11institutetext: Institute for Theoretical Physics and Center for Extreme
Matter and Emergent Phenomena
Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The
Netherlands22institutetext: Institute Lorentz for Theoretical Physics, Leiden
University
P.O. Box 9506, Leiden 2300RA, The Netherlands33institutetext: Institut für
Theoretische Physik, Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt
am Main, Germany44institutetext: Institute for Theoretical Physics, TU Wien,
Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria55institutetext: Guangdong
Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China66institutetext:
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China77institutetext: Institute of Theoretical Physics, Jagiellonian
University, S. Lojasiewicza 11, PL 30-348 Krakow, Poland

# QNEC2 in deformed holographic CFTs

C. Ecker 4 D. Grumiller 5,6,7 H. Soltanpanahi 4 and P. Stanzer<EMAIL_ADDRESS>frankfurt.de<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We use the quantum null energy condition in strongly coupled two-dimensional
field theories (QNEC2) as diagnostic tool to study a variety of phase
structures, including crossover, second and first order phase transitions. We
find a universal QNEC2 constraint for first order phase transitions with
kinked entanglement entropy and discuss in general the relation between the
QNEC2-inequality and monotonicity of the Casini–Huerta $c$-function. We then
focus on a specific example, the holographic dual of which is modelled by
three-dimensional Einstein gravity plus a massive scalar field with one free
parameter in the self-interaction potential. We study translation invariant
stationary states dual to domain walls and black branes. Depending on the
value of the free parameter we find crossover, second and first order phase
transitions between such states, and the $c$-function either flows to zero or
to a finite value in the infrared. We present evidence that evaluating QNEC2
for ground state solutions allows to predict the existence of phase
transitions at finite temperature.

††preprint: TUW–20–01

##  Introduction

While the main job description of holography is to teach us how quantum
gravity works, its favorite pastime is to enlighten us about strongly coupled
quantum field theories (QFTs). Starting with the seminal work by Ryu and
Takayanagi (RT) Ryu:2006bv  on the holographic computation of entanglement
entropy (EE), the last 1.5 decades have led to a cross-fertilization between
quantum information and holography, yielding numerous new tools and insights,
see e.g. Harlow:2014yka ; Hayden:2016cfa ; VanRaamsdonk:2016exw ;
Almheiri:2020cfm  and refs. therein.

In the present work we focus on a particular such tool, namely the Quantum
Null Energy Condition (QNEC) Bousso:2015mna  with the aim of using it to
diagnose strongly coupled QFTs, their phase structure, and their ultraviolet
(UV) and infrared (IR) behavior. We always assume that the QFT under
consideration is two-dimensional, has a conformal field theory (CFT2) fixed
point in the UV and a holographic description in terms of a three-dimensional
gravity theory with asymptotically anti-de Sitter (AdS3) solutions.

QNEC locally constrains the expectation value of null projections of the
energy-momentum tensor $\langle T_{kk}\rangle$ in terms of lightlike
variations (denoted by prime) of EE $S$

$\mathrm{QNEC:}\quad 2\pi\,\langle
T_{kk}\rangle\geq\frac{1}{\sqrt{h}}\,S^{\prime\prime}\qquad\qquad\langle
T_{kk}\rangle:=\langle T_{\mu\nu}k^{\mu}k^{\nu}\rangle\qquad\forall k^{2}=0$
(1)

where $h$ is the determinant of the induced metric at the boundary of the
entangling region (for details see Bousso:2015mna ). Proofs of QNEC exist for
free bosonic Bousso:2015wca  and fermionic Malik:2019dpg  theories, for CFTs
with holographic duals Koeller:2015qmn  and for interacting QFTs in spacetime
dimension $d>2$ Balakrishnan:2017bjg ; Ceyhan:2018zfg . Recently the notion of
QNEC has been extended also to non-relativistic theories
PhysRevLett.123.121602 .

In CFT2 QNEC takes the stronger form

$\mathrm{QNEC_{2}:}\quad 2\pi\,\langle T_{kk}\rangle\geq
S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}$ (2)

where $c$ is the central charge of the CFT2 and the additional positive
contribution $\frac{6}{c}(S^{\prime})^{2}$ follows from the conformal
transformation properties of EE Wall:2011kb . The lightlike variations denoted
by prime are defined as follows. At the spacetime point where the left hand
side of the QNEC2 inequality is evaluated one of the two endpoints of the
entangling interval is anchored, while the second one can be chosen
arbitrarily. The first endpoint is deformed into the null direction $k^{\mu}$,
parametrized by an infinitesimal parameter $\lambda$. EE then depends on the
deformation parameter $\lambda$ and prime means derivative with respect to it.
We shall be more explicit about this construction and how to evaluate QNEC2 in
our review in section 2. For sake of brevity, when there is no chance of
confusion, sometimes we refer to the expression on the right hand side of the
inequality (2) and sometimes to the inequality itself as QNEC2.

The first part of our discussion will be general, where we review some known
properties of QNEC2 and address also some novel ones, related to kinked EE,
first order phase transitions and monotonicity of the Casini–Huerta
$c$-function. Later on we focus on bulk matter in AdS3 described by a massive
scalar field $\phi$, which on the field theory side corresponds to a
deformation of the CFT2 by a scalar operator $\mathcal{O}_{\phi}$. For
concrete examples we assume a family of scalar field potentials with a single
free parameter, similar to those studied in Gubser:2008ny ; Janik:2015iry ;
Attems:2016ugt ; Janik:2016btb ; Attems:2017ezz ; Janik:2017ykj , which lead
to a non-trivial phase structure. The study of this phase structure and in
particular of crossovers, second and first order phase transitions using QNEC2
and EE is one of the main goals of our work.

The paper is structured as follows: in section 2 we summarize main aspects of
QNEC2, holographic EE and AdS3/CFT2, in particular a convexity constraint on
kinked EE and the relation between QNEC2 and monotonicity of the Casini–Huerta
$c$-function; in section 3 we review the holographic model, its formulation
through a superpotential, its holographic renormalization, domain wall and
black brane solutions; in section 4 we present results for the thermodynamic
quantities, including free energy, entropy density and the speed of sound for
different choices of the potential leading to different kinds of phase
transitions; in section 5 we present the results for the holographic EE and
the Casini–Huerta $c$-function, perturbatively for small and large entangling
intervals and numerically in between; in section 6 we present results for
QNEC2, in particular for the ground state, where we see evidence for first and
second order phase transitions at finite temperature; in section 7 we conclude
with a summary and an outlook to generalizations.

##  QNEC2 and AdS3/CFT2

In this section we review salient features of QNEC2, including holographic
aspects, and present also novel features. In section 2.1 we give a lightning
review of AdS3/CFT2. In section 2.2 we display a uniformized result for
(holographic) EE valid for all states dual to vacuum solutions of the Einstein
equations. In section 2.3 we summarize the proof that QNEC2 saturates for all
such states. In section 2.4 we recall relevant features when QNEC2 does not
saturate, in particular the half-saturation effect for quenches. In section
2.5 we present a shortcut to QNEC2 for boost invariant states. In section 2.6
we show how to determine efficiently the QNEC2 combination of EE variations
for general states. In section 2.7 we demonstrate that QNEC2 poses a convexity
constraint on kinks in EE. Finally, in section 2.8 we show a relationship
between QNEC2 and the Casini–Huerta $c$-function.

###  AdS3/CFT2

We work mostly on the gravity side and are not specific about the dual QFT,
except that it must have a UV fixed point corresponding to a CFT2 without
gravitational anomaly and with central charge
$c=\frac{3\ell_{\textrm{\tiny{AdS}}}}{2G_{3}}\gg 1$, where
$\ell_{\textrm{\tiny{AdS}}}$ is the AdS radius (which we set to one) and
$G_{3}$ is Newton’s constant. The inequality is necessary for the validity of
the (super-)gravity approximation. The CFT is put either on a torus, a
cylinder or the plane. On the cylinder we use standard coordinates
$\operatorname{d}\\!s^{2}=-\operatorname{d}\\!t^{2}+\operatorname{d}\\!\varphi^{2}$
with $\varphi\sim\varphi+2\pi$, and on the plane we use
$\operatorname{d}\\!s^{2}=-\operatorname{d}\\!t^{2}+\operatorname{d}\\!x^{2}$.

The gravity theory we consider is AdS3 Einstein gravity with scalar matter,
reviewed in detail in the next section. In the absence of matter the boundary
conditions are the seminal ones by Brown and Henneaux Brown:1986nw  and the
solutions to this theory are given by the Bañados metrics Banados:1998gg

$\operatorname{d}\\!s^{2}=\frac{\operatorname{d}\\!z^{2}-\operatorname{d}\\!x^{+}\operatorname{d}\\!x^{-}}{z^{2}}+{\cal
L}^{+}(x^{+})\big{(}\operatorname{d}\\!x^{+}\big{)}^{2}+{\cal
L}^{-}(x^{-})\big{(}\operatorname{d}\\!x^{-}\big{)}^{2}-z^{2}{\cal
L}^{+}(x^{+}){\cal
L}^{-}(x^{-})\,\operatorname{d}\\!x^{+}\operatorname{d}\\!x^{-}$ (3)

where we used lightcone coordinates $x^{\pm}=t\pm\varphi$. The solution ${\cal
L}^{\pm}=-\tfrac{1}{4}$ (${\cal L}^{\pm}=0$) describes global (Poincaré patch)
AdS3, while constant positive ${\cal L}^{\pm}$ yield non-extremal BTZ black
holes with horizons located at $r_{\pm}=|\sqrt{\cal L^{+}}\pm\sqrt{\cal
L^{-}}|$, temperature $T=(r_{+}^{2}-r_{-}^{2})/(2\pi r_{+})$ and angular
velocity $\Omega=r_{-}/r_{+}$.

The holographic dictionary relates the Bañados geometries (3) to CFT2 states
$|{\cal L}^{+},{\cal L}^{-}\rangle$ (global AdS3 corresponds to the vacuum
state $|0\rangle$). Key relations for us are the expectation values of the
flux components of the CFT2 energy-momentum tensor expressed in terms of the
metric functions ${\cal L}^{\pm}$,

$2\pi\big{<}{\cal L}^{+},{\cal L}^{-}\big{|}T_{\pm\pm}(x^{\pm})\big{|}{\cal
L}^{+},{\cal L}^{-}\big{>}=\frac{c}{6}\,{\cal L}^{\pm}(x^{\pm})\,.$ (4)

The fact that all Bañados geometries are locally diffeomorphic to each other
leads to corresponding uniformization results on the CFT-side. We summarize
below this uniformization for EE, which is a necessary ingredient for QNEC2.

###  Uniformized result for holographic entanglement entropy

There is a simple and uniform result for EE in CFT2 for all states dual to
Bañados geometries, namely

$S(x_{1}^{\pm},\,x_{2}^{\pm})=\frac{c}{6}\,\ln\big{(}\ell^{+}(x_{1}^{+},\,x_{2}^{+})\ell^{-}(x_{1}^{-},\,x_{2}^{-})/\ell_{\textrm{\tiny{cut}}}^{2}\big{)}$
(5)

where $x_{1}^{\pm}$ and $x_{2}^{\pm}$ are the two endpoints of the entangling
interval, $\ell_{\textrm{\tiny{cut}}}$ is a UV cutoff (that tends to zero when
the cutoff is removed) and the functions
$\ell^{\pm}=\psi_{1}^{\pm}(x^{\pm}_{1})\psi_{2}^{\pm}(x^{\pm}_{2})-\psi_{2}^{\pm}(x^{\pm}_{1})\psi_{1}^{\pm}(x^{\pm}_{2})$
are bilinears in solutions to Hill’s equation, $\psi^{\pm\,\prime\prime}-{\cal
L}^{\pm}\,\psi^{\pm}=0$, subject to unit Wronskians,
$\psi_{1}^{\pm}\psi_{2}^{\pm\,\prime}-\psi_{2}^{\pm}\psi_{1}^{\pm\,\prime}=\pm
1$.

On the gravity side the geometric reason for this uniformization is that all
solutions to the vacuum Einstein equations are locally AdS3 and therefore
there is a diffeomorphism that maps any such geometry locally to Poincaré
patch AdS3 with coordinates $x^{\pm}_{\textrm{\tiny{PP}}}$ and
$z_{\textrm{\tiny{PP}}}$ Roberts:2012aq ; Sheikh-Jabbari:2016znt . The
coordinate transformation involves the solutions to Hill’s equation above,
$x^{\pm}_{\textrm{\tiny{PP}}}=\int\operatorname{d}\\!x^{\pm}/\psi^{\pm\,2}-z^{2}\psi^{\mp\,\prime}/[\psi^{\pm\,2}\psi^{\mp}(1-z^{2}/z_{h}^{2})]$
and $z_{\textrm{\tiny{PP}}}=z/[\psi^{+}\psi^{-}(1-z^{2}/z_{h}^{2})]$, where
$z_{h}=[\psi_{a}^{+}\psi_{b}^{-}/(\psi_{c}^{+\,\prime}\psi_{d}^{-\,\prime})]^{1/2}$
($a,b=1,2$ and $c,d=1,2$ in some permutation) is the locus of one of the
Killing horizons; since we want to map the outside causal patch to Poincaré
patch AdS3 for us $z_{h}$ is always the black hole event horizon, so we have
to choose $a,b,c,d$ accordingly.

For Poincaré patch AdS (${\cal L}^{\pm}=0$) Hill’s equation is solved by
$\psi_{2}^{+}=1=\psi_{1}^{-}$, $\psi_{1}^{+}=x^{+}$ and $\psi_{2}^{-}=x^{-}$
leading to $\ell^{\pm}=\pm x_{1}^{\pm}\mp x_{2}^{\pm}$, thereby recovering the
well-known result for EE of a constant-time interval
$\ell=|x_{2}^{+}-x_{1}^{+}|=|x_{1}^{-}-x_{2}^{-}|$ in a CFT2 on the plane
Holzhey:1994we

$S_{\textrm{\tiny{PP}}}=\frac{c}{3}\,\ln\frac{\ell}{\ell_{\textrm{\tiny{cut}}}}\,.$
(6)

By slight abuse of notation we refer to logarithmic behavior in the entangling
interval as ‘area law’. Following the RT prescription Ryu:2006bv  the result
(6) is obtained on the gravity side from the length of a geodesic anchored at
the endpoints of the entangling interval.

For non-extremal BTZ black holes or black branes (${\cal
L}^{\pm}=\textrm{const.}>0$) Hill’s equation is solved by
$\psi_{1}^{\pm}=\exp{(\sqrt{{\cal L}^{\pm}})}/(4{\cal L}^{\pm})^{1/4}$ and
$\psi_{2}^{\pm}=\mp\exp{(-\sqrt{{\cal L}^{\pm}})}/(4{\cal L}^{\pm})^{1/4}$
leading to $\ell^{+}=\sinh{[\sqrt{{\cal L}^{+}}(x_{2}^{+}-x_{1}^{+})}]$ and
$\ell^{-}=\sinh{[\sqrt{{\cal L}^{-}}(x_{1}^{-}-x_{2}^{-})}]$, thereby
recovering as special case (non-rotating BTZ, ${\cal L}^{+}={\cal
L}^{-}=\pi/\beta$) the result for EE of thermal states in a CFT2
Calabrese:2004eu

$S_{\textrm{\tiny{thermal}}}=\frac{c}{3}\,\ln\Big{(}\frac{\beta}{\pi\ell_{\textrm{\tiny{cut}}}}\,\sinh\frac{\pi\ell}{\beta}\Big{)}$
(7)

where $\beta$ is inverse temperature and
$\ell=|x_{2}^{+}-x_{1}^{+}|=|x_{1}^{-}-x_{2}^{-}|$ is again the constant-time
interval defining the entangling region. In the large $\ell$ limit EE (7)
obeys the volume law

$S_{\textrm{\tiny{thermal}}}(\ell\gg
1)=\frac{c\pi\ell}{3\beta}+\textrm{subleading}\,.$ (8)

###  QNEC2 saturation for vacuum-like states

By virtue of the uniformized result (5) for EE it is straightforward to prove
that the QNEC2 inequality (2) saturates for all CFT2 states dual to Bañados
geometries (3) Ecker:2019ocp . Namely, defining a function resembling a
vertex-operator, $V=\exp{(6S/c)}=V^{+}V^{-}$ with
$V^{\pm}=\ell^{\pm}/\ell_{\textrm{\tiny{cut}}}$, leads to the right hand side
of QNEC2,
$cV^{\pm\,\prime\prime}/(6V^{\pm})=S^{\prime\prime}+\tfrac{6}{c}\,(S^{\prime})^{2}$,
where prime denotes $\operatorname{d}\\!/\operatorname{d}\\!x_{1}^{+}$ for the
upper sign and $\operatorname{d}\\!/\operatorname{d}\\!x_{1}^{-}$ for the
lower sign. On the other hand, the explicit form of the vertex-functions
$V^{\pm}$ shows that they obey Hill’s equation, $V^{\pm\,\prime\prime}={\cal
L}^{\pm}V^{\pm}$. Using the holographic dictionary (4) then establishes the
equality

$2\pi\big{<}{\cal L}^{+},{\cal L}^{-}\big{|}T_{\pm\pm}(x^{\pm})\big{|}{\cal
L}^{+},{\cal
L}^{-}\big{>}=S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}$ (9)

which is the saturated version of the QNEC2 inequality (2).

