while holding other factors constant.

The causal analysis also identifies valid adjustment sets for estimating
causal effects with the back-door criterion. It helps determine which
variables need to be controlled for in order to obtain an unbiased estimate of
the causal effect of an intervention. We could utilize the back-door path from
the causal variable (e.g., store size) to the outcome variable (e.g., sales
performance) that contains an arrow entering the causal variable. In the
retail store example, a potential back-door path could be through the variable
“promotional activities.” The criterion requires that all back-door paths
between the causal variable and the outcome are blocked.

An adjustment set is a set of variables that, when conditioned on (adjusted
for), blocks all back-door paths. To satisfy the back-door criterion, the
adjustment set must contain variables that are ancestors of the causal
variable (store size) in the causal graph but not descendants of the causal
variable or intermediates on the causal path between the causal variable and
the outcome. The adjustment set should not include variables that are affected
by the intervention (store size). Including post-treatment variables in the
adjustment set can introduce bias into the estimation of the causal effect.

In practice, identifying a valid adjustment set may require domain knowledge,
subject expertise, and careful consideration of the causal relationships among
variables. Techniques such as propensity score matching, stratification, or
regression adjustment can be used to implement the adjustment set and estimate
the causal effect while satisfying the back-door criterion.

If we further assess the impact of promotional activity on sales performance.
We can construct a counterfactual scenario where the promotional activity was
not conducted (denoted as “ _do_(promotional activity = false)”). This allows
us to compare the actual sales performance with the hypothetical sales
performance that would have occurred if the promotional activity had not taken
place. If the actual sales performance is higher than the counterfactual
outcome, it suggests a positive causal effect, indicating that the promotional
activity had a beneficial impact on sales. Conversely, if the actual sales
performance is similar or lower than the counterfactual outcome, it indicates
a minimal or negative causal effect.

##  3 Discussion

In general, causality analysis requires more data than traditional forecasting
methods because it aims to identify causal relationships between variables.
While traditional forecasting methods may only require a few variables to make
predictions, causal models typically require more variables and a deeper
understanding of the underlying mechanisms driving the data. This can make it
difficult for forecasters who have limited data or data that is noisy or
incomplete or not accessible due to privacy concerns. A possible direction is
to design a time series data collecting protocol so that time series data can
be properly used for causal analysis.

The book demonstrates the advances of Pearl’s causality framework with
Bayesian graphs, but in general causal inference also involves more complex
computations than traditional forecasting methods. Graphical models and DAGS,
which are commonly used in Pearl’s framework to represent causal
relationships, can be computationally intensive to build and analyze. Causal
inference algorithms, such as the _do-calculus_ and backdoor criterion, can
also be complex to apply and may require specialized software. This could
still be a challenge for forecasters without a strong background in statistics
and computer science.

Any causality framework relies on several assumptions, such as the assumption
of faithfulness and the requirement for no unmeasured confounding variables in
Pearl’s causality. Violations of these assumptions can lead to inaccurate
forecasts and may be difficult for forecasters to identify and address.
Additionally, the framework may not be directly applicable to all forecasting
scenarios, particularly when more machine learning models are getting
attracted to the forecasting domain. See [6] for possible solutions.

Computer software is also essential for causal analysis. I found an open-
source Python library “Causalnex” [1] a good companion. It aims for building,
testing, and applying Bayesian networks for causal modeling and inference. The
library includes tools for time series modeling, with support for
incorporating external variables, handling missing data, and performing
counterfactual analysis. Causalnex also provides visualization tools for
exploring causal relationships and model outputs.

Overall, Pearl’s causality framework offers the ladder for time series
forecasting and Bayesian graphs that allows for the explicit modeling of
counterfactuals, uncertainty estimation, and incorporating prior knowledge to
estimate causal effects in different scenarios. There exist issues including
the definition of identifiability and the choice of priors in both low and
high dimensional regimes [3]. A plausible approach is to consider the causal
machine learning methods [2, 4]. Furthermore, there are many avenues for
causal inference in forecasting with generative models, given the wide
applicability of simulations in forecasting models. Forecasters must carefully
consider the data requirements, computational complexity, need for domain
expertise, interpretability, and assumptions of the framework before adopting
it. Addressing these challenges may require specialized expertise,
computational resources, and careful consideration of the specific forecasting
problem at hand.