A related way to understand QNEC2 saturation is to analyze the transformation
properties of EE under bulk diffeomorphisms or boundary conformal
transformations generated by some [anti-]holomorphic function $\xi(x^{+})$
[$\xi(x^{-})$]. As shown by Wall EE transforms like an anomalous scalar
Wall:2011kb .

$\delta_{\xi}S=\xi S^{\prime}-\frac{c}{12}\,\xi^{\prime}$ (10)

The first term is the usual Lie-derivative expression for a scalar field and
the second, anomalous, term results from dilatation relative to the cutoff.
The QNEC2 combination then transforms with the same infinitesimal Schwarzian
derivative

$\delta_{\xi}\Big{(}S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\Big{)}=\xi\,\Big{(}S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\Big{)}^{\prime}+2\xi^{\prime}\,\Big{(}S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\Big{)}-\frac{c}{12}\,\xi^{\prime\prime\prime}$
(11)

as the boundary stress tensor. This means that whenever QNEC2 saturates for
one particular state/geometry it also saturates for all states/geometries
related by conformal transformations/diffeomorphisms, which explains the idea
of the proof above.

###  QNEC2 non-saturation and half-saturation in presence of bulk matter

When bulk matter is present QNEC2 does not saturate in general, and in
particular it never saturates when the RT surface intersects regions with bulk
matter Khandker:2018xls . A sufficient condition for QNEC2 to hold is that
bulk matter obeys the null energy condition Koeller:2015qmn , which will
always be the case in the present work.

An intriguing aspect discovered (but not explained) in Ecker:2019ocp  is that
there is universal half-saturation of QNEC2 for quenches (modelled on the
gravity side by Vaidya-type of metrics) in the sense that the ratio of left-
and right-hand sides of the QNEC2 inequality (2) approaches $\tfrac{1}{2}$ in
the limit of large entangling intervals. Evidence for half-saturation was
extracted from numerical and perturbative calculations. This phenomenon was
explained more recently in a work by Mezei and Virrueta Mezei:2019sla  who
studied quantum quenches and QNEC2 constraints imposed on them. Their
beautifully simple explanation of the ratio $\tfrac{1}{2}$ is that it
corresponds to $\tfrac{1}{d}$ in CFTd, i.e., one over the spacetime dimension
of the CFT.111 Their results not only explain the half-saturation observed in
Ecker:2019ocp  for CFT2, but also the ‘curious ratio $0.25$’ mentioned in an
AdS5/CFT4 context in the first numerical study of QNEC, see the caption of
Figure 3 in Ecker:2017jdw .  To be more explicit we recall their key
statement. Shortly after the quench at time $t=0$ EE has the expansion222 The
state is assumed to be time reflection symmetric at $t=0$.

$S(t,\,\ell)=s_{0}(\ell)+s_{2}(\ell)\,t^{2}+{\cal O}(t^{4})$ (12)

where $\ell$ is the entangling interval. The quantity $s_{0}$ drops out in the
QNEC2 evaluation since before the quench the Vaidya metric is a special case
of a Bañados geometry, so only $s_{2}$ enters there. The QNEC2 inequality (2)
then leads to the bound $s_{2}\leq 2\pi(e+p)/2$, where $e$ is the energy
density and $p$ the pressure of the state under consideration (their sum
$(e+p)/2$ corresponds to $\langle T_{kk}\rangle$). Mezei and Virrueta were
able to holographically prove a stronger bound for states dual to Vaidya
metrics in CFTd, viz. $s_{2}\leq 2\pi(e+p)/(2d)$, which explains the half-
saturation for CFT2. (An intrinsic CFT2 proof of this statement has not been
found so far, but probably exists.)

When bulk matter is present EE and the right hand side of QNEC2 in general can
only be determined numerically, except for special states and in large- or
small-interval limits.

###  QNEC2 for boost invariant states

In typical QFTs there is at least one boost invariant state, namely the
Poincaré invariant vacuum. Whenever we have such a boost invariant state there
is a simple way to obtain QNEC2 by calculating EE as function of the interval
length $\ell$ and taking a suitable combination of derivatives thereof. We
show now how this works.

The null deformation of the interval requires the evaluation of EE for slices
where time is not constant. However, if the state under consideration is boost
invariant we can always boost to the rest frame and determine EE on a constant
time slice in that frame.

$S(\lambda,\,\ell\pm\lambda)=S(0,\,\sqrt{(\ell\pm\lambda)^{2}-\lambda^{2}})=:S_{0}(\sqrt{(\ell\pm\lambda)^{2}-\lambda^{2}})$
(13)

The expression on the left hand side denotes EE as a function of the temporal
and spatial extent of the null deformed entangling interval in the original
frame, while the right hand side is EE in the rest frame, which we denote as
$S_{0}$.

Expanding $S_{0}$ as function of proper length in powers of $\lambda$ up to
second order establishes the desired relation between QNEC2 and derivatives of
EE with respect to the interval length.

$\frac{\operatorname{d}\\!^{2}S_{0}}{\operatorname{d}\\!\lambda^{2}}\bigg{|}_{\lambda=0}+\frac{6}{c}\,\bigg{(}\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\lambda}\bigg{)}^{2}\bigg{|}_{\lambda=0}=\frac{\operatorname{d}\\!^{2}S_{0}}{\operatorname{d}\\!\ell^{2}}-\frac{1}{\ell}\,\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\ell}+\frac{6}{c}\,\bigg{(}\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\ell}\bigg{)}^{2}$
(14)

Thus, if one knows EE as function of the interval length $\ell$ for a boost
invariant state then the right hand side of (14) yields the right hand side of
the QNEC2 inequality (2) for this state.

The simplest holographic example is Poincaré patch AdS3. The result (6) for EE
yields

$\frac{\operatorname{d}\\!^{2}S_{\textrm{\tiny{PP}}}}{\operatorname{d}\\!\ell^{2}}-\frac{1}{\ell}\,\frac{\operatorname{d}\\!S_{\textrm{\tiny{PP}}}}{\operatorname{d}\\!\ell}+\frac{6}{c}\,\bigg{(}\frac{\operatorname{d}\\!S_{\textrm{\tiny{PP}}}}{\operatorname{d}\\!\ell}\bigg{)}^{2}=-\frac{c}{3\ell^{2}}-\frac{c}{3\ell^{2}}+\frac{6}{c}\,\bigg{(}\frac{c}{3\ell}\bigg{)}^{2}=0\,.$
(15)

This result provides an alternative proof of QNEC2 saturation for the state
dual to Poincaré patch AdS3, since also the boundary stress tensor vanishes
for this state.

A more interesting example is a situation where the QFT flows from a CFT2 in
the UV with central charge $c=c_{\textrm{\tiny{UV}}}$ to a different CFT2 in
the IR with central charge $c_{\textrm{\tiny{IR}}}$ so that for large values
of the interval $\ell$ we have EE obeying an area law

$\lim_{\ell\gg 1}S_{0}=\frac{c_{\textrm{\tiny{IR}}}}{3}\,\ln\ell+{\cal O}(1)$
(16)

but with the IR value of the central charge. We then obtain from (14) the
QNEC2 expression

$\frac{\operatorname{d}\\!^{2}S_{0}}{\operatorname{d}\\!\lambda^{2}}\bigg{|}_{\lambda=0,\,\ell\gg
1}+\frac{6}{c}\,\bigg{(}\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\lambda}\bigg{)}^{2}\bigg{|}_{\lambda=0,\,\ell\gg
1}=-\frac{2c_{\textrm{\tiny{IR}}}}{3\ell^{2}}\,\Big{(}1-\frac{c_{\textrm{\tiny{IR}}}}{c_{\textrm{\tiny{UV}}}}\Big{)}+\dots$
(17)

where the ellipsis denotes terms that vanish more quickly than the terms
displayed in the limit of large intervals. This means that the well-known
inequality $c_{\textrm{\tiny{UV}}}\geq c_{\textrm{\tiny{IR}}}$ can be
considered as a consequence of the QNEC2 inequality (2).

For small $\ell$ we expect EE to be close to the Poincaré patch AdS3 result
(6),

$S_{0}=\frac{c}{3}\,\ln\ell+\sum_{n=0}^{\infty}\,s_{n}\ell^{n}$ (18)

with some coefficients $s_{n}$ that depend on the boost invariant state.
According to (14) QNEC2 contains a piece that diverges at small $\ell$.

$\frac{\operatorname{d}\\!^{2}S_{0}}{\operatorname{d}\\!\lambda^{2}}\bigg{|}_{\lambda=0}+\frac{6}{c}\,\bigg{(}\frac{\operatorname{d}\\!S_{0}}{\operatorname{d}\\!\lambda}\bigg{)}^{2}\bigg{|}_{\lambda=0}=\frac{3s_{1}}{\ell}+{\cal
O}(1)$ (19)

So despite of being close to AdS3 in general there is a large correction to
QNEC2 at small $\ell$. The QNEC2 inequality (2) with $T_{kk}=0$ requires non-
positivity of the coefficient $s_{1}$.

$s_{1}\leq 0$ (20)

###  QNEC2 with paper and pencil

Having discussed the definition and main properties of QNEC2 we elaborate on
how to calculate the right hand side of the QNEC2 inequality (2)
holographically. For concreteness and because this is the only case considered
in our work we focus on states dual to geometries with two commuting Killing
vectors (either stationary axi-symmetric or stationary homogeneous
geometries). The presence of two commuting Killing vectors means that there is
always an adapted set of coordinate systems where these Killing vectors read
$\partial_{t}$ and $\partial_{x}$. We use such a coordinate system from now
on. The metric
$\operatorname{d}\\!s^{2}=g_{\mu\nu}(z)\,\operatorname{d}\\!x^{\mu}\operatorname{d}\\!x^{\nu}$
then depends only on the holographic (‘radial’) coordinate $z$.

Thanks to RT we just need to calculate the lengths of a one-parameter family
of geodesics, where the family parameter $\lambda$ corresponds to the null
deformation of the entangling region required to generate the QNEC2
expression. At each step we can work perturbatively in $\lambda$ to second
order, since in the end we take at most two derivatives with respect to
$\lambda$ and set it to zero afterwards. Using the spatial coordinate $x$ as
affine parameter the geodesic Lagrangian (dot means derivative with respect to
$x$)

${\cal
L}(\dot{t},\,\dot{z},\,z)=\sqrt{g_{tt}(z)\dot{t}^{2}+g_{zz}(z)\dot{z}^{2}+g_{xx}(z)+g_{tz}(z)\dot{t}\dot{z}+g_{tx}(z)\dot{t}+g_{zx}(z)\dot{z}}$
(21)

yields the area

${\cal
A}(\lambda,\,\ell,\,z_{\textrm{\tiny{cut}}})=2\int\limits_{0}^{(\ell+\lambda)/2-\omega}\operatorname{d}\\!x\,{\cal
L}(\dot{t},\,\dot{z},\,z)$ (22)

where $\ell$ is the length of the entangling interval, $\lambda$ is the
aforementioned deformation parameter, $\omega$ is a specific function of the
cutoff $z_{\textrm{\tiny{cut}}}$ in the radial coordinate and the overall
factor $2$ comes about because we integrate the geodesic from its turning
point in the bulk to the anchor point at the cutoff surface and use the fact
that the geodesic is mirror symmetric around $x=0$.

Since the Lagrangian ${\cal L}$ is $x$-independent we have the usual Noether-
charge associated with $x$-translation invariance.

$Q_{1}=\dot{z}\,\frac{\partial\cal
L}{\partial\dot{z}}+\dot{t}\,\frac{\partial\cal L}{\partial\dot{t}}-\cal L$
(23)

It is convenient to evaluate the Noether charge $Q_{1}$ at the turning point
$z=z_{\ast}$. The Lagrangian ${\cal L}$ is also $t$-independent, which yields
a second Noether charge.

$Q_{2}=\frac{\partial\cal L}{\partial\dot{t}}$ (24)

The two Noether charges allow to express the velocity of the time coordinate
and the velocity of the radial coordinate as functions of radial coordinate
and turning point. However, it is more convenient to relabel their dependence
on the Noether charges as dependence on the turning point $z_{\ast}$ and a
specific combination $\Lambda$ of the two Noether charges that vanishes when
the deformation parameter goes to zero, $\lambda\to 0$.

$\dot{t}=\Lambda\,h(z,\,z_{\ast},\,\Lambda)\qquad\qquad\dot{z}=f(z,\,z_{\ast},\,\Lambda)$
(25)

For the special case of diagonal metrics $\Lambda$ is given by the ratio of
the Noether charges, $\Lambda=Q_{2}/Q_{1}$.

The temporal part of the deformed entangling interval is obtained by
integrating $\operatorname{d}\\!t$.

$\frac{\lambda}{2}=\int\limits_{0}^{\lambda/2}\operatorname{d}\\!t=\int\limits_{z_{\ast}}^{0}\operatorname{d}\\!z\,\frac{\dot{t}}{\dot{z}}=\Lambda\,\int\limits_{z_{\ast}}^{0}\operatorname{d}\\!z\,\frac{h(z,\,z_{\ast},\,\Lambda)}{f(z,\,z_{\ast},\,\Lambda)}$
(26)

The spatial part of the deformed entangling interval is obtained by
integrating $\operatorname{d}\\!x$.

$\frac{\ell+\lambda}{2}=\int\limits_{0}^{(L+\lambda)/2}\operatorname{d}\\!x=\int\limits_{z_{\ast}}^{0}\frac{\operatorname{d}\\!z}{\dot{z}}=\int\limits_{z_{\ast}}^{0}\frac{\operatorname{d}\\!z}{f(z,\,z_{\ast},\,\Lambda)}$
(27)

Finally, the area integral (22) can be recast as

${\cal
A}=2\int\limits_{z_{\ast}}^{z_{\textrm{\tiny{cut}}}}\operatorname{d}\\!z\,\frac{{\cal
L}(\Lambda\,h(z,\,z_{\ast},\,\Lambda),\,f(z,\,z_{\ast},\,\Lambda),\,z)}{f(z,\,z_{\ast},\,\Lambda)}$
(28)

where $z_{\textrm{\tiny{cut}}}$ denotes the cutoff on the radial coordinate;
for concreteness we assume that the limit $z_{\textrm{\tiny{cut}}}\to 0^{+}$
corresponds to removing the cutoff. Evaluating the temporal interval integral
(26) yields $\Lambda$ as function of $\lambda$ and $z_{\ast}$. Since we can
drop terms of order ${\cal O}(\lambda^{3})$ and to leading order $\Lambda$ is
already linear in $\lambda$, we can expand the functions $h$ and $f$ before
integrating, which usually simplifies these integrals considerably. The
evaluation of the spatial interval integral (27) allows to express the turning
point $z_{\ast}$ in terms of the interval length $\ell$ and the deformation
parameter $\lambda$. Again one can expand in $\Lambda$ and keep only the first
few terms, dropping everything of order ${\cal O}(\lambda^{3})$. These results
allow to express the area integral (28) entirely in terms of $\ell$ and
$\lambda$, so performing this integral then yields deformed EE as function of
the interval length $\ell$ and the deformation parameter $\lambda$.