## References

  * [1] Paul Beaumont et al.  “CausalNex: A Toolkit for Causal Reasoning with Bayesian Networks”, 2021  URL: https://github.com/quantumblacklabs/causalnex
  * [2] Jean Kaddour et al.  “Causal Machine Learning: A Survey and Open Problems”, 2022  DOI: 10.48550/arXiv.2206.15475
  * [3] Fan Li, Peng Ding and Fabrizia Mealli  “Bayesian Causal Inference: A Critical Review”  In _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 381.2247 Royal Society, 2023, pp. 20220153  DOI: 10.1098/rsta.2022.0153
  * [4] Yuejiang Liu et al.  “Towards Robust and Adaptive Motion Forecasting: A Causal Representation Perspective”  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 17081–17092  URL: https://openaccess.thecvf.com/content/CVPR2022/html/Liu_Towards_Robust_and_Adaptive_Motion_Forecasting_A_Causal_Representation_Perspective_CVPR_2022_paper.html
  * [5] Spyros Makridakis, Evangelos Spiliotis and Vassilios Assimakopoulos  “M5 Accuracy Competition: Results, Findings, and Conclusions”  In _International Journal of Forecasting_ 38.4, 2022, pp. 1346–1364  DOI: 10.1016/j.ijforecast.2021.11.013
  * [6] Bernhard Schölkopf  “Causality for Machine Learning”  In _Probabilistic and Causal Inference: The Works of Judea Pearl_ 36 New York, USA: Association for Computing Machinery, 2022, pp. 765–804  URL: https://doi.org/10.1145/3501714.3501755


# INVESTIGATION OF NON-EQUILIBRIUM IONIZATION PLASMA DURING A GIANT FLARE OF
UX ARIETIS TRIGGERED WITH MAXI AND OBSERVED WITH NICER

Miki Kurihara Department of Astronomy, Graduate School of Science, The
University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Japan Aerospace Exploration
Agency, Institute of Space and Astronautical Science,
3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan  Wataru Buz
Iwakiri Chiba University, Inage-ku, Chiba, Chiba, 263-8522 Japan  Masahiro
Tsujimoto Japan Aerospace Exploration Agency, Institute of Space and
Astronautical Science,
3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan  Ken Ebisawa
Department of Astronomy, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Japan Aerospace Exploration
Agency, Institute of Space and Astronautical Science,
3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan  Shin Toriumi
Japan Aerospace Exploration Agency, Institute of Space and Astronautical
Science,
3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan  Shinsuke
Imada Department of Earth and Planetary Science, Graduate School of Science,
The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Yohko Tsuboi Department of
Physics, Faculty of Science and Engineering, Chuo University,
1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan  Kazuki Usui  Department of
Astronomy, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Kavli Institute for the Physics
and Mathematics of the Universe,
5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan  Keith C. Gendreau
Astrophysics Science Division, NASA’s Goddard Space Flight Center,
Greenbelt, MD 20771, USA  Zaven Arzoumanian  Astrophysics Science Division,
NASA’s Goddard Space Flight Center,
Greenbelt, MD 20771, USA

###### Abstract

We detected a giant X-ray flare from the RS-CVn type binary star UX Ari using
MAXI on 2020 August 17 and started a series of NICER observations 89 minutes
later. For a week, the entire duration of the flare was covered with 32
snapshot observations including the rising phase. The X-ray luminosity reached
2$\times$1033 erg s-1 and the entire energy release was $\sim 10^{38}$ erg in
the 0.5–8.0 keV band. X-ray spectra characterized by continuum emission with
lines of Fe XXV He$\alpha$ and Fe XXVI Ly$\alpha$ were obtained. We found that
the temperature peaks before that of the flux, which suggests that the period
of plasma formation in the magnetic flare loop was captured. Using the
continuum information (temperature, flux, and their delay time), we estimated
the flare loop size to be $\sim 3\times 10^{11}$ cm and the peak electron
density to be $\sim 4\times 10^{10}$ cm-3. Furthermore, using the line ratio
of Fe XXV and Fe XXVI, we investigated any potential indications of deviation
from collisional ionization equilibrium (CIE). The X-ray spectra were
consistent with CIE plasma throughout the flare, but the possibility of an
ionizing plasma away from CIE was not rejected in the flux rising phase.

††software:  HEAsoft (HEASARC, 2014), Xspec (Arnaud, 1996), AtomDB (Smith et
al., 2001; Foster et al., 2017)

##  1 Introduction

Solar and stellar flares are considered to be caused by the same physical
mechanisms but with different scales (e.g.; Fletcher et al. 2011; Benz & Güdel
2010). The accumulated magnetic energy is impulsively released through
magnetic reconnection and plasma instability (Priest & Forbes, 2002; Shibata &
Magara, 2011). In the standard model for the eruptive flares, known as the
CSHKP model (Carmichael, 1964; Sturrock, 1966; Hirayama, 1974; Kopp & Pneuman,
1976), the released energy is partially converted into the kinetic energy of
charged particles at the reconnection site, which then bombard the flare loop
top and cause an impulsive increase in temperature. The particles move
downward along the loop and collide with the dense chromosphere. The
evaporated matter fills the loop, causing the flux to peak. After the initial
rise, the temperature decreases conductively and radiatively.

Because of this dynamic process, it is expected and often observed that the
temperature peak precedes the flux peak (e.g.; Maggio et al. 2000; Pillitteri
et al. 2022; Stelzer et al. 2022). This rising phase is the time when physical
constraints of flares can be obtained independently from and more stringently
than the decay phase. Reale (2007) proposed a model to describe the behavior
of X-ray light curves during flares and derived analytical formulae relating
observables with physical properties of flares and applied them to
observations. In X-ray observations of stellar flares, the observables are
often limited to values obtained from the continuum emission (such as
temperature and volume emission measure; hereafter called “continuum
observables”), but those from the line emission (“line observables”) will
carry independent information about the flare dynamics. However, this has been
hampered by the lack of spectral resolution or photon statistics (Favata et
al., 2000; Maggio et al., 2000; Stelzer et al., 2002).

Of particular interest are line ratios that deviate from collisional
ionization equilibrium (CIE), as it is an expected consequence of the dynamic
nature of flares, thus providing new keys to understanding flare physics
(Imada et al., 2011; Reale & Orlando, 2008; Bradshaw & Mason, 2003). There are
some observational claims (Kawate et al., 2016; Imada, 2021) of non-
equilibrium ionization (NEI) plasma in solar flares, but such observations are
limited despite the continuous and intensive observations of the Sun. The
largest obstacle is the short duration of NEI conditions expected to last only
for $\mathcal{O}(10^{2}~{}\mathrm{s})$.

Stellar flare observations could be more advantageous. The electron
temperature is $\sim 10$ times larger than the Sun during flares (Feldman et
al., 1995), which makes the equilibrium time scales longer for the same
electron density, as more charged ions dominate the population and the
ionization/recombination rate coefficient decreases. In order to exploit this
advantage, we need X-ray observations from the rising phase of stellar flares
using a telescope with a large collecting area and the capability to
distinguish major lines of different charges.

We break through this limitation by combining two X-ray instruments onboard
the International Space Station (ISS): the Monitor of All-sky X-ray Image
(MAXI; Matsuoka et al. 2009) and the Neutron star Interior Composition
Explorer (NICER; Gendreau et al. 2016). MAXI updates all-sky X-ray images
every 92 minutes, which has led to 167 detections of stellar flares from 30
objects since 2009\. NICER has an ability to conduct fast maneuvers and
collect photons with a larger effective area and a better spectral resolution
than MAXI.

We have developed two systems, in which MAXI detects transient events and
NICER responds in the shortest achievable time. One is OHMAN (Orbiting High-
energy Monitor Alert Network), in which the triggering from MAXI to NICER is
automated within the ISS. OHMAN has been in operation since 2023 but is
effective only for very bright sources. The other is MANGA (MAXI And NICER
Ground Alert) since 2017, in which a human decision is inserted based on the
MAXI data downlinked to the ground. MANGA still makes human-triggered
observations faster than any previous approaches. In our study, we use MANGA
for stellar flare observations (e.g., Iwakiri et al. 2018; Sasaki et al.
2021).