For most practical purposes these three integrals cannot be performed by hand
since the functions $h$ and $f$ can be quite complicated, even when the metric
functions are known in closed form. However, in the limit of small $\ell$ or
large $\ell$ drastic simplifications occur that can allow to perform the first
two integrals. The third integral diverges when the cutoff is removed, so it
is practical to use instead a renormalized area

${\cal
A}_{\textrm{\tiny{ren}}}(\lambda,\,\ell)=2\int\limits_{z_{\ast}}^{0}\operatorname{d}\\!z\,\bigg{(}\frac{{\cal
L}(\Lambda\,h(z,\,z_{\ast},\,\Lambda),\,f(z,\,z_{\ast},\,\Lambda),\,z)}{f(z,\,z_{\ast},\,\Lambda)}-\frac{\operatorname{d}\\!A_{\textrm{\tiny{ct}}}(z)}{\operatorname{d}\\!z}\bigg{)}-2A_{\textrm{\tiny{ct}}}(z_{\ast})\\\
={\cal A}_{0}(\ell)+\lambda\,{\cal A}_{1}(\ell)+\frac{\lambda^{2}}{2}\,{\cal
A}_{2}(\ell)+{\cal O}(\lambda^{3})$ (29)

where $A_{\textrm{\tiny{ct}}}(z)$ is a counter-term added in such a way that
the additional term in the renormalized area is independent from the interval
length $\ell$ and the deformation parameter $\lambda$, i.e., it only depends
on the cutoff $z_{\textrm{\tiny{cut}}}$. The integral in (29) is now finite
and has $0$ as one of its boundaries, which considerably simplifies its
evaluation. Inserting the solutions for $\Lambda$ and $z_{\ast}$ in terms of
$\ell$ and $\lambda$ yields the functions ${\cal A}_{i}(\ell)$ in the second
line in (29).

The RT-formula

$S(\lambda,\,\ell)=\frac{{\cal A}(\lambda,\,\ell)}{4G_{3}}=\frac{{\cal
A}_{\textrm{\tiny{ren}}}(\lambda,\,\ell)}{4G_{3}}+\lambda\textrm{-independent
terms}$ (30)

establishes the final expression appearing on the right hand side of QNEC2

$\frac{\operatorname{d}\\!^{2}S}{\operatorname{d}\\!\lambda^{2}}\bigg{|}_{\lambda=0}+\frac{6}{c}\,\bigg{(}\frac{\operatorname{d}\\!S}{\operatorname{d}\\!\lambda}\bigg{)}^{2}\bigg{|}_{\lambda=0}=\frac{c}{6}\,\big{(}{\cal
A}_{2}(\ell)+{\cal A}_{1}(\ell)^{2}\big{)}$ (31)

where we replaced Newton’s constant by the central charge,
$\tfrac{1}{4G_{3}}=\tfrac{c}{6}$. In later sections we shall provide some
examples where ${\cal A}_{2}(\ell)$ and ${\cal A}_{1}(\ell)$ are calculated in
the limits of small and/or large entangling interval $\ell$.

For boost invariant states, like domain wall solutions, we can instead
determine EE as function of the interval length $\ell$ and apply (14). In the
algorithm above this means that $\lambda$ and $\Lambda$ can be set to zero and
the time integral (26) need not be calculated, which makes the calculation a
bit shorter.

###  QNEC2 constraint on kinked entanglement

Sometimes holographic EE leads to two or more branches of geodesics
Myers:2012ed ; Liu:2012eea , so there can be a critical interval value where
one jumps from one of these branches to another. If this happens then EE as a
function of the interval $\ell$ has a kink, and there is a first order phase
transition (the converse is not necessarily true: there can be first order
phase transitions without kinks in EE). We show now that QNEC2 imposes a
constraint on the behavior of EE near such a kink.

Let us assume EE as function of the entangling region has the following form

$S(\lambda,\,\ell+\lambda)=f_{L}(\lambda,\,\ell+\lambda)\,\theta\big{(}f_{L}(\lambda,\,\ell+\lambda)\big{)}+f_{R}(\lambda,\,\ell+\lambda)\,\theta\big{(}f_{R}(\lambda,\,\ell+\lambda)\big{)}$
(32)

where $f_{L}$ and $f_{R}$ are sufficiently smooth functions of the spatial
length of the entangling interval $\ell$ and the null deformation parameter
$\lambda$ used in QNEC2. The subscripts $L,R$ refer to ‘left’ and ‘right’ of
the kink at $\ell+\lambda=\ell_{0}$. Taylor expanding these functions without
loss of generality yields ($\alpha_{L}\neq\alpha_{R}$)

$\displaystyle f_{L}$
$\displaystyle=\alpha_{L}(\lambda)(\ell_{0}-\ell-\lambda)+{\cal
O}(\lambda^{2})+{\cal O}(\ell+\lambda-L_{0})^{2}$ (33a) $\displaystyle f_{R}$
$\displaystyle=\alpha_{R}(\lambda)(\ell+\lambda-\ell_{0})+{\cal
O}(\lambda^{2})+{\cal O}(\ell+\lambda-L_{0})^{2}\,.$ (33b)

The expression $\ell+\lambda$ is the proper length of the entangling interval,
up to irrelevant higher order terms, since
$\sqrt{(\ell+\lambda)^{2}-\lambda^{2}}=\ell+\lambda+{\cal O}(\lambda^{2})$.

With the assumptions above EE (32) is smooth except at
$\ell+\lambda=\ell_{0}$. In the following we investigate the QNEC2 combination
of first and second derivatives of EE with respect to the null deformation
parameter $\lambda$.

Consider the first derivative of EE. If the derivative acts on the step
function we obtain an expression of the form $x\,\delta(x)$ which vanishes.

$\frac{\partial S}{\partial\lambda}=\frac{\partial
f_{L}}{\partial\lambda}\,\theta(f_{L})+\frac{\partial
f_{R}}{\partial\lambda}\,\theta(f_{R})$ (34)

This means that the first derivative of EE is piecewise continuous.

Consider now the second derivative of EE evaluated at vanishing $\lambda$.

$\frac{\partial^{2}S}{\partial\lambda^{2}}\bigg{|}_{\lambda=0}=\bigg{(}\frac{\partial
f_{L}}{\partial\lambda}\bigg{)}^{2}\,\delta(f_{L})+\bigg{(}\frac{\partial
f_{R}}{\partial\lambda}\bigg{)}^{2}\,\delta(f_{R})+\textrm{piecewise\;continuous}$
(35)

Since we are interested in the behavior near the kink we use now the Taylor
expansions (33) as well as the identity
$\delta(\alpha(x-x_{0}))=\delta(x-x_{0})/|\alpha|$ and obtain the QNEC2
combination

$\frac{\partial^{2}S}{\partial\lambda^{2}}\bigg{|}_{\lambda=0}+\frac{6}{c}\,\bigg{(}\frac{\partial
S}{\partial\lambda}\bigg{)}^{2}\bigg{|}_{\lambda=0}=(\alpha_{R}^{0}-\alpha_{L}^{0})\,\delta(\ell-\ell_{0})+\textrm{piecewise\;continuous}$
(36)

where $\alpha_{L,R}^{0}=|\alpha_{L,R}(\lambda=0)|$. At the kink
$\ell=\ell_{0}$ the QNEC2 combination has an infinite peak due to the
$\delta$-function. The sign of this peak must be negative since otherwise the
QNEC2 inequality (2) would be violated. Therefore, consistency with QNEC2
imposes the convexity condition

$\alpha_{L}^{0}>\alpha_{R}^{0}\,.$ (37)

For ground states the inequality (37) is a consequence of RG-flow
monotonicity. In the next subsection we make the relation between QNEC2 and
RG-flows more precise.

###  Relation between QNEC2 and Casini–Huerta $c$-function

If one views EE as function of the interval length $\ell$ mechanically as a
trajectory then QNEC2 is the natural acceleration associated with this
trajectory. It is then suggestive to ponder whether the velocity associated
with this trajectory also has a physical meaning. At least for boost invariant
states, which we shall refer to as ground states, the answer is affirmative —
the velocity is the Casini–Huerta $c$-function Casini:2004bw ; Casini:2006es ;
Myers:2012ed . We show now how this relationship works, denoting EE in a
general frame by $S$ and in the rest frame by $S_{0}$.

The Casini–Huerta $c$-function for a deformed CFT2

$c(\ell)=3\ell\,\frac{\operatorname{d}\\!S_{0}(\ell)}{\operatorname{d}\\!\ell}$
(38)

is proportional to the first derivative with respect to the null deformation
parameter (to reduce sign clutter we assume here an outgoing null deformation)

$\frac{\operatorname{d}\\!S(\lambda,\,\ell+\lambda)}{\operatorname{d}\\!\lambda}\bigg{|}_{\lambda=0}=\frac{\operatorname{d}\\!S_{0}(\sqrt{(\ell+\lambda)^{2}-\lambda^{2}})}{\operatorname{d}\\!\lambda}\bigg{|}_{\lambda=0}=\frac{\operatorname{d}\\!S_{0}(\ell)}{\operatorname{d}\\!\ell}=\frac{1}{3\ell}\,c(\ell)\,.$
(39)

So up to an overall factor of $1/(3\ell)$ the Casini–Huerta $c$-function (38)
is indeed the ‘velocity’ of EE in essentially the same sense that QNEC2 is its
‘acceleration’. It obeys the monotonicity relation

$\frac{\operatorname{d}\\!c(\ell)}{\operatorname{d}\\!\ell}\leq 0\,.$ (40)

Integrating this monotonicity inequality for EE using the definition (38)
yields a bound on ground state EE

$S_{0}(\ell)\leq\frac{\hat{c}}{3}\,\ln\frac{\ell}{\ell_{\textrm{\tiny{cut}}}}\qquad\qquad\hat{c},\,\ell_{\textrm{\tiny{cut}}}\in\mathbb{R}^{+}$
(41)

which means in particular that at large $\ell$ ground state entanglement
cannot grow faster than the area law (6).

Consistently, the first derivative of the Casini–Huerta $c$-function (up to a
factor $3\ell$)

$\frac{1}{3\ell}\,\frac{\operatorname{d}\\!c(\ell)}{\operatorname{d}\\!\ell}=\frac{\operatorname{d}\\!^{2}S_{0}(\ell)}{\operatorname{d}\\!\ell^{2}}-\frac{1}{\ell}\,\frac{\operatorname{d}\\!S_{0}(\ell)}{\operatorname{d}\\!\ell}+\frac{6}{c(\ell)}\,\bigg{(}\frac{\operatorname{d}\\!S_{0}(\ell)}{\operatorname{d}\\!\ell}\bigg{)}^{2}$
(42)

yields essentially the QNEC2 combination, as evident from our result (14) for
boost invariant states.

Thus, for boost invariant states the right hand side of the QNEC2 inequality
can be re-interpreted as the rate by which the Casini–Huerta $c$-function
changes in the UV. The monotonicity property of this $c$-function (40) is then
equivalent to the QNEC2 inequality (2) with $\langle T_{kk}\rangle=0$ for
Poincaré invariant states, except that $c$ is replaced by $c(\ell)$. (By the
constant $c$ we always mean the UV-value $\lim_{\ell\to 0}c(\ell)$ of the
Casini–Huerta $c$-function, which in our context is the Brown–Henneaux central
charge.) Monotonicity of the Casini–Huerta $c$-function (40) is stronger than
QNEC2 since $c\geq c(\ell)$.

For general states that are not boost invariant the function (38) need not be
monotonically decreasing. For instance, thermal states dual to BTZ black
branes have EE (7) leading to a monotonically increasing function (38),
$c(\ell)=\frac{c\pi\ell}{\beta}\,\coth\frac{\pi\ell}{\beta}$. Nevertheless,
the QNEC2 inequality (2) holds in full generality even for states that are not
boost invariant.

##  Holographic model

In our work we consider deformed holographic CFTs that are dual to three-
dimensional Einstein gravity with a minimally coupled massive self-interacting
scalar field. In this section we summarize the gravity side of the holographic
model that we study in the rest of this work.

In section 3.1 we present the Einstein–Klein–Gordon action, including its
boundary terms and the explicit form of the scalar potential in terms of a
superpotential. In section 3.2 we derive solutions corresponding to the ground
state of the dual field theory. In section 3.3 we discuss solutions
corresponding to thermal states in the dual field theory.

###  Action

The action of the gravity system

$\Gamma=\frac{1}{16\pi
G_{3}}\int_{\mathcal{M}}\\!\\!\operatorname{d}\\!^{3}x\sqrt{-g}\,\Big{(}R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\Big{)}+\frac{1}{8\pi
G_{3}}\int_{\partial\mathcal{M}}\\!\\!\\!\\!\operatorname{d}\\!^{2}x\sqrt{-\gamma}\,K+I_{\textrm{ct}}$
(43)

entails the Ricci scalar $R$ of the bulk geometry on a manifold $\cal M$ with
boundary $\partial\cal M$ and bulk metric $g_{\mu\nu}$, the trace of the
extrinsic curvature $K$ of the boundary geometry with induced metric
$\gamma_{ij}$ and the counter-term $I_{\textrm{ct}}$ that renders the
variational principle well-defined (and also the on-shell action finite).

For asymptotically AdS3 spacetimes the potential needs to have the small
$\phi$ expansion

$V(\phi)=-2+\frac{1}{2}m^{2}\phi^{2}+v_{4}\phi^{4}+\ldots$ (44)

where we assumed the $\mathbb{Z}_{2}$ symmetry $V(\phi)=V(-\phi)$ for
simplicity. By the usual AdS/CFT dictionary the conformal weight $\Delta$ of
the dual operator is related to the mass of the scalar field as
$m^{2}=\Delta(\Delta-2)$. We restrict to potentials that globally can be
written in terms of a superpotential $W$.

$\displaystyle
V(\phi)=-\frac{1}{2}W(\phi)^{2}+\frac{1}{2}W^{\prime}(\phi)^{2}$ (45)

Technical advantages of this choice are that one can find the ground state
(domain wall) solution more easily, the counter-terms $I_{\textrm{ct}}$ are
known explicitly in terms of the superpotential (see appendix B) and neither
in Fefferman–Graham- nor in Gubser-gauge [defined in (55)] the near boundary
solution has logarithmic terms. The latter is related to the absence of the
trace anomaly (again, see appendix B). This makes the numerical analysis of
geodesics and QNEC2 on such backgrounds more stable.

It turns out that simple superpotentials characterised by a single parameter

$\displaystyle W(\phi)=-2-\frac{1}{4}\,\phi^{2}-\frac{\alpha}{8}\,\phi^{4}$
(46)

reveal already rich physical features. The associated potential (45) reads

$V(\phi)=-2-\frac{3}{8}\,\phi^{2}-\frac{1}{32}\,\phi^{4}-\frac{\alpha\left(1-4\alpha\right)}{32}\,\phi^{6}-\frac{\alpha^{2}}{128}\,\phi^{8}\,.$
(47)

See Figure 1 for three examples. The conformal weight of the dual operator is
then given by $\Delta=\tfrac{1}{2}$ or $\Delta=\tfrac{3}{2}$, since
$m^{2}=-\tfrac{3}{4}$, and the value of the quartic self-interaction constant
is fixed to $v_{4}=-\frac{1}{32}$. In the present work we always make the
choice $\Delta=\tfrac{3}{2}$ and consider exclusively the potential (47) for
our examples. We comment on generalizations to arbitrary $m^{2}$ (respecting
the Breitenlohner–Freedman bound Breitenlohner:1982bm ; Breitenlohner:1982jf )
and potentials of other form than (46) in sections 6.2 and 7.