We present here the result of the MANGA observation made with the shortest
response time to date for stellar flares. We detected a giant flare from UX
Ari with MAXI and initiated NICER observations well before the flux peak. UX
Ari is an RS-CVn type binary system consisting of a K0 IV and G5 V star with a
6.4 day orbital period (Hummel et al., 2017). The source is known to cause
occasional flares in X-rays (Franciosini et al., 2001; Tsuboi et al., 2016;
Güdel et al., 1999; Tsuru et al., 1989). The peak flare luminosity of the
present data reaches 2$\times$1033 erg s-1 in the 0.5–8.0 keV band, which is
$\sim$10 times larger than any flares observed with X-ray pointing
observations of this source. The goal of this study is to examine the plasma
development throughout the flare and investigate any hints of the off-CIE
plasma using the X-ray spectra. Throughout the paper, we use AtomDB (Smith et
al., 2001; Foster et al., 2017) for plasma calculations. The quoted
uncertainties are for 90% statistical error.

##  2 Observations

###  2.1 Instrument

NICER is a payload onboard the ISS equipped with an X-ray Timing Instrument
(XTI). XTI is an assembly of 56 sets of X-ray concentrators (Okajima et al.,
2016) and silicon drift detectors (Prigozhin et al., 2016), of which 52 are in
routine use. The energy range is 0.2–12.0 keV. The effective area and the
energy resolution in FWHM at 6 keV are $\sim$ 600 cm2 and 137 eV,
respectively.

NICER is useful for wide astrophysical applications. In particular, the
operational flexibility, the large effective area, and the fast detector
readout allow spectroscopic observations of eruptive events uncompromised by
limitations of photon statistics and detector dynamic range. These features
make NICER particularly well suited for stellar flare observations (Namekata
et al., 2020; Sasaki et al., 2021; Hamaguchi et al., 2023).

###  2.2 Observation and data reduction

We captured a flux increase of UX Ari based on the MAXI Nova-Alert system
(Negoro et al., 2016) recorded on 2020 August 17 11:51:54 (UT). Using the
MANGA system, the first NICER observation was executed 89 minutes later. NICER
continued observations for a week (ObsIDs: 3100380101–3100380106) until the
X-ray count rate returned to the pre-flare level. NICER visited the source 32
times, which we refer to as snapshot 0 to 31. The on-source exposure ranged
from 101 to 1286 s with an average of 752 s and the photon counts ranged from
$2.2\times 10^{4}$ to $2.0\times 10^{6}$ with an average of $4.0\times 10^{5}$
counts. Thanks to the large effective area of NICER, it can accumulate
sufficient counts even with short exposure times.

We applied level 2 processing using the nicerl2 tasks to incorporate the
latest calibration and event screening and produced level 3 products using the
nicerl3 tasks in HEASoft version 6.32.1 (HEASARC, 2014). The level 3 products
include the telescope and detector response files, binned source spectrum, and
background spectrum based on the SCORPEON model, which we generated and used
for each snapshot.

##  3 Analysis

###  3.1 Light curve and spectra

Fig. 1 (a) shows the X-ray light curve of the data set. Snapshot 0 captures
the rising part of the flare, 1 captures the flux peak, and the rest shows the
decaying part. There is a slight re-brightening during 14–16 and 26–29. The
peak luminosity and the e-folding time of the NICER light curve are $2\times
10^{33}$ erg s-1 and $2.4\times 10^{4}$ s, respectively, deriving the total
released energy in 0.5–12 keV to be $\sim 10^{38}$ erg.

Figure 1: (a) NICER (black) and MAXI (grey) light curves. MAXI count rate is
scaled to match with NICER. The labels 0–31 are given for each NICER snapshot.
(b) NICER X-ray spectra with different colors for different snapshots. The
intensity of each spectrum is offset incrementally by 0.5 dex for clarity.

Fig. 1 (b) shows the X-ray spectra for the first ten snapshots (0–9) and two
selected later ones(19 and 29). The spectra are characterized by continuum
emission with lines. The most conspicuous feature is found at the Fe K band,
in which Fe XXV He$\alpha$ and Fe XXVI Ly$\alpha$ emission lines are
identified at 6.7 and 7.0 keV.