Figure 1: Potential (47) for three different holographic RG flows.

Extrema of the potential are obtained when either the superpotential has an
extremum, $W^{\prime}(\phi)=0$, leading to the real solution $\phi=\phi_{0}=0$
(for negative $\alpha$ there is an additional extremum with real scalar field
$\phi=\phi_{n}=1/\sqrt{-\alpha}$), or when the superpotential obeys
$W^{\prime\prime}(\phi_{\pm})=W(\phi_{\pm})$, leading to the (potentially)
real solutions

$\phi_{\pm}^{2}=6-\frac{1}{\alpha}\pm\frac{\sqrt{1-24\alpha+36\alpha^{2}}}{\alpha}\,.$
(48)

The extremum corresponding to $\phi_{0}$ is always a maximum. The extrema
corresponding to $\phi_{+}$ are maxima and exist only for
$\alpha>\tfrac{1}{6}\,(2+\sqrt{3})$. The extrema corresponding to $\phi_{-}$
are maxima for $\alpha<0$ and minima for
$\tfrac{1}{6}\,(2+\sqrt{3})<\alpha<1$. The extrema corresponding to $\phi_{n}$
are always minima. In the range $0<\alpha<\tfrac{1}{6}\,(2+\sqrt{3})$ the only
extremum is at $\phi_{0}$. We are going to exploit this information when
investigating holographic RG flows between UV and IR conformal fixed points.

###  Ground states

Poincaré2-invariant domain wall solutions to the Einstein–Klein–Gordon
equations

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi\,\partial_{\nu}\phi-\frac{1}{4}(\partial\phi)^{2}g_{\mu\nu}-\frac{1}{2}Vg_{\mu\nu}$
(49a) $\displaystyle\nabla^{2}\phi$ $\displaystyle=\frac{\partial
V}{\partial\phi}$ (49b)

corresponding to the ground state of the dual deformed CFT can be found
easily. The domain wall parametrization of the metric

$\displaystyle\operatorname{d}\\!s^{2}=\operatorname{d}\\!\rho^{2}+e^{2A(\rho)}\left(-\operatorname{d}\\!t^{2}+\operatorname{d}\\!x^{2}\right)$
(50)

reduces the equations of motion (49) in terms of the superpotential to first-
order equations.

$\displaystyle\frac{\operatorname{d}\\!A(\rho)}{\operatorname{d}\\!\rho}=-\frac{1}{2}W(\phi(\rho))\qquad\qquad\frac{\operatorname{d}\\!\phi(\rho)}{\operatorname{d}\\!\rho}=\frac{\operatorname{d}\\!W(\phi(\rho))}{\operatorname{d}\\!\phi(\rho)}$
(51)

The asymptotic region in this parametrization is at $\rho\rightarrow\infty$.
Integrating the equations of motion (51) with the superpotential (46) yields

$\displaystyle\phi(\rho)$ $\displaystyle=\frac{{j}e^{-\rho/2}}{\sqrt{1-\alpha
j^{2}e^{-\rho}}}$ (52a) $\displaystyle A(\rho)$
$\displaystyle=\Big{(}1-\frac{1}{16\alpha}\Big{)}\rho-\frac{j^{2}}{16\left(e^{\rho}-\alpha
j^{2}\right)}+\frac{\log\left(e^{\rho}-\alpha j^{2}\right)}{16\alpha}\,$ (52b)

where the integration constant $j$ can be identified with the source of the
dual operator. For real field configurations the inequality $\alpha j^{2}<1$
holds.

Using the near boundary expansion of this solution one can show that the
expectation value of the boundary stress tensor and the operator dual to the
scalar field (126)-(128) vanish.

$\displaystyle\langle T_{ij}\rangle=0=\langle{O_{\phi}}\rangle$ (53)

This shows that this solution has vanishing free energy, $F=-\langle
T_{xx}\rangle$, and thus corresponds to the ground state of the dual field
theory. We shall see that even this elementary background exhibits remarkable
features in EE and QNEC2 studies.

###  Thermal states

We are also interested in more general (thermal) states and their entanglement
and QNEC2 properties. To describe their gravity duals we make the ansatz

$\displaystyle\operatorname{d}\\!s^{2}=e^{2A}\left(-H\operatorname{d}\\!t^{2}+\operatorname{d}\\!x^{2}\right)+e^{2B}\,\frac{\operatorname{d}\\!r^{2}}{H}$
(54)

where $A,B$ and $H$ are functions of the radial coordinate $r$ only. The
ansatz (54) encodes solutions that are invariant under spacetime translations.
If $H$ has a simple zero at some $r=r_{h}$, then the geometry is a black brane
with regular event- and Killing horizon at $r=r_{h}$, which gives rise to
finite temperature and entropy density of the dual field theory state.

The ansatz (54) has a residual gauge freedom, namely reparametrizations of the
radial coordinate. We fix this freedom by using Gubser gauge Gubser:2008ny ,
where the radial coordinate is identified with the corresponding value of the
scalar field.

$r:=\phi(r)$ (55)

The equations of motion (49),

$\displaystyle
H\left(B^{\prime}-2A^{\prime}\right)-H^{\prime}+e^{2B}V^{\prime}$
$\displaystyle=0$ (56) $\displaystyle
2\left(A^{\prime}B^{\prime}-A^{\prime\prime}\right)-1$ $\displaystyle=0$ (57)
$\displaystyle H^{\prime\prime}+(2A^{\prime}-B^{\prime})H^{\prime}$
$\displaystyle=0$ (58) $\displaystyle 2A^{\prime}H^{\prime}+H\left(4A^{\prime
2}-1\right)+2e^{2B}V$ $\displaystyle=0$ (59)

can be rephrased as a single master equation

$2G\left(GV^{\prime}+V\right)G^{\prime\prime}=\left(6GV^{\prime}+2V\right)G^{\prime
2}+\left(4G^{3}V^{\prime}+2G^{2}V^{\prime\prime}+4G^{2}V+3GV^{\prime}+V\right)G^{\prime}$
(60)

for the master field $G(\phi):=A^{\prime}(\phi)$, where prime denotes
derivatives with respect to $\phi$.

For a given potential $V$ a solution of (60) allows to express $A,B$ and $H$
in terms of integrals of simple functions of $G$ and $G^{\prime}$ only. The
solution for $A$ can be obtained by integrating the definition of $G$

$A(\phi)=A(\phi_{h})+\int\limits_{\phi_{h}}^{\phi}\mathrm{d}\phi^{\prime}G(\phi^{\prime})\,.$
(61)

The solution for $B$ follows from integrating (57)

$B(\phi)=B(\phi_{h})+\log\frac{G(\phi)}{G(\phi_{h})}+\int\limits_{\phi_{h}}^{\phi}\frac{\mathrm{d}\phi^{\prime}}{2G(\phi^{\prime})}\,.$
(62)

Knowing $B$ allows to express $H$ algebraically by combining (56), (57) and
(59)

$H(\phi)=-\frac{e^{2B}(V+GV^{\prime})}{G^{\prime}}\,.$ (63)

For certain simple choices of $V$ it is possible to solve the second order
ordinary differential equation (60) in closed form Gubser:2008ny , but in
general the master equation (60) needs to be solved numerically. In that case
it is useful to extract the divergent asymptotic behavior of the master field
$G$ inherited from the asymptotic behavior of $A$

$A(\phi)=\frac{\log(\phi)}{\Delta-2}+\ldots\quad\implies\quad
G(\phi)=\frac{1}{(\Delta-2)\phi}+\tilde{G}(\phi)$ (64)

where $\tilde{G}(\phi)$ remains finite at the boundary $\phi\to 0$. As
discussed in Gubser:2008ny  such a near boundary behavior of the fields
corresponds to a relevant deformation of the CFT2, namely

$\displaystyle\mathcal{L}=\mathcal{L}_{\textrm{CFT}_{2}}+j^{2-\Delta}\mathcal{O}_{\phi}\
.$ (65)

To find the equation of state, we set the source $j$ of the dual operator to
one in units of AdS radius, $j=1$. This leaves the horizon value of the scalar
field $\phi_{h}$ (which is equal to the horizon radius in Gubser gauge) as the
only free parameter.

##  Thermodynamics

In this section we analyze the thermodynamics of the system for different
choices of the scalar field (super-) potential resulting in different types of
phase transitions. We do this for a deformation by an operator with fixed
conformal weight $\Delta=\tfrac{3}{2}$, and modify the quartic term of the
superpotential (46) by choosing different values for $\alpha$. The entropy
density and the temperature of the system can be expressed in terms of horizon
data

$s=\frac{1}{4G_{3}}\,e^{A(\phi_{h})}\qquad\qquad
T=\frac{1}{4\pi}\,e^{A(\phi_{h})+B(\phi_{h})}\left|V^{\prime}(\phi_{h})\right|\,.$
(66)

Figure 2 displays free energy density $f$, entropy density over temperature
$s/T$ and the speed of sound squared $c_{s}^{2}$ for three different
potentials (we set Newton’s constant $G_{3}=1$ from now on in all numerical
results). By ‘density’ we mean that we divide by the trivial but infinite
volume along the black brane. The free energy of the boundary theory is
determined holographically from the on-shell action $\Gamma$ and given by the
purely spatial component of the boundary stress-tensor, which in turn is the
pressure

$\displaystyle f=-\langle T_{xx}\rangle=-p$ (67)

while the speed of sound squared of the boundary theory can be computed from
horizon- or boundary-data as

$\displaystyle c_{s}^{2}=\frac{\operatorname{d}\\!\
\ln{T}}{\operatorname{d}\\!\
\ln{s}}=\frac{\operatorname{d}\\!p}{\operatorname{d}\\!e}$ (68)

where

$e=\langle T_{tt}\rangle$ (69)

is the energy density of the boundary theory.

At large temperatures the entropy density becomes proportional to the
temperature. This indicates that in this limit the corresponding states in all
cases are close to the CFT2.

Figure 2: Free energy density (left), entropy density over temperature
(center), speed of sound squared (right). Blue: crossover ($\alpha=0$).
Orange: second order phase transition ($\alpha=0.16$). Magenta: first order
phase transition ($\alpha=0.32$). Vertical gray: locus of phase transition.
Dashed: thermodynamically unstable solutions with imaginary speed of sound.

For $\alpha<0.1605$ the system undergoes a smooth crossover as the temperature
increases.333Hereafter we refer to this specific value as $\alpha\approx 0.16$
for brevity. For $\alpha=0$ the speed of sound has a global minimum at
$T\approx 0.01$ and we call this the critical temperature of the crossover.
For this choice, the free energy, the entropy density and the speed of sound
are smooth functions of temperature. For $\alpha\approx 0.16$ we find a second
order phase transition. The speed of sound vanishes at the critical
temperature $T_{c}\approx 0.024$ and the entropy density shows critical
behavior close to $T_{c}$

$s(T)=s_{0}+s_{1}\left(\frac{T-T_{c}}{T_{c}}\right)^{1-\gamma}$ (70)

where we estimate $\gamma\approx 0.66$ for the critical exponent (see Figure
3) and $s_{1}\approx 0.02$.

Figure 3: Entropy density close to second order phase transition. Orange dots:
numerical results. Green line: fit (70) with $\gamma=0.66$ and $s_{1}=0.02$.

For $\alpha>0.16$ the system has a first order phase transition. On the
gravity side it follows from the existence of three different black brane
solutions around the critical point. They have the same Hawking temperature
but different free energies. In this family we choose $\alpha=0.32$ as
example, which leads to a first order phase transition at $T_{c}\approx 0.04$
between a large and small black brane geometry. The imaginary speed of sound
in the middle branch (dashed line in Figure 2) corresponds to unstable
quasinormal modes in the sound channel which is known as spinodal instability
Chomaz:2003dz . The dynamics of the first order phase transition was studied
in detail in 4D Janik:2017ykj  and 5D Attems:2017ezz  and a static black brane
geometry with inhomogenous horizon and Hawking temperature $T=T_{c}$
Janik:2017ykj ; Bea:2020ees  identified as the thermodynamically stable
classical solution. This solution is a mixed state dual to an inhomogeneous
arrangement of large and small black branes of equal temperature $T_{c}$.

We checked that different choices for $\Delta$ lead to qualitatively
equivalent thermodynamic properties. In particular, the critical exponent
$\gamma$ turns out to be remarkably universal. By computing $\gamma$ for
various values of the conformal scaling dimension, for different forms of the
potential and for different numbers of spacetime dimensions we always find
values consistent with $\gamma=2/3$ discovered in Gubser:2008ny  for $d=4$. In
Table 1 we summarize the different cases we have checked. In each row we show
the boundary dimension $d$ in the first column and the (super)potential in the
second column. For each row we sampled (equi-distantly) nine different values
of conformal weights of the dual scalar operator in the interval given in the
third column. The last column shows the mean and standard deviation of the
critical exponent $\gamma$ defined in (70) averaged over the sample of nine
different conformal weights.

$d$ | (super)potential | $\Delta$ | $\gamma$
---|---|---|---
2 | $W(\phi)=-2+\frac{\Delta-2}{2}\,\phi^{2}+{\widetilde{\alpha}}\,\phi^{4}$ | $[1,2]$ | 0.6681 $\pm$ 0.0041
2 | $V(\phi)=-2\cosh(\frac{\phi}{\sqrt{3}})+\frac{3\Delta^{2}-6\Delta+2}{6}\,\phi^{2}+{\widetilde{\alpha}}\,\phi^{4}$ | $[1,2]$ | 0.6658 $\pm$ 0.0062
3 | $W(\phi)=-4+\frac{\Delta-3}{2}\,\phi^{2}+{\widetilde{\alpha}}\,\phi^{4}$ | $[\frac{3}{2},3]$ | 0.6670 $\pm$ 0.0045
3 | $V(\phi)=-6\cosh(\frac{\phi}{\sqrt{3}})+\frac{\Delta^{2}-3\Delta+2}{2}\,\phi^{2}+{\widetilde{\alpha}}\,\phi^{4}$ | $[\frac{3}{2},3]$ | 0.6635 $\pm$ 0.0055
4 | $W(\phi)=-6+\frac{\Delta-4}{2}\,\phi^{2}+{\widetilde{\alpha}}\,\phi^{4}$ | $[2,4]$ | 0.6667 $\pm$ 0.0078
4 | $V(\phi)=-12\cosh(\frac{\phi}{\sqrt{2}})+\frac{\Delta^{2}-4\Delta+6}{2}\phi^{2}+{\widetilde{\alpha}}\phi^{4}$ | $[2,4]$ | 0.6691 $\pm$ 0.0080
Table 1: Means and standard deviations of critical exponents averaged over
conformal weights

From Table 1 we see that our hypothesis that the critical exponent $\gamma$ is
independent from the dimension, the choice of potential and the conformal
weight is supported by the data. Using the whole data set of 54 data points
yields the critical exponent averaged over dimensions, potentials and
conformal weights.

$\bar{\gamma}=0.6667\pm 0.0062$ (71)

The averaged critical exponent (71) is in excellent agreement with our
hypothesis that the true critical exponent always is given by $\gamma=2/3$.

##  Entanglement entropy

In this section we calculate EE for our theory holographically using the RT
formula. In some cases this means we need to numerically determine geodesics.

In section 5.1 we summarize briefly the main formulas for the special class of
geometries discussed in sections 3.2 and 3.3. In section 5.2 we focus on
ground state entanglement, which already exhibits non-trivial features such as
the appearance of different branches that we discuss in detail. In section 5.3
we calculate EE holographically for thermal states, finding a rich phase
structure.

###  Entanglement entropy from geodesics

We adopt the algorithm described in section 2.6 for QNEC2 to calculate EE
holographically, basically by setting the deformation parameter $\lambda$ in
that section to zero, specializing to static metrics

$\displaystyle\operatorname{d}\\!s^{2}=g_{tt}(z)\,\operatorname{d}\\!t^{2}+g_{zz}(z)\,\operatorname{d}\\!z^{2}+g_{xx}(z)\,\operatorname{d}\\!x^{2}\,.$
(72)

EE for an interval with a separation $\ell$ along the spatial direction $x$
holographically is given by the geodesic length [times a factor $1/(4G_{3})$]
Ryu:2006bv

$S_{\textrm{EE}}(\ell)=\frac{1}{2G_{3}}\int\limits_{z_{\ast}}^{z_{\textrm{\tiny{cut}}}}\operatorname{d}\\!z\,\frac{\sqrt{g_{zz}(z)}}{\sqrt{1-\frac{g_{xx}(z_{\ast})}{g_{xx}(z)}}}$
(73)

where we assumed that the asymptotic AdS boundary is at $z=0$, the cutoff
surface that regulates UV divergences is at $z=z_{\textrm{\tiny{cut}}}$, and
the geodesic hanging from the boundary into the bulk has a turning point at
$z=z_{\ast}$. The quantity ${\cal A}_{0}(\ell)/(4G_{3})$ in (29) is a
renormalized version of EE (73) where the cutoff term is subtracted. Using
renormalized EE has the technical advantage that the integral extends from $0$
to $z_{\ast}$ and thus is a bit simpler to perform and also more stable to
evaluate numerically, if required. Whenever we evaluate EE (73) numerically we
set Newton’s constant to unity, $G_{3}=1$.

The Noether charge (23) establishes
$\dot{z}=\sqrt{g_{xx}(z)/g_{zz}(z)}\,\sqrt{g_{xx}(z)/g_{xx}(z_{\ast})-1}$,
which allows to evaluate the spatial interval integral (27). The ensuing
relation between the turning point $z_{\ast}$ and the spatial separation
$\ell$

$\displaystyle\ell=2\int\limits_{0}^{z_{\ast}}\frac{\operatorname{d}\\!z}{g_{xx}(z)}\,\frac{\sqrt{g_{zz}(z)g_{xx}(z_{\ast})}}{\sqrt{1-\frac{g_{xx}(z_{\ast})}{g_{xx}(z)}}}$
(74)

may not be one to one and thus can lead to several branches, as we shall see
explicitly in some of the examples below.

We are going to evaluate the integrals above numerically for arbitrary
interval lengths $\ell$ and perturbatively in the limits of small and large
$\ell$ (the former works since we need to know only the asymptotic expansion
of the metric, while the latter works only without numerics if we have closed
form expressions for the metric). For the numerical evaluation one can use
simple methods, such as Mathematica’s NIntegrate.

The perturbative treatment at small $\ell$ schematically works as follows. The
turning point has to be close to the AdS boundary, so we parametrize it using
a parameter $\delta$ that tends to zero when the AdS boundary is approached
(e.g. in the coordinates used above we could use $z_{\ast}=\delta$) and make
expansions in $\delta$ for all quantities. This simplifies the integrals so
that often they can be performed in closed form. The integral (74) then
establishes a series in $\delta$ for the turning point $z_{\ast}$ as function
of the interval length $\ell$. Plugging this series into the renormalized
version (29) of the area integral (73) by virtue of the RT formula then leads
to a generalized power series for EE

$S(\ell\ll
1)=\frac{c}{3}\,\ln\ell+\sum_{n=0}^{\infty}s_{n}\ell^{n}+S_{\textrm{\tiny{cut}}}$
(75)

where the universal leading order term comes from the asymptotic AdS behavior
and $S_{\textrm{\tiny{cut}}}$ is an $\ell$-independent term depending on the
UV cutoff that drops out in renormalized EE. The Taylor–Maclaurin coefficients
$s_{n}$ are determined from relatively simple integrals, as described above.
We are going to display the first few coefficients for the examples studied in
subsection 5.2 below.

At large $\ell$ the situation is similar, but technical details regarding the
evaluation of integrals are quite different, essentially because one can get
arbitrarily close to branch cuts in integrands. There are basically three
different scenarios at large $\ell$.

In the first one the radial coordinate, and thus the turning point, is
unbounded, by which we mean that there is neither a center nor a horizon that
would provide an IR cutoff on the radial coordinate. If this happens the
situation is similar to the small $\ell$ expansion, except that the integrals
now extend into the deep IR. As a technical consequence the final expression
for EE is not a generalized power series (75) but can have a more complicated
functional behavior on the (large) interval $\ell$. For example, as we shall
see in the next subsection the leading order term can start with a double log,
$S(\ell\gg 1)\propto\ln\ln\ell+{\cal O}(1)$.

In the second scenario there is an IR cutoff at $z=z_{\textrm{\tiny{cen}}}$
due to a center (which may be a regular center or feature a curvature
singularity). The turning point is then close to this IR cutoff, so we
introduce a small parameter $\delta$, e.g.
$z_{\ast}=z_{\textrm{\tiny{cen}}}\,(1-\delta)$. While the integrals now remain
bounded in the IR, the relation between the expansion parameter $\delta$ and
the (large) interval length $\ell$ again leads to an expression for EE that is
not a generalized power series of the form (75). In the example studied in the
next subsection the leading behavior is going to be $S(\ell\gg 1)\propto
S_{0}+1/\ln\ell+{\cal O}(\ln\ln\ell/\ln^{2}\ell)$ with some constant $S_{0}$.
The physical reason behind these drastic changes of EE is the IR behavior of
the Casini–Huerta $c$-function.

Finally, in the third scenario there is an IR cutoff at $z=z_{h}$ due to a
black brane horizon with inverse temperature $\beta$. Again the turning point
is close to this IR cutoff, so we use e.g. $z_{\ast}=z_{h}\,(1-\delta)$. In
this scenario the small parameter $\delta$ turns out to be suppressed
exponentially in the (large) interval length,
$\delta\sim\exp{(-2\pi\ell/\beta)}$. If one is interested only in the leading
order term in EE then no complicated integrals need to be evaluated and the
final result is the volume law (8) plus exponentially suppressed corrections.
If EE is calculated for QNEC2 purposes the story is conceptually the same, but
technically slightly more complicated since the structure of the integrals is
more delicate. We discuss this rather explicitly in appendix C for the domain
wall and black brane examples studied in section 6 below.

###  Ground states

We calculate now EE for ground states dual to the domain wall solutions (52).
These states are Poincaré invariant and have vanishing temperature, so a
useful cross-check on the correctness of the results is positivity and
monotonicity of the Casini–Huerta $c$-function, see section 2.8. There are
three qualitatively different possibilities for the parameter $\alpha$ in the
potential (47): Case 0, where $\alpha=0$, Case I, where $0<\alpha<1$ and Case
II, where $\alpha<0$.

Geometrically, the difference between these three cases is as follows. Case 0
develops a curvature singularity at large negative values of the radial
coordinate $\rho$ in the domain wall metric (50). Case I has a curvature
singularity at the finite value $\rho=\ln\alpha$. Case II has a second locally
asymptotically AdS3 region at large negative values of $\rho$, with AdS-radius
$\ell^{\textrm{\tiny{IR}}}_{\textrm{\tiny{AdS}}}$ given by

$\frac{\ell^{\textrm{\tiny{IR}}}_{\textrm{\tiny{AdS}}}}{\ell^{\textrm{\tiny{UV}}}_{\textrm{\tiny{AdS}}}}=\frac{1}{1-1/(16\alpha)}$
(76)

which is smaller than its UV pendant. From these geometric considerations we
expect the Casini–Huerta $c$-function to flow to the trivial value
$c_{\textrm{\tiny{IR}}}\to 0$ for Cases 0 and I, while for Case II it should
flow to the non-trivial value
$c_{\textrm{\tiny{IR}}}=c_{\textrm{\tiny{UV}}}/(1-1/(16\alpha))$ where
$c_{\textrm{\tiny{UV}}}=c$ is the Brown–Henneaux central charge. Below we
show, among other things, that these expectations turn out to be correct.

We consider first Case 0. The metric function (52b) simplifies to
$A=\rho-\exp(-\rho)/8$. The integrals in section 2.6 444In order to compare
with sections 2.6 and 5.1 we need to redefine the radial coordinate as
$z=\exp{(-\rho)}$. The respective turning points
$z_{\ast}=\exp{(-\rho_{\ast})}$ and UV cutoffs
$z_{\textrm{\tiny{cut}}}=\exp{(-\rho_{\textrm{\tiny{cut}}})}$ are related
correspondingly.

$\displaystyle\ell e^{A_{\ast}}$
$\displaystyle=2\int\limits_{\rho_{\ast}}^{\infty}\operatorname{d}\\!\rho\,\Big{(}\frac{1}{\sqrt{1-y}}-\sqrt{1-y}\Big{)}$
$\displaystyle A_{\ast}$ $\displaystyle:=A(\rho_{\ast})$ (77)
$\displaystyle{\cal A}_{\textrm{\tiny{ren}}}(\ell)$ $\displaystyle=\ell
e^{A_{\ast}}-2\rho_{\ast}+2\int\limits_{\rho_{\ast}}^{\infty}\operatorname{d}\\!\rho\,\big{(}\sqrt{1-y}-1\big{)}$
$\displaystyle y$ $\displaystyle:=\exp{\big{(}2A_{\ast}-2A(\rho)\big{)}}$ (78)

can be evaluated perturbatively for small $\ell$ (using
$\rho_{\ast}=-\ln\delta$ with $\delta\ll 1$, see appendix C)

$S_{0}(\ell\ll
1)=\frac{c}{3}\,\ln\ell-\frac{c\pi}{192}\,\ell+\frac{c(64-3\pi^{2})}{73728}\,\ell^{2}-\frac{c\pi(\pi^{2}-5)}{1179648}\,\ell^{3}+{\cal
O}(\ell^{4})+S_{\textrm{\tiny{cut}}}$ (79)

and for large $\ell$ (using $\rho_{\ast}=\ln\delta$ with $\delta\ll 1$)

$S_{0}(\ell\gg 1)=\frac{c}{3}\,\ln\ln\ell+c\,\ln 2-\frac{c\,\ln
2}{\ln\ell}+{\cal O}(1/\ln^{2}\ell)+S_{\textrm{\tiny{cut}}}\,.$ (80)

The Casini–Huerta $c$-function (38) in these limits

$\displaystyle c(\ell\ll 1)$
$\displaystyle=c\,\Big{(}1-\frac{\pi}{64}\,\ell+\frac{64-3\pi^{2}}{12288}\,\ell^{2}+{\cal
O}(\ell^{3})\Big{)}$ (81a) $\displaystyle c(\ell\gg 1)$
$\displaystyle=\frac{c}{\ln\ell}+\frac{3c\ln 2}{\ln^{2}\ell}+{\cal
O}(1/\ln^{3}\ell)$ (81b)

is indeed a strictly monotonically decreasing positive function of $\ell$. The
large $\ell$ behavior in (81) shows that the central charge flows to zero in
the IR in such a way that for an RG scale given by the (large) interval $\ell$
we have $c_{\textrm{\tiny{UV}}}=c_{\textrm{\tiny{IR}}}\,\ln\ell+\dots$. For
Case I the small $\ell$ computation is very similar to Case 0 and yields

$S_{I}(\ell\ll 1)=S_{0}(\ell\ll
1)-\frac{c\alpha\,\ell^{2}}{192}+\Big{(}\frac{11\alpha}{98304}-\frac{\alpha^{2}}{1536}\Big{)}\,\pi
c\,\ell^{3}+{\cal O}(\ell^{4})$ (82)

which has a smooth limit to (79) for vanishing $\alpha$. At large $\ell$ there
is a qualitative change: there is a minimal value of the radial coordinate
$\rho_{\textrm{\tiny{min}}}=\ln\alpha$, as opposed to Case 0 where $\rho$
could take arbitrarily negative values. This implies a different large $\ell$
behavior of EE

$S_{I}(\ell\gg
1)=-\frac{c}{3}\,\ln\alpha-\frac{c}{48\alpha\,\ln\ell}+\frac{c\,(32\alpha-1)}{768\alpha^{2}}\,\frac{\ln\ln\ell}{\ln^{2}\ell}+{\cal
O}(1/\ln^{2}\ell)+S_{\textrm{\tiny{cut}}}$ (83)

that has no smooth limit to (80) for vanishing $\alpha$. The fact that EE does
not grow at large $\ell$ can be understood geometrically from the presence of
a center at $\rho=\rho_{\textrm{\tiny{min}}}$: since most of the geodesic is
close to this center at large values of $\ell$ and the center is geometrically
a point, there is practically no contribution to the length of the geodesic,
which means that changing $\ell$ from a large to an even larger value almost
does not affect the geodesic length. It would be interesting to find a field
theoretic explanation of this plateau behavior of EE at large $\ell$.

Also for Case I the Casini–Huerta $c$-function (38) is a positive strictly
monotonically decreasing function of $\ell$ in both limits. The relation
between UV and IR values now reads
$c_{\textrm{\tiny{UV}}}=c_{\textrm{\tiny{IR}}}\,16\alpha\,\ln^{2}\ell+\dots\,\,$.

Case II, $\alpha<0$, is the only situation where we flow to another CFT fixed
point in the IR. It is possible to evaluate all geodesic integrals exactly in
the limit of large negative $\alpha$, yielding a result for EE

$S_{II}(\ell\gg 1)=\frac{c_{\textrm{\tiny{IR}}}}{3}\,\ln\ell+\dots$ (84)

that obeys the area law (6), but with a central charge that is smaller than
the UV result

$c_{\textrm{\tiny{IR}}}=\frac{c_{\textrm{\tiny{UV}}}}{1-1/(16\alpha)}\,.$ (85)

The ellipsis in (84) denotes terms that are $\ell$-independent or vanish at
large $\ell$, as well as terms that vanish like $\alpha^{-3}$. Despite the
assumption of large absolute values of $\alpha$, the IR value of the central
charge (85) is correct at arbitrary negative values of $\alpha$. Consistently,
the ratio of IR and UV central charges coincides with the ratio of IR and UV
AdS radii (76). The right plot in Figure 4 shows an example of this case where
$\alpha=-1$. The red solid line is the numerical results and the green dashed
line is the IR expression given in (85). By contrast, the left plot in Figure
4 displays the Casini–Huerta $c$-function for the Case I example
$\alpha=0.32$.

Figure 4:  Left plot: Casini–Huerta $c$-function for $\alpha=0.32$. Blue and
red solid lines: short and long RT surfaces on left and right hand sides of
kink at $\ell^{\ast}$. Orange and green dashed lines: perturbative results for
small (82) and large (83). Right plot: Casini–Huerta $c$-function for
$\alpha=-1$. Blue solid: numerical result. Orange dashed line: UV result,
$c_{\textrm{\tiny{UV}}}=3/2$. Green dashed line: IR result (85),
$c_{\textrm{\tiny{IR}}}=24/17\approx 1.41$.

In order to get EE beyond the small or large interval expansions we resort to
numerics and study three different values $\alpha=0.32,\,0.16,\,0$ of the free
parameter in the potential (47), i.e. two examples of Case I and Case 0. Since
Case II does not lead to interesting phase transitions we do not continue
studying it numerically. The central value of the radial coordinate can be
seen in the left panel of Figure 5 where we plot $e^{2A(z)}$ for the three
cases together with the corresponding metric function $1/z^{2}$ of pure AdS3.
For small values of the radial coordinate $z$ all geometries converge to AdS3.
For $\alpha>0$ the geometry degenerates to a central point at
$z=z_{\textrm{\tiny{cen}}}=\tfrac{1}{\alpha}$ where
$e^{2A(z_{\textrm{\tiny{cen}}})}=0$.

Figure 5: Left: Scale factor in domain wall geometry (50) for different
potentials. Right: Renormalized EE for four cases in left plot.

In the right panel of Figure 5 we show renormalized EE as function of $\ell$.
As expected from the discussion in section 2.8 we find that EE is a monotonic
function of $\ell$.