The development of the line pair traces the decreasing plasma temperature. The
intensity ratio of Fe XXVI Ly$\alpha$ against Fe XXV He$\alpha$ monotonically
decreases with time, suggesting that the Fe25+ (H-like) population is taken
over by Fe24+ (He-like) in the charge state distribution. At the end of the
flare (snapshot 29), the Fe XXV He$\alpha$ feature almost disappears and the
Fe L features at $\sim$1 keV increases, suggesting that Fe24+ is taken over by
Fe23+ (Li-like) and less charged ions.

###  3.2 Spectral modeling

####  3.2.1 Phenomenological model

We first tried a phenomenological model to explore the characteristics of the
data set. We used the first eight snapshots with sufficient statistics for
fitting in the energy range of interest (5–10 keV). We assessed the level of
the persistent emission using the last part of the datasets (snapshots 24–31
except for 27–29 with a possible rebrightening) when the flare seemed settled.
Its contribution to the first part of the flare (snapshots 0–7) is smaller
than a fraction of $\sim$ 0.1 in the energy band of interest. We do not use
time-sliced spectra within each snapshot, as we found that they exhibit no
significant change between the former and the latter halves.

The model is composed of two components: a Bremsstrahlung model for the
continuum and Gaussian models for the lines. A total of nine Gaussian lines
are added: three for Fe XXVI Lyman series (Ly$\alpha$, Ly$\beta$, and
Ly$\gamma$), three for Fe XXV $n=2\rightarrow 1$ series (He$\alpha$,
He$\beta$, and He$\gamma$), and He$\alpha$ lines of other He-like ions of Cr
XXIII, Mn XXIV, and Ni XXVII.

The fitting was conducted simultaneously for the two components. The free
parameters for the Bremsstrahlung component are the electron temperature
($T_{\mathrm{e}}$) and the emission measure ($EM$). For the Gaussian
components, we fitted their normalization individually and the energy shift
collectively within $\pm$1% around the center. The line widths were fixed to
null. For the Lyman series, the energy averaged between ($n$p) ${}^{2}P_{1/2}$
$\rightarrow$ (1s) ${}^{2}S_{1/2}$ and ($n$p) ${}^{2}P_{3/2}$ $\rightarrow$
(1s) ${}^{2}S_{1/2}$ weighted by their statistical weights is used as the line
center, where $n\in\left\\{2,3,4\right\\}$. For the He series, the energy for
($n$p1s) ${}^{1}P_{1}$ $\rightarrow$ (1s)2 ${}^{1}S_{0}$ is used among several
fine-structure lines that spread over $<$1% of the line energy for Cr, Mn, Fe,
and Ni. All fits were statistically acceptable. Fig. 2 depicts the results of
the fitting.

####  3.2.2 Physical models

We next applied three physical models: CIE, ionizing, and recombining plasmas.
They are respectively implemented as apec, nei, and rnei models in the xspec
spectral fitting package (Arnaud, 1996). The latter two are off-CIE plasmas in
opposite directions. The ionizing plasma model assumes the initial condition
in which ions are neutral and electrons have a temperature higher than the
neutral ions. The degree of Coulomb relaxation is parameterized by the
ionization parameter $\tau=\int_{0}^{t}n_{\mathrm{e}}(t^{\prime})dt^{\prime}$,
in which $n_{\mathrm{e}}(t^{\prime})$ is the time-varying electron density and
$t$ is the time from the initial state. The recombining plasma model assumes
the opposite initial condition that ions are ionized for a given initial
temperature and electrons have a lower temperature. Their relaxation is also
parameterized by $\tau$. Recombining plasma conditions might be observed if
the cooling via thermal conduction occurs more rapidly than charge state
equilibration after impulsive heating of the plasma.

The free parameters for CIE plasma model are the electron temperature
($T_{\mathrm{e}}$), abundance ($Z_{\odot}$) relative to the cosmic value
(Anders & Grevesse, 1989), and emission measure ($EM$). For the ionizing and
recombining plasma, the ionization parameter $\tau$ is also added. The initial
temperature of the recombining plasma was fixed to the value obtained in the
phenomenological fitting of snapshot 0. All fits were statistically
acceptable. The results of the fitting are illustrated in Fig. 2.