Asymptotic AdS3 boundary conditions in the UV (= small $z$) yield the area law
(6) at small $\ell$. Geometrically this follows from the fact that RT surfaces
with small $\ell$ reside predominantly in the asymptotically AdS3 part of the
geometry. This can be seen in the right plot of Figure 5, where at small
$\ell$ all curves collapse to the universal scaling of the vacuum state dual
to empty AdS3 (blue curve), compatible with the perturbative results (79) and
(82).

The IR (= large $z$) properties of the geometry determine the large $\ell$
behavior of EE. For cases where the geometry ends at a finite value of $z$ EE
develops a plateau in the sense that increase of entanglement with $\ell$ is
suppressed, as evident from the perturbative result (83).

The UV and IR regimes are well-described by the perturbative small and large
$\ell$ results above. We studied numerically the region at intermediate $\ell$
and found that its details depend crucially on the value of the parameter
$\alpha$. Namely, while EE is a monotonic and continuous function of $\ell$ in
all cases, there can be kinks, meaning that left and right derivatives can be
different at some values of $\ell$. The geometric reason for such kinks is the
existence of several saddle points in the area functional, which give rise to
several extremal surfaces with the same $\ell$ but different areas. For
$\alpha<\alpha^{\ast}\approx 0.277$ the extremalization has a unique solution,
but for each $\alpha>\alpha^{\ast}$ there exists a finite range of $\ell$
where the area functional has three different saddle points. EE is associated
only with the dominant saddle point, defined as the one with the smallest
surface area. At a critical interval $\ell^{\ast}$ two of the saddle points
exchange dominance. As a consequence the first derivative of EE is
discontinuous, which is an example for kinked entanglement discussed in
section 2.7.

Figure 6:  Left: EE in intermediate region. Right: Minimal surfaces in domain
wall geometry.

We illustrate this situation in Figure 6 for the domain wall with
$\alpha=0.32$. In the left plot we show the renormalized EE as a function of
$\ell$ in the interesting region around the kink at $\ell^{\ast}$. In the
range $3.870<\ell<4.058$ the area functional has three saddle points resulting
in three different branches for the surface area shown in blue, red and gray.
Dominant branches are shown as solid lines and sub-dominant ones with dashed
lines. The red branch ceases to exist for $\ell<3.870$ and the blue one for
$\ell>4.058$. At $\ell^{\ast}\approx 3.988$ the saddle points for the blue and
red curve exchange dominance, while the third saddle point represented by the
gray dashed line remains always sub-dominant. The plot on the right is a bulk
picture of the corresponding RT surfaces. The black line indicates
$z_{\textrm{\tiny{cen}}}$ where the geometry degenerates to a point and the
dark gray region at $z>z_{\textrm{\tiny{cen}}}$ is not part of the geometry.
Solid red and blue curves are representatives of minimal surfaces in the
respective dominant branches. They can exist in the red and blue colored
regions only. The dashed red and blue curves are subdominant surfaces with
smallest and larges separation in the two branches marked by red and blue dots
in the left plot.

Curiously, the gray area remains untouched by dominant surfaces and is only
traced out by sub-dominant ones. Thus, modulo $x$-translations of the interval
these gray areas are entanglement shadows Balasubramanian:2014sra , i.e. bulk
regions which no RT-surface can probe. However, permitting $x$-translations of
the interval casts light on these shadows. This means that a field theory
observer trying to reconstruct the bulk geometry will not be able to do so
just by extending their entangling intervals to all possible values of
interval length $\ell$; instead, they also have to shift the central position
of the interval in order to reconstruct the gray areas in the bulk.

###  Thermal states

We consider now EE of thermal states that are dual to black brane geometries
discussed in section 3.3, first perturbatively and then numerically.

The perturbative analysis is most interesting in the large $\ell$ limit, since
for small $\ell$ the existence of a horizon at finite temperature has almost
no effect on EE. We consider additionally the limit of large temperature,
which means that geometry is dominantly a BTZ black brane. This implies that
the scalar field provides only a small perturbation of the thermal background,
so one can solve the Klein–Gordon equation on a BTZ black brane background and
then consider leading order backreactions on that background. We shall do this
analysis in detail in section 6.3 below when analyzing QNEC2. The result (103)
for renormalized area yields the anticipated volume law (8) plus corrections
that are suppressed exponentially like $e^{-2\pi\ell/\beta}$, as expected from
Fischler:2012ca .

In the remainder of this section we discuss numerical results for EE in Gubser
gauge (54)-(55), but using as radial coordinate $z=\phi^{2}/\phi_{h}^{2}$
where $\phi_{h}$ is the value of the scalar field at the horizon. The metric
then expands near the boundary $z=0$ as

$\displaystyle\operatorname{d}\\!s^{2}=\frac{\phi_{h}^{4}\,\big{(}-\operatorname{d}\\!t^{2}+\operatorname{d}\\!x^{2}\big{)}+\operatorname{d}\\!z^{2}}{z^{2}}+\textrm{subleading}\,.$
(86)

Renormalized EE and interval length are given by

$\displaystyle
S_{\textrm{\tiny{ren}}}=\frac{1}{2G_{3}}\int\limits^{z_{\ast}}_{z_{\textrm{\tiny{cut}}}}\left(\frac{e^{2B}}{H\left(1-e^{2(A_{\ast}-A)}\right)}\right)^{1/2}\operatorname{d}\\!z+\frac{1}{2G_{3}}\frac{\log(z_{\textrm{\tiny{cut}}})}{2-\Delta}$
(87)
$\displaystyle\ell=2\int\limits_{0}^{z_{\ast}}\frac{1}{e^{2A}}\left(\frac{e^{2(B+A_{\ast})}}{H\left(1-e^{2(A_{\ast}-A)}\right)}\right)^{1/2}\operatorname{d}\\!z\,.$
(88)

We evaluated the metric functions $A$, $B$ and $H$ and the two integrals above
numerically.

Consider first again the large separation behaviour of EE. As stated above, EE
obeys the volume law (8) with exponentially small corrections suppressed by
$e^{-2\pi\ell/\beta}$, which provides a simple cross-check on the correctness
of the numerics. For purposes of displaying results it is useful to plot EE
density

$\displaystyle\sigma(\ell,T):=\frac{\operatorname{d}\\!S_{\textrm{\tiny{ren}}}}{\operatorname{d}\\!\ell}$
(89)

since it is approximately constant at large values of $\ell$. In Figure 7 we
show the temperature dependence of $\sigma(\ell,T)$ for several intervals. We
use the potential (47) with $\alpha=0.32$, which leads to a first order phase
transition between large and small black branes at $T=T_{c}$. As expected from
the volume law, for large separations $\sigma(\ell,T)$ approaches the entropy
density for all black brane geometries. The region around the critical
temperature is more involved and shows that at the point where large black
brane branch and spinodal unstable branch join together the agreement between
EE and thermal entropy can be seen only for very large intervals.

Figure 7: Comparing entropy density $s/T$ (Gray solid) with EE density
$\sigma/T$ (Colored dashed) defined in (89) for intervals
$\ell=5,\,10,\,15,\,20$.

In Figure 8 we show the renormalized EE as a function of the interval $\ell$
for different thermal states in the theory with $\alpha=0.32$ which has a
first order phase transition. A selection of black brane solutions is shown as
colored points in the left panel. The labels for each of these points
correspond to the horizon value of the scalar field. We use the same colors
for the renormalized EE in the right panel.

Figure 8: Left panel: Entropy density over temperature for first order phase
transition ($\alpha=0.32$). Dots: horizon value of scalar field,
$\phi_{h}=1,3,5,8,10$. Right panel: EE computed using (87) for different black
brane solutions as function of interval $\ell$. Colors correspond to solutions
marked with points in left panel.

In the following we summarize the main features of EE in theories with first
order phase transitions. For small intervals the turning point of the geodesic
is close to the boundary where the metric is essentially locally AdS3.
Therefore, for all thermal states EE approaches the same value in this regime.
For large intervals EE obeys the volume law. This is what one expects since
most of the geodesic is close and parallel to the horizon as discussed around
Figure 7. In the insertion of the right panel in Figure 8 we show a novel
feature of EE for thermal states at low temperatures dual to small black
branes. The RT surface is a multivalued function for a range of intervals,
which is similar to the case we studied extensively in previous subsection for
the ground state of the same theory. We found this phenomenon for
$\phi_{h}>5.6$, which is on the spinodal branch but close to the joining point
to the small black brane branch. The smaller the value of $\phi_{h}$, the
larger the critical value of the interval at which the transition between the
RT surfaces occurs and the smaller the range of intervals with multivalued
surfaces. We did not find this phenomenon in theories with crossover or second
order phase transitions.

In Figure 9 we show the renormalized EE as a function of $\ell$ for various
thermal states in a theory with second order phase transition ($\alpha=0.16$).
In contrast to the previously discussed case with first order phase
transition, in a theory with second order phase transition EE is never kinked
or discontinuous, but always a monotonic and smooth function of $\ell$ and
$T$. From the result for $\phi_{h}=8$ (dotted blue line) for example we can
see that for large $\ell$ EE has only a very mild, plateau-like, volume law
scaling at low $T$. The reason for this is the small value of the entropy
density (66) which gives the leading contribution to the slope of EE (89) at
large $\ell$. On the other hand EE shows a very rapid volume law scaling at
large $T$, because the corresponding value of the thermal entropy is large. An
example for this is the red curve for $\phi_{h}=1$ in the right plot of Figure
9.

Figure 9: Left panel: Entropy density over temperature for second order phase
transition ($\alpha=0.16$). Dots: horizon value of scalar field,
$\phi_{h}=1,3,5,8,10$. Right panel: EE computed using (87) for different black
brane solutions as function interval $\ell$. Colors correspond to solutions
marked with points in left panel.

Finally, in Figure 10 we show the renormalized EE as a function of $\ell$ in a
theory with crossover ($\alpha=0$). Also in this case EE is monotonic and
smooth. As discussed in subsection 5.2 $\alpha=0$ corresponds to Case 0 in
which the ground state EE is not suppressed at large $\ell$, but grows like
$\frac{c}{3}\ln\ln\ell$ instead, cf. (80).

Figure 10: Left panel: Entropy density over temperature for crossover
($\alpha=0$). The dots show the horizon value of the scalar field at each
point, $\phi_{h}=1,3,5,8,10$. Right panel: EE computed by using the (87) for
different black hole solutions as function of separation. Colors correspond to
solutions marked with points in left panel.

As a consequence at low temperatures the volume law scaling is approached
extremely slow. This can be seen for example from the result for $\phi_{h}=8$
(dotted blue line) shown in the right Figure 10. The scaling of EE at large
$T$ (red line in right Figure 10) is qualitatively similar to the other two
cases discussed above. This universal (theory independent) behavior at large
$T$ follows from (103) obtained perturbatively in subsection 6.3.

We conclude this section with a brief summary table showing renormalized EE
for various states dual to geometries that we studied in the limits of small
and large intervals. QFT refers specifically to the deformed CFT that we
study; the subscript refers to the sign of the parameter $\alpha<1$ in the
scalar potential (47).

State | Geometry | Small interval $\ell\ll 1$ | Large interval $\ell\gg 1$
---|---|---|---
CFT ground | PP AdS | $\frac{c}{3}\,\ln\ell$ | $\frac{c}{3}\,\ln\ell$
QFT0 ground | Dom. wall | $\frac{c}{3}\,\ln\ell-\frac{c\pi}{192}\,\ell+{\cal O}(\ell^{2})$ | $\frac{c}{3}\,\ln\big{(}8\,\ln\frac{\ell}{8}\big{)}+{\cal O}(\ln^{-2}\ell)$
QFT+ ground | Dom. wall | $\frac{c}{3}\,\ln\ell-\frac{c\pi}{192}\,\ell+{\cal O}(\ell^{2})$ | $-\frac{c}{3}\,\ln(\alpha+\frac{1}{16\ln\ell})+{\cal O}(\ln^{-2}\ell)$
QFT- ground | Dom. wall | $\frac{c}{3}\,\ln\ell-\frac{c\pi}{192}\,\ell+{\cal O}(\ell^{2})$ | $\frac{c}{3(1-1/(16\alpha))}\,\ln\ell+{\cal O}(1)$
CFT thermal | BTZ | $\frac{c}{3}\,\ln\ell+{\cal O}(\ell^{2})$ | $\frac{c\pi\ell}{3\beta}+{\cal O}(e^{-2\pi\ell/\beta})$
Table 2: Renormalized EE for various systems at small and large intervals.

##  Quantum Null Energy Condition

In this section we present our results for QNEC2, starting with the analysis
of vacuum states in section 6.1, followed by remarks on how to predict phase
transitions using QNEC2 in section 6.2, and concluded by the analysis of
finite temperature states in section 6.3.

###  Ground states

For boost-invariant ground states small- and large-$\ell$ expansions of QNEC2
follow directly from the corresponding expansions of EE in section 5.2.
Applying (14) to the small $\ell$-expansion (82) yields

$\lim_{\ell\ll
1}\Big{(}S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\Big{)}=-\frac{c\pi}{64\ell}+\frac{c(128-3\pi^{2})}{18432}-\frac{53\pi+9\pi^{3}}{1179648}\,c\,\ell\\\
-\frac{c\alpha}{24}+\frac{229\pi}{98304}\,c\alpha\,\ell-\frac{5\pi}{512}\,c\alpha^{2}\ell+{\cal
O}(\ell^{2})\,.$ (90)

The result (90) is valid both for Case 0, I and II (in the former case only
the first line contributes). The first term in (90) is negative and thrice the
linear $\ell$-term in EE (79), compatible with the general results (19) and
(20). For the large $\ell$ expansion we need to discriminate between Case 0
($\alpha=0$), where we apply (14) to (80)

$\lim_{\ell\gg
1}\Big{(}S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\Big{)}=-\frac{2c}{3\ell^{2}\,\ln\ell}-\frac{c(6\ln
2-1)}{3\ell^{2}\,\ln^{2}\ell}+{\cal O}(1/(\ell^{2}\,\ln^{3}\ell))$ (91)

Case I ($0<\alpha<1$), where we apply (14) to (83)

$\lim_{\ell\gg
1}\Big{(}S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\Big{)}=-\frac{c}{24\alpha\,\ell^{2}\,\ln^{2}\ell}+{\cal
O}(1/(\ell^{2}\,\ln^{3}\ell))$ (92)

and Case II ($\alpha<0$), where we apply (17) to (85)

$\lim_{\ell\gg
1}\Big{(}S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\Big{)}=-\frac{2c_{\textrm{\tiny{IR}}}}{3\ell^{2}(1-16\alpha)}+\dots$
(93)

At intermediate values of $\ell$ we computed QNEC2 by numerically integrating
(73) for a sufficiently large number of values for $z^{\ast}$ which together
with (74) allows to express EE as function of $\ell$. In the following we do
this for several different examples and compare the results to the
perturbative expressions.

As a first example we chose $\alpha=0.32$ which lies comfortably in the
interesting regime $\alpha>\alpha^{\ast}\approx 0.27736$ where EE can be
kinked and QNEC2 discontinuous. In Figure 11 we show QNEC2 as function of
$\ell$ for the same setup and color coding as in Figure 6. Blue and red curves
again correspond to two different saddle points in the area functional. Dotted
orange and green curves are the perturbative small- and large-$\ell$ formulas.

Perturbation theory works remarkably well, except in a finite region close to
the critical separation $\ell^{\ast}$ where the discontinuity is located. As
shown in section 2.7 QNEC2 does not only jump but also has a divergence in
form of a negative delta function at $\ell^{\ast}$ which we indicate by the
black dashed line. While the precise value of $\ell^{\ast}$ and size of the
jump need to be determined numerically, we emphasize that the existence of the
delta-divergence can only be deduced from analytic considerations. For
$\ell<\ell^{\ast}$ QNEC2 is not monotonic, but has a local maximum that leads
to a finite gap.

Figure 11: QNEC2 for the domain wall solution with $\alpha=0.32$. Blue and red
solid lines: numerical results for QNEC2 in the small and large $\ell$
branches left and right of the critical interval $\ell^{*}$. Orange and green
dashed lines: perturbative results for small $\ell$ and large $\ell$ given in
(90) and (92), respectively.

The second example is $\alpha=0.16$ (see left Figure 12). In this case the
extremal surfaces are unique and EE is a smooth function of $\ell$. This
results in QNEC2 being smooth as well, even though is develops a shoulder
around $\ell\approx 10$. The transition from the UV- to the IR-scaling regime
clearly happens in this region.

Figure 12: QNEC2 for the domain wall solution with $\alpha=0.16$ (left) and
$\alpha=0$ (right). Red solid lines: numeric result for QNEC2. Orange and
green dashed lines: perturbative results for small $\ell$ and large $\ell$
given in (90) and (92), respectively.

Finally, the right plot in Figure 12 shows the case $\alpha=0$. Compared to
the other examples QNEC2 does not have any distinguished features in this
case. The extremal surfaces are unique, EE and QNEC2 are smooth and it is not
possible to localize the transition between UV- and IR scaling from the
numerical results.

###  Predicting phase transitions from ground state QNEC2

A striking consequence of our numerical analysis for intermediate $\ell$ is
the curious fact that QNEC2 in ground states has features that allow to
characterize the thermodynamic phase structure of the theory. If ground state
QNEC2 is non-monotonic in $\ell$, which happens in our one-parameter family of
theories for $\alpha>0.2$, then the theory has a first order phase transition
at finite $T$. Increasing $\alpha$ makes the local minimum that is responsible
for the non-monotonicity of QNEC2 more pronounced until it eventually turns
into a discontinuity for $\alpha\approx 0.277$. For $\alpha>0.277$ the
phenomenon of multiple RT-surfaces and the related jump (plus delta function)
in QNEC2 occur. Also in this case the theory always has a first order phase
transition. This is illustrated in Figure 13, showing ground state QNEC2 as
function of $\ell$ for different values of $\alpha$.

Figure 13: Ground state QNEC2 as function of $\ell$ for $0.2\leq\alpha\leq
0.3$.

Thus, spotting non-monotonicity or a jump in QNEC2 for the ground state allows
to predict a first order phase transition at some finite temperature.

Similarly, QNEC2 for the ground state is monotonic and has a smooth, but
clearly visible and localized transition region between the UV- and IR-scaling
regime in case the theory has a second order phase transition at finite $T$.
We show an example for this in the left plot of Figure 12. If the theory at
finite temperature has a smooth crossover then ground state QNEC2 is a smooth,
monotonic and featureless function of the interval $\ell$ without a clearly
localized transition region between the UV- and IR-scaling regime. An example
for this is shown in the right plot of Figure 12.

This means ground state QNEC2 can be used as diagnostic tool to identify and
locate phase transitions. While the identification of ground state QNEC2 and
thermodynamic phase structure seems remarkable, we find it not to be precisely
one-to-one. There are for example cases in which the theory has a first order
phase transition but QNEC2 is monotonic in $\ell$. In our model this is the
case for $0.16<\alpha\leq 0.2$. But the converse statement “if ground state
QNEC2 is non-monotonic then the theory has a first order phase transition” we
always find to be true. We have checked for the potentials listed in Table 1
(two-dimensional cases) that similar features arise rather generically
regardless of the value of the conformal weight, which suggests they are not
artefacts of our particular choice of model.

###  Thermal states

We consider now QNEC2 for thermal states that are dual to black brane
geometries discussed in section 3.3, using metrics of the form

$\operatorname{d}\\!s^{2}=\frac{1}{z^{2}}\,\Big{(}-A(z)\,\operatorname{d}\\!t^{2}+\frac{\operatorname{d}\\!z^{2}}{A(z)}+S(z)\,\operatorname{d}\\!x^{2}\Big{)}\,.$
(94)

Again we start with a perturbative analysis, where we are mostly interested in
the large $\ell$ limit. With no loss of generality we scale the temperature
such that $2\pi T=1$ so that our only rermaining scale is the interval length
$\ell$.

The exact expression for the scalar field

$\phi(z)=\epsilon\,\frac{4\sqrt{\pi}}{\Gamma(\tfrac{1}{4})^{2}}\,\sqrt{z}\,K\Big{(}\frac{1-z}{2}\Big{)}+{\cal
O}(\epsilon^{2})$ (95)

contains the backreaction parameter $\epsilon\ll 1$ and the elliptic integral
$K$. The metric functions with no loss of generality are given by

$A(z)=1-z^{2}+\epsilon^{2}\,A_{2}(z)+{\cal O}(\epsilon^{3})\qquad\qquad
S(z)=1+\epsilon^{2}S_{2}(z)+{\cal O}(\epsilon^{3})$ (96)

where for convenience we fixed the horizon to be located at $z_{h}=1$. The
near boundary expansions of these three functions are given by

$\displaystyle\phi(z)$
$\displaystyle=\epsilon\,\sqrt{z}\,\Big{(}1-\frac{4\pi^{2}}{\Gamma(\tfrac{1}{4})^{4}}\,z+\frac{1}{8}\,z^{2}+{\cal
O}(z^{3})\Big{)}$ (97) $\displaystyle A_{2}(z)$
$\displaystyle=-\frac{1}{4}\,z\,\ln z+a_{2}z^{2}+{\cal O}(z^{3})$ (98)
$\displaystyle S_{2}(z)$ $\displaystyle=-\frac{1}{4}\,z\,\ln
z+\frac{3\pi^{2}}{\Gamma(\tfrac{1}{4})^{4}}\,z^{2}+{\cal O}(z^{3})$ (99)

where we set to zero some of the leading and subleading coefficients, again
with no loss of generality.

The flux components of the boundary stress tensor

$2\pi\langle
T_{kk}\rangle=\frac{1}{4G_{3}}+\frac{\epsilon^{2}}{4G_{3}}\,\bigg{(}\frac{3\pi^{2}}{\Gamma(\tfrac{1}{4})^{4}}-a_{2}\bigg{)}+{\cal
O}(\epsilon^{3})$ (100)

can be expressed in terms of the near horizon quantity $S_{2}(1)$ due to the
existence of the radially conserved quantity

$Q=-\frac{S(z)^{3/2}}{2z}\frac{\operatorname{d}\\!}{\operatorname{d}\\!z}\frac{A(z)}{S(z)}\qquad\qquad\frac{\operatorname{d}\\!Q}{\operatorname{d}\\!z}=0$
(101)

which yields $Q=1+\epsilon^{2}\,[3\pi^{2}/\Gamma(\tfrac{1}{4})^{4}-a_{2}]$ in
the near boundary expansion and $Q=1+\tfrac{1}{2}\,\epsilon^{2}S_{2}(1)$ at
the horizon. Therefore, the left hand side of the QNEC2 inequality (2) reads

$2\pi\langle
T_{kk}\rangle=\frac{1}{4G_{3}}+\frac{\epsilon^{2}}{8G_{3}}\,S_{2}(1)+{\cal
O}(\epsilon^{3})\,.$ (102)

To get the right hand side of the QNEC2 inequality (2) we follow the procedure
in section 2.6. See appendix C for details on the integrals. The
$\lambda$-dependent result for the renormalized area is given by

${\cal A}^{\textrm{\tiny ren}}(\ell\gg
1)=(\ell+\lambda)\,\Big{(}1+\frac{\epsilon^{2}}{2}\,S_{2}(1)\Big{)}+\frac{\lambda^{2}\epsilon^{2}}{16}\,\big{(}S_{2}^{\prime}(1)-4S_{2}(1)\big{)}+{\cal
O}(e^{-\ell})+{\cal O}(\lambda^{3})+{\cal O}(\epsilon^{3})$ (103)

where prime means derivative with respect to $z$.

From the result (103) for the renormalized area we obtain the expected volume
law (8) plus corrections that are suppressed exponentially like $e^{-\ell}$ or
at least cubic in the backreaction parameter $\epsilon$. Moreover, we obtain
expressions for the quantities denoted by ${\cal A}_{1}$ and ${\cal A}_{2}$ in
(29).

${\cal A}_{1}(\ell)=1+\frac{\epsilon^{2}}{2}\,S_{2}(1)+\dots\qquad\qquad{\cal
A}_{2}(\ell)=\frac{\epsilon^{2}}{8}\,\big{(}S_{2}^{\prime}(1)-4S_{2}(1)\big{)}+\dots$
(104)

The ellipses denote terms suppressed by $e^{-\ell}$ or by $\epsilon^{3}$.
Therefore, the right hand side of the QNEC2 inequality (2) reads

$\partial_{\lambda}^{2}S+\frac{6}{c}\,\big{(}\partial_{\lambda}S\big{)}^{2}=\frac{c}{6}+\frac{c\epsilon^{2}}{12}\,S_{2}(1)+\frac{c\epsilon^{2}}{48}\,S_{2}^{\prime}(1)+{\cal
O}(e^{-\ell})+{\cal O}(\epsilon^{3})$ (105)

where for clarity we wrote explicitly the $\lambda$-derivatives and kept the
notation that prime denotes $z$-derivative.

Using $4G_{3}=6/c$ we see that left (102) and right (105) hand sides of QNEC2
almost cancel and the QNEC2 inequality turns into a convexity condition for
the function $S_{2}$ at the horizon.

$2\pi\langle
T_{kk}\rangle-\partial_{\lambda}^{2}S-\frac{6}{c}\,\big{(}\partial_{\lambda}S\big{)}^{2}=-\frac{c\epsilon^{2}}{48}\,S_{2}^{\prime}(1)\geq
0$ (106)

We have checked both numerically and analytically that this condition is
satisfied. It is also possible to argue on general grounds from the second law
of black hole mechanics that the inequality (106) has to be satisfied.555This
goes back to the genesis of the QNEC2 inequality, which was associated with
the quantum focusing conjecture and the generalized second law Bousso:2015mna
. Since the QNEC2 inequality must be respected the fact that (106) holds is
unsurprising. Namely, one can view our backreaction calculation as a
consequence of a dynamical process where initially $\epsilon$ was zero and
then a small amount of scalar matter is thrown into the black hole. The
quantity $S_{2}(1)$ is proportional to the additional entropy density
generated in this process and thus has to be positive. From a field theory
perspective it also has to be monotonically increasing with energy $\sim 1/z$.
This means that $S_{2}$ expressed as function of $z$ must be monotonically
decreasing, which is precisely the inequality (106).

We turn now to the evaluation of QNEC2 at arbitrary values of the interval
length $\ell$. In boost invariant ground states it is possible to write QNEC2
entirely in terms of $\ell$-derivatives of EE. For this a simple shooting
algorithm is sufficient to determine EE as function of $\ell$ and subsequently
QNEC2. The situation in black brane geometries is more complicated and one has
to evaluate genuine lightlike derivatives of EE. We do this using the
relaxation approach of Ecker:2019ocp  with either a known geodesic in AdS3 or
previously obtained numerical solutions as initial guess.

These ansatz geodesics are then relaxed to discrete one-parameter families of
geodesics with one endpoint shifted by lightlike vectors
$k^{\mu}_{i,\pm}=i\delta(1,±1)$. Since we only consider spatially homogeneous
and time-reversal invariant states, QNEC2 does not depend on the orientation
of the null vector and also not on which endpoint we vary. We typically
produce a family of seven geodesics with $i={−3,−2,...,3}$ and set the size of
the increment to $\delta=10^{-5}$. From the length of these geodesics we
compute the corresponding EEs and generate a third order polynomial fit
$S≈c_{0}+c_{1}\delta+c_{2}\delta^{2}+c_{3}\delta^{3}$ from which we extract
first and second derivative at $\delta=0$. More details on the numerical
implementation can be found in Ecker:2018jgh .

We recall that $\alpha$ determines the thermodynamic phase structure of the
model and can be tuned to realize either a first or second order phase
transition or a smooth crossover. Like for EE we study now QNEC2 in specific
examples for each of these cases. It is useful to first look at the left hand
side of QNEC2 (2) before comparing it with the right hand side. In Figure 14
we show $\langle T_{kk}\rangle$ as function of $T$ for our three different
values of $\alpha$.

Figure 14: $\langle T_{kk}\rangle$ as function of temperature for different
values of $\alpha$.

As expected $\langle T_{kk}\rangle$ displays features that are characteristic
for the specific kind of phase transitions in the respective model. Since
$\langle T_{kk}\rangle$ is a positive, but not necessarily single-valued
function of $T$, all examples we consider satisfy in addition to QNEC2 also
the classical null energy condition.

For $\alpha=0.32$ (left plot in Figure 14) $\langle T_{kk}\rangle$ is multi-
valued around the critical temperature $T_{c}$, where it has a finite jump.
Thermodynamically unpreferred branches are dashed and preferred ones solid.
For $T<T_{c}$ the energy momentum tensor is close to the vacuum value $\langle
T_{kk}\rangle=0$. The ‘critical’ state at $T=T_{c}$ requires special
attention. As shown in Janik:2017ykj ; Attems:2019yqn ; Bea:2020ees  there
exists an infinite number of degenerate phases at $T=T_{c}$, which on the
gravity side have inhomogenous horizons and can be seen as a mixture of small
and large black branes. We expect this to also be the case in our two-
dimensional model. Because we limit our study to homogeneous states, we cannot
make precise statements about EE and QNEC2 at $T=T_{c}$, only that they will
inherit the inhomogeneous structure of the bulk geometry.

In the middle of Figure 14 we plot $\langle T_{kk}\rangle$ for $\alpha=0.16$
where the system has a second order phase transition. In this case $\langle
T_{kk}\rangle$ is a single-valued, continuous and monotonic function of $T$.
For $T<T_{c}$ $\langle T_{kk}\rangle$ remains close to zero and the transition
between UV- and IR-scaling regime is tightly localized around $T=T_{c}$. For
$T>T_{c}$, i.e., in the large black hole branch $\langle T_{kk}\rangle$
depends strongly on $T$ and approaches at large $T$ the universal form given
by (102).

Finally in the right plot of Figure 14 we show $\langle T_{kk}\rangle$ for
$\alpha=0$. We recall that we defined $T_{c}\approx 0.01$ as the temperature
where the speed of sound has a local minimum. Here the curve does not have any
distinct features at $T=T_{c}$ and the transition between UV- and IR-regime is
not localized.

Let us now discuss the results for QNEC2. We begin with $\alpha=0.32$ for
which the theory has a first order phase transition at $T=T_{c}$ and QNEC2 as
function of $T$ has a jump. If we choose $\ell$ in the range
$3.870<\ell<4.058$ there is, like for domain walls, an additional
discontinuity due to the existence of multiple RT-surfaces. In Figure 15 we
show the results for $\ell=1$, which is well below this range. Blue and gray
lines are left and right hand side of QNEC2 and solid (dotted) lines indicate
thermodynamically preferred (unpreferred) solutions.

Figure 15: QNEC2 as function of $T$ for black brane solution with
$\alpha=0.32$ and $\ell=1$. Blue lines show the QNEC2 expression
$\frac{1}{2\pi}\left(S^{\prime\prime}+\tfrac{6}{c}(S^{\prime})^{2}\right)$ and
gray lines show $\langle T_{kk}\rangle$, where solid (dashed) parts of the
curves correspond to thermodynamically stable (unstable) phases. In addition,
the dashed green curve shows the value of QNEC2 of the corresponding $T=0$
domain wall solution.

For $T<T_{c}$, i.e., in the small black brane branch, QNEC2 grows only very
slowly with $T$ and remains very close to the $T=0$ limit given by the domain
wall result shown as dashed green line.

In Figure 16 we show the case $\ell=4$ in which QNEC2 has an especially rich
structure.

Figure 16: QNEC2 as function of $T$ for black brane solution with
$\alpha=0.32$ and $\ell=4$. Blue and red lines show the QNEC2 expression
$\frac{1}{2\pi}\left(S^{\prime\prime}+\tfrac{6}{c}(S^{\prime})^{2}\right)$ and
gray lines show $\langle T_{kk}\rangle$, where solid (dashed) parts of the
curves correspond to thermodynamically stable (unstable) phases. Red and blue
segments correspond to different branches of RT surfaces that dominate at
small and large separations, respectively. In addition, the dashed green curve
shows the value of QNEC2 of the corresponding $T=0$ domain wall solution.