Figure 2: Spectra and best-fit phenomenological model (upper-most panel) and
the residuals of the fit (other panels) for the phenomenological model
(black), CIE plasma model (purple), ionizing plasma model (blue), and
recombining plasma model (red) for snapshots 0–7.

##  4 Discussion

###  4.1 Continuum observables

We first explored the continuum observables following the previous work by
Reale (2007). We used the result of the phenomenological spectral fitting (§
3.2.1). Fig. 3 (a and b) shows the development of the best-fit values of
$T_{\mathrm{e}}$ and $EM$. The peak of $EM$ (snapshot 1) is delayed from that
of $T_{\mathrm{e}}$ (snapshot 0) in time for $\sim 5\times 10^{3}$ s, as in
Fig. 1 of Reale (2007). We assume that the density peak coincides with the
$EM$ peak in snapshot 1. It is notable that $T_{\mathrm{e}}$ and $EM$ do not
monotonically decrease. They stagnate or increase around snapshots 3–4, which
may suggest repeated heating.

Figure 3: Evolution of best-fit values for $T_{\mathrm{e}}$, $T_{\mathrm{Z}}$,
$EM$, and line intensities of Fe XXV He$\alpha$ and Fe XXVI Ly$\alpha$ in the
phenomenological model. The discrepancy between the electron temperature
$T_{\mathrm{e}}$ and the ionization temperature $T_{\mathrm{Z}}$ suggests an
off-CIE plasma.

Based on the estimated delay time ($\Delta t_{\mathrm{lag}}\sim 5\times
10^{3}$ s), we can infer the flare loop half-length $L$ and the peak density
$n_{\rm{M}}$ by using Eqn. (13) and (9) in Reale (2007):

$\left(\frac{L}{10^{9}~{}\mathrm{cm}}\right)=2.5\frac{\psi}{\ln{\psi}}\left(\frac{T_{0}}{10^{7}~{}\mathrm{K}}\right)^{\frac{1}{2}}\left(\frac{\Delta
t_{\mathrm{lag}}}{10^{3}~{}\mathrm{s}}\right)$ (1)
$\left(\frac{n_{\mathrm{M}}}{10^{10}~{}\mathrm{cm^{-3}}}\right)=13\left(\frac{T_{\mathrm{M}}}{10^{7}~{}\mathrm{K}}\right)^{2}\left(\frac{L}{10^{9}~{}\mathrm{cm}}\right)^{-1}$
(2)

where $\psi=\frac{T_{0}}{T_{\mathrm{M}}}$ is the ratio of the maximum
temperature $T_{0}$ and temperature at the density peak $T_{\mathrm{M}}$, and
$\Delta t_{\mathrm{lag}}$ is the time lag between $T_{\mathrm{e}}$ and $EM$.
These relations are derived from the assumption that the density reaches its
maximum when the radiative cooling rate equals the conductive cooling rate.
$L$ is estimated to be $\sim 3\times 10^{11}$ cm ($\sim$4 $R_{\odot}$), which
is much larger than the typical value for solar flares ($\sim 10^{9}$ cm) and
even comparable to the scales of the binary system; the radii of UX Ari Aa and
Ab are 5.6 $R_{\odot}$ and 1.6 $R_{\odot}$, respectively, and the binary
separation is $\sim 0.1$ AU$\sim 20~{}R_{\odot}$ (Hummel et al., 2017). The
density maximum gives a fairly similar value $n_{\mathrm{M}}\sim 4\times
10^{10}$ cm-3 to solar coronal plasma density in flaring regions (Aschwanden &
Benz, 1997). Since the plasma is confined by the magnetic flux tube, the
magnetic field strength ($B$) can be estimated from the pressure balance.
Assuming the plasma to be an ideal gas, $B>\sqrt{16\pi
n_{\mathrm{M}}T_{\mathrm{M}}}\sim 200$ G. By additionally using the measured
$EM$ value, the aspect ratio between the loop length and the loop cross
section radius is estimated to be $\sim$ 0.6. These values are of the same
order as those in other giant stellar flares (Favata & Schmitt, 1999; Maggio
et al., 2000; Franciosini et al., 2001; Reale et al., 2004; Reale, 2007;
Sasaki et al., 2021; Pillitteri et al., 2022).

###  4.2 Line observables

####  4.2.1 Phenomenological model

We next examined the line observables. Using the phenomenological spectral
fitting (§ 3.2.1), the development of the line intensities of Fe XXV
He$\alpha$ and Fe XXVI Ly$\alpha$ is shown in Fig. 3 (c). Their intensities
are well-constrained thanks to the photon-rich spectra. In order to examine
any deviation from CIE, the line ratio of Fe Ly$\alpha$ and He$\alpha$ was
derived for each snapshot. The Ly$\alpha$ against He$\alpha$ is larger in
snapshot 1 than snapshot 2 despite a similar $T_{\mathrm{e}}$ value. Here, we
used the line ratio of the same elements to avoid systematics caused by
elemental abundance.

We calculated the expected line ratio as a function of $T_{\mathrm{e}}$ for
CIE, ionizing, and recombining plasmas (Fig. 4). The starting condition for
the ionizing/recombining plasma is that all Fe ions are neutral/ionized. The
line intensities were extracted from the synthesized X-ray spectra calculated
for $\tau=10^{12}-10^{13}$ cm-3 s with geometrical spacing. The Ly$\alpha$ and
He$\alpha$ line complexes that are unresolved with NICER were combined.

Based on the calculated results, we plotted the observed results. Snapshot 0
significantly deviates from CIE in the ionizing direction. It approaches CIE
for snapshot 1, but deviates again in the ionizing direction for snapshot 3.
The later snapshots are consistent with CIE within uncertainties. The
ionization temperature ($T_{\mathrm{Z}}$) derived from the line ratio assuming
CIE is compared with the electron temperature from the continuum emission
($T_{\mathrm{e}}$) in Fig. 3 (a). The two temperatures are in disagreement in
snapshots 1 and 3.

Figure 4: Line ratio of Fe XXVI Ly$\alpha$ over Fe XXV He$\alpha$. Different
cases are shown in different line styles: CIE (solid), recombining (dashed),
and ionizing (dotted) plasmas of varying $\log_{10}{\tau}$ (cm-3 s) from 12.0
to 13.0 with a step of 0.2. Data points are plotted with red error bars and
the snapshot number.

####  4.2.2 Physical model

Fig. 5 shows the development of the best-fit values of the parameters of the
three physical models. All the parameters change similarly for the three
models, suggesting that the off-CIE features in the spectra are insignificant.
The best-fit $\tau$ values for the two off-CIE plasma models are $\gtrsim
1\times 10^{13}$ cm-3 s, which is sufficiently large to reach equilibrium.

Figure 5: Evolution of best-fit values for $T_{\mathrm{e}}$, $EM$, and
abundance of the three physical models, and $\tau$ in the physical models. For
$\tau$, lower limits are illustrated as triangle symbols.

The phenomenological model suggests an off-CIE plasma in the ionizing
direction in snapshots 0 and 3 (Fig. 4), whereas the physical ionizing plasma
model suggests CIE condition. This discrepancy arises from the different
assessments of $T_{\mathrm{e}}$. In the former,
$T_{\mathrm{e}}=12.5\pm^{1.7}_{1.2}$ and $8.7\pm^{0.9}_{0.8}$ keV for
snapshots 0 and 3. In the latter, $T_{\mathrm{e}}=10.8\pm^{0.5}_{0.4}$ and
$~{}7.2\pm^{0.3}_{0.4}$ keV. The physical models determine $T_{\mathrm{e}}$
using not only continuum information but also line information. Because the
energy range of the current dataset does not cover the expected cut-off of the
Bremsstrahlung continuum, the $T_{\mathrm{e}}$ constraint by the continuum
emission alone is loose. As a result, two fitting solutions of high
$T_{\mathrm{e}}$/low $\tau$ or low $T_{\mathrm{e}}$/high $\tau$ are allowed.

In order to investigate the coupling nature of the two parameters, we made a
contour plot in the ($T_{\mathrm{e}}$, $\tau$) space for snapshots 0 and 3
using the ionizing plasma model (Fig. 6). The contours indicate that either
solution is acceptable: CIE plasma of $\tau\gtrsim 5\times 10^{12}$ cm-3 s
with $T_{\mathrm{e}}$ more stringently constrained than $\tau$ (thus
vertically long part of the contours) and the ionizing plasma of $\tau\sim
1\times 10^{12}$ cm-3 s with $\tau$ more stringently constrained than
$T_{\mathrm{e}}$ (thus horizontally long part of the contours). The best-fit
parameter pair of the physical ionizing plasma model resides in the former,
whereas that of the phenomenological model resides in the latter. We conclude
that the present spectra do not allow us to distinguish between these two
possibilities.

Figure 6: Contour plots of confidence intervals on ($T_{\mathrm{e}}$, $\tau$)
for snapshots 0 and 3. Dashed, dashed-and-dotted, and solid lines correspond
to 68%, 90%, and 99%. The best-fit values for the phenomenological model and
the physical ionizing plasma model are shown with markers. The $\tau$ of the
phenomenological model is derived from Fig. 4.

The development of the light curve favors CIE solution. Assume that there is
no continued heating in the rising phase and the density increases linearly in
time as

$n(t)=n_{0}(t-t_{\mathrm{start}}),$ (3)

where $t_{\mathrm{start}}$ is the flare start time that is assumed to be the
MAXI trigger time (Fig. 1 a). The normalization $n_{0}$ satisfies
$n(t_{1})=n_{\mathrm{M}}=4\times 10^{10}$ cm-3, in which $t_{1}$ is the time
of the snapshot 1. Then, the ionization parameter at snapshot 0 ($t_{0}$) is
$\tau_{0}=\int_{t_{\mathrm{start}}}^{t_{0}}n(t)dt\sim 10^{14}$ cm-3 s, which
is large enough to reach an ionization equilibrium.

##  5 Conclusion

Based on the MAXI trigger, we observed a giant X-ray flare from UX Ari using
NICER. Thanks to the rapid trigger response made possible by the MANGA system,
we successfully captured the rising phase of the flux. Photon-rich spectra
were obtained throughout the flare covered by 32 snapshot observations with
NICER.

Using the continuum information (temperature, flux, and their peak delays), we
constrained the flare loop size $\sim 3\times 10^{11}$ cm and the peak
electron density $\sim 4\times 10^{10}$ cm-3, which are consistent with other
giant stellar flares. Furthermore, using the line information (ratio of the Fe
XXV He$\alpha$ and Fe XXVI Ly$\alpha$ lines), we examined any hints of off-CIE
plasma. We fitted the 5–10 keV spectra with a phenomenological model and three
physical models of CIE, ionizing, and recombining plasmas. The X-ray spectra
are consistent with CIE plasma throughout the flare, but the ionizing plasma
away from CIE also explains the spectra in the flux rising phase.

The present study demonstrated the possibility of conducting X-ray
spectroscopy studies of the rising part of stellar flares without relying on
luck to capture one during long exposures. In the near future, we expect that
two improvements can be made. One is the X-ray observation beyond 10 keV using
NuSTAR (Vievering, 2019) to constrain the electron temperature
$T_{\mathrm{e}}$ more precisely. The other is high-resolution spectroscopy
using XRISM. The microcalorimeter will resolve the line complex with more
constraining power for off-CIE signatures.

## Acknowledgments

We appreciate comments by Hiroya Yamaguchi at ISAS. This research used data
and/or software provided by the High Energy Astrophysics Science Archive
Research Center (HEASARC), which is a service of the Astrophysics Science
Division at NASA/GSFC. This work was supported by JSPS KAKENHI Grant Nos.
JP20KK0072 (PI: S. Toriumi), JP21H01124 (PI: T. Yokoyama), JP21H04492 (PI: K.
Kusano), JP17K05392 (PI: Y. Tsuboi). Also, Y. T. acknowledges the support by
Chuo University Grant for Special Research.

NICER (Gendreau et al., 2016), MAXI (Matsuoka et al., 2009)

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