This example has in addition to the phase transition at $T=T_{c}$ another
transition at $T=T^{\ast}\approx 0.8784\,T_{c}$ due to the existence of
multiple RT-surfaces. Here the two saddle points for the RT-surfaces displayed
in red and blue exchange dominance which leads to a jump in QNEC2. From the
analysis in section 2.7 we know there is actually also a negative delta
function which we indicate by the dashed black line. Curiously, QNEC2 is not a
monotonic function of $T$ in this case. Starting from small $T$ where it
nicely fits the corresponding domain wall result shown as dashed green curve,
QNEC2 decreases monotonically until it falls at $T=T^{\ast}$ to the smaller
value of the blue branch. This short blue branch is then monotonically
increasing until the jump at $T=T_{c}$. Following the blue curve at $T>T_{c}$
QNEC2 grows monotonically and approaches (105) in the limit of large $T$. We
emphasize that the two critical temperatures $T^{\ast}$ and $T_{c}$ are
different in nature. The former is related to the existence of multiple RT
surfaces, while the latter is related to the thermal phase transition.
Additionally we find $T^{\ast}<T_{c}$, because multiple RT surfaces only exist
in the small black brane branch with $T<T_{c}$.

The third case we analyze is $\ell=8$ shown in Figure 17.

Figure 17: QNEC2 as function of $T$ for black brane solution with
$\alpha=0.32$ and $\ell=8$. Blue lines show the QNEC2 expression
$\frac{1}{2\pi}\left(S^{\prime\prime}+\tfrac{6}{c}(S^{\prime})^{2}\right)$ and
gray lines show $\langle T_{kk}\rangle$, where solid (dashed) parts of the
curves correspond to thermodynamically stable (unstable) phases. In addition,
the dashed green curve shows the value of QNEC2 of the corresponding $T=0$
domain wall solution.

This case is again outside (this time above) the regime where multiple RT-
surfaces exist. Hence there is only the discontinuity at the thermal phase
transition. Curiously in the thermodynamically disfavoured branch (dashed
line) QNEC2 has two self-intersections, one at $T<T_{c}$ and another (barely
visible) close to the turning point at $T>T_{c}$. At these intersections QNEC2
takes the same value in two different phases which makes QNEC2 not a good
probe to distinguish different phases. As one can deduce from Figure 8 (right)
for $\ell=8$ we are for $T<T_{c}$ always in the non-gapped plateau like IR-
scaling regime where QNEC2 is almost saturated. This is different to the
aforementioned case $\ell=4$ in which QNEC2 for $T<T_{c}$ is in the gapped UV-
scaling regime and not close to saturation. For the cases $\alpha=0.16$ and
$\alpha=0$ QNEC2 is a monotonic and continuous function of both $T$ and
$\ell$.

Figure 18: QNEC non-saturation ratio ${\cal R}_{\textrm{\tiny QNEC}}$ as
function of temperature $T>T_{c}$ for various values of interaction strength
$\alpha$ and interval length $\ell$

Finally, we quantify the QNEC non-saturation by plotting in Figure 18 the
dimensionless ratio

${\cal R}_{\textrm{\tiny
QNEC}}:=1-\frac{S^{\prime\prime}+\frac{6}{c}\,(S^{\prime})^{2}}{2\pi\,\langle
T_{kk}\rangle}$ (107)

for various values of $\ell$ and the three values of $\alpha$ discussed above,
for temperatures greater than the critical. When QNEC2 saturates ${\cal
R}_{\textrm{\tiny QNEC}}$ vanishes and otherwise is positive, so this quantity
is a good measure for how far we are from saturation. As evident from the
plots, at large temperature the ratio ${\cal R}_{\textrm{\tiny QNEC}}$ tends
to zero, which means that QNEC is approximately saturated at large
temperature. We see this analytically by comparing with (106) which also
yields a vanishing ratio in the limit of small $\epsilon$.

##  Conclusion

After summarizing in section 2 general aspects of QNEC2 in deformed
holographic CFTs we focused on a specific example from section 3 onwards,
namely a deformed CFT2 dual to Einstein gravity with a massive scalar field
(43) and asymptotically AdS3 boundary conditions. For sake of specificity we
fixed the conformal weight of the operator dual to the scalar field to
$\Delta=\frac{3}{2}$ and considered a one-parameter family of potentials (47),
which led already to a rich phase structure, with crossovers, second- and
first-order phase transitions that we analyzed in detail in sections 4-6 using
QNEC2 (and associated quantities like EE and the Casini–Huerta $c$-function)
as diagnostic tool. For small and large intervals we were able to provide
closed-form expressions, while for intermediate intervals we relied on
numerics.

Unexpectedly our numerical studies reveal that QNEC2 in ground states displays
certain features that are characteristic for the thermodynamic phase structure
of the theory. For example, if ground state QNEC2 is non-monotonic in $\ell$,
then the theory has a first order phase transition at finite temperature.
Furthermore, our examples show that QNEC2 as function of $T$ can have a very
rich and non-trivial structure that includes multiple discontinuities caused
by thermal phase transitions and multiple RT surfaces. Using QNEC2 as tool for
detecting phase transitions is similar in spirit to using entanglement as a
probe of confinement Klebanov:2007ws , but the type of phase transition
discussed in our work differs from the confinement-deconfinement phase
transition

We address now some generalizations of our results. Even without changing the
model one could consider more general states than the ones considered in the
present work, by dropping the assumption of stationarity and/or translation
invariance. In full generality this can be done only numerically, but the
numerical routines to determine QNEC2 are typically not hard to implement and
computationally not very demanding.

The most straightforward model modification is to change the mass of the
scalar field and the potential. In this way one could scan the model space and
study QNEC2 as function of the conformal weight $\Delta$ and additional
parameters in the potential, which may reveal novel features. A particularly
interesting set of examples would be critical states at $T=T_{c}$ with first
order phase transition Janik:2017ykj , which we did not analyze in this paper.
Another extension would be to consider more general potentials related to
exotic holographic RG flows classified in Kiritsis:2016kog .

Adding a Maxwell field leads to a wider class of models with even richer phase
structure and phenomenology, like holographic superconductors Gubser:2008px ;
Hartnoll:2008vx  or other holographic condensed matter models
Charmousis:2010zz , that can again be analyzed along the lines of the present
work, using QNEC2 as diagnostic tool. Investigating the effects of
hyperscaling violation Gouteraux:2011ce ; Huijse:2011ef  and breaking of
translation invariance Andrade:2013gsa  from a QNEC2 perspective could be
rewarding as well.

It could also be of interest to venture beyond the supergravity approximation
and consider $1/c$ corrections to bulk and boundary theories, see section 5 in
Ecker:2019ocp  and refs. therein for such corrections to QNEC2.

Finally, it will be interesting to consider holographic correspondences beyond
asymptotically AdS3/deformed CFT2. There are two different types of
generalizations (which can also be combined): either one keeps the dimension,
but relaxes the asymptotic AdS3 behavior, e.g. by considering flat space
holography and the associated generalization of QNEC2 Grumiller:2019xna , or
one goes to higher dimensions and uses QNEC (1) instead of QNEC2 as diagnostic
tool for phase transitions and other phenomenological aspects.

## Note added

While finishing our manuscript reference Baggioli:2020cld  appeared on the
arXiv, which uses a similar logic as in section 6.2 to identify quantum phase
transitions. Instead of QNEC they use a generalized version of the
Casini–Huerta $c$-function Chu:2019uoh  as diagnostic tool.

## Acknowledgement

This work was supported by the Austrian Science Fund (FWF), projects P
28751-N27, P 30822-N27, P 32581-N27 and W 1252-N27. CE was in addition
supported by the Delta-Institute for Theoretical Physics (D-ITP) that is
funded by the Dutch Ministry of Education, Culture and Science (OCW). HS was
partly supported by Guangdong Major Project of Basic and Applied Basic
Research No. 2020B0301030008, the National Natural Science Foundation of China
with Grant No.12035007, Science and Technology Program of Guangzhou No.
2019050001.

##  Appendix A Near-boundary analysis and UV/IR relation

Like in the main text we fix the conformal weight of the operator dueal to the
scalar field to $\Delta=\frac{3}{2}$. The metric functions in Fefferman–Graham
gauge

$\operatorname{d}\\!s^{2}=\frac{\operatorname{d}\\!z^{2}-F(z)\operatorname{d}\\!t^{2}+G(z)\operatorname{d}\\!x^{2}}{z^{2}}$
(108)

are expanded near the boundary as (generalized) Taylor–Maclaurin series of the
radial coordinate $z$.

$F(z)=\sum_{i=0}^{\infty}f_{i}\,z^{i}\qquad\qquad
G(z)=\sum_{i=0}^{\infty}g_{i}\,z^{i}\qquad\qquad\phi(z)=z^{1/2}\sum_{i=0}^{\infty}\phi_{i}\,z^{i}$
(109)

The leading order coefficients are given by $f_{0}=g_{0}=1$ while $\phi_{0}=j$
is the source of the scalar field. The one-point functions are related to the
normalizable modes $f_{2}$ and $\phi_{1}$. Using on-shell conditions the first
few terms are given by

$\displaystyle F(z)=1-\frac{\phi_{0}^{2}}{4}z+f_{2}z^{2}+\mathcal{O}(z^{3})$
(110) $\displaystyle
G(z)=1-\frac{\phi_{0}^{2}}{4}z-\Big{(}\frac{\phi_{0}^{4}}{16}+\frac{3\phi_{0}\phi_{1}}{4}+f_{2}\Big{)}\,z^{2}+\mathcal{O}(z^{3})$
(111)
$\displaystyle\phi(z)=z^{1/2}\left(\phi_{0}+\phi_{1}z+\mathcal{O}(z^{2})\right)\,.$
(112)

By contrast, the metric functions in Gubser gauge $\phi(r)=j^{2-\Delta}r$
where we set $j=1$ from now on

$\displaystyle\operatorname{d}\\!s^{2}=e^{2A}\left(-H\operatorname{d}\\!t^{2}+\operatorname{d}\\!x^{2}\right)+e^{2B}\,\frac{\operatorname{d}\\!r^{2}}{H}$
(113)

have asymptotic expansions involving also logarithmic terms.

$A(r)=-2\,\log r+\sum_{i=0}^{\infty}a_{2i}\,r^{2i}\qquad
B(r)=\log\frac{2}{r}+\sum_{i=0}^{\infty}b_{2i}\,r^{2i}\qquad
H(r)=\sum_{i=0}^{\infty}h_{2i}\,r^{2i}$ (114)

The non-normalizable modes are fixed as $a_{0}=b_{0}=0$, $h_{0}=1$. The
expectation value of the boundary stress tensor and the operator dual to the
scalar field are related to the normalizable modes $a_{2}$ and $h_{4}$. Using
on-shell conditions the first few terms are given by

$\displaystyle A(r)=-2\,{\log r}+a_{2}r^{2}+\mathcal{O}\left(r^{4}\right)$
(115) $\displaystyle
B(r)=\log\frac{2}{r}-\Big{(}a_{2}+\frac{1}{8}\Big{)}r^{2}+\mathcal{O}\left(r^{4}\right)$
(116) $\displaystyle
H(r)=1+h_{4}r^{4}\Big{(}1-\frac{1+24a_{2}}{12}\,r^{2}+\mathcal{O}\left(r^{4}\right)\Big{)}\,.$
(117)

In Gubser gauge the UV and IR data are related as follows. Knowing the
functions $A$ and $B$ one can integrate (58) to find

$\displaystyle
H(r)=H(r_{0})+C\int\limits_{r_{0}}^{r}e^{-2A(r^{\prime})+B(r^{\prime})}\operatorname{d}\\!r^{\prime}$
(118)

where $H(r_{0})$ and $C$ are constants of integration. By choosing
$r_{0}=r_{h}$ and noting that $H$ is the blackening function the expression
(118) reduces to

$\displaystyle
H(r)=C\int_{r_{h}}^{r}e^{-2A(r^{\prime})+B(r^{\prime})}\operatorname{d}\\!r^{\prime}$
(119)

Differentiating with respect to $r$ and evaluating at the horizon yields

$\displaystyle H^{\prime}(r_{h})=C\,e^{-2A(\phi_{h})+B(\phi_{h})}\,.$ (120)

Comparing equations (120) and (56) at the horizon determines the integration
constant.

$\displaystyle C=e^{2A(\phi_{h})+B(\phi_{h})}V^{\prime}(r_{h})=-16\pi
G_{3}\,s\,T$ (121)

Finally, expanding both sides in equation (120) near the boundary yields
$C=2h_{4}$, which combined with (121) establishes a relation between boundary
and horizon data.

$\displaystyle h_{4}=\frac{1}{2}e^{2A(\phi_{h})+B(\phi_{h})}V^{\prime}(r_{h})$
(122)

##  Appendix B Holographic renormalization

In general holographic renormalization is based on the details of the theory,
namely the conformal weight of the scalar field and the scalar potential. As
in the rest of our work we fix $\Delta=\frac{3}{2}$. It turns out that the
counter-terms are

$\displaystyle I_{\textrm{ct}}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G_{3}^{2}}\int_{\partial
M}\operatorname{d}\\!^{2}x\sqrt{\gamma}\,\Big{[}\ell_{\textrm{\tiny{AdS}}}\log(\rho)\,R+\frac{W}{\ell_{\textrm{\tiny{AdS}}}}\Big{]}$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi G_{3}^{2}}\int_{\partial
M}\operatorname{d}\\!^{2}x\sqrt{\gamma}\,\Big{[}{\ell_{\textrm{\tiny{AdS}}}}\log(\rho)\,R-\frac{2}{\ell_{\textrm{\tiny{AdS}}}}-\frac{1}{4\ell_{\textrm{\tiny{AdS}}}}\,\phi^{2}-\frac{\alpha}{8\ell_{\textrm{\tiny{AdS}}}}\,\phi^{4}\Big{]}$

where $\gamma$ is the determinant of the boundary metric and in the second
line we use the superpotential (46). Using the superpotential as counter-term
fixes the ambiguous coefficient of the finite $\phi^{4}$ term to the unique
value that gives zero free energy for the ground state dual to the domain wall
geometry. The near boundary solution for static and stationary solutions with
fixed AdS is given in appendix A.

The one point functions are given as functional derivatives of the generating
functional

$\displaystyle\langle{\mathcal{O}_{\phi}}\rangle=\lim_{z\rightarrow
0}\frac{z^{-\Delta}}{\sqrt{-\gamma}}\frac{\delta\Gamma}{\delta\phi}\qquad\qquad\langle
T_{ij}\rangle=2\lim_{z\rightarrow
0}\frac{z^{-1}}{\sqrt{-\gamma}}\frac{\delta\Gamma}{\delta\gamma^{ij}}$ (124)

with the holographically renormalized action

$\displaystyle\Gamma=I_{\textrm{\tiny{bulk}}}+I_{\textrm{\tiny{GHY}}}+I_{\textrm{\tiny{ct}}}$
(125)

that consists of the bulk action $I_{\textrm{\tiny{bulk}}}$, the
Gibbons–Hawking–York boundary term $I_{\textrm{\tiny{GHY}}}$ and the
holographic counter-term $I_{\textrm{\tiny{ct}}}$.

The near boundary expansion from appendix A yields the one point functions

$\displaystyle\langle{\mathcal{O}_{\phi}}\rangle$ $\displaystyle=\frac{1}{8\pi
G_{3}}\,\Big{(}\frac{1}{2}\phi_{1}-\frac{\alpha}{4}{\phi_{0}^{3}}\Big{)}$
(126) $\displaystyle\langle T_{tt}\rangle$ $\displaystyle=\frac{1}{8\pi
G_{3}}\,\Big{(}-2f_{2}+\frac{1+2\alpha}{16}\phi_{0}^{4}-\phi_{0}\phi_{1}\Big{)}$
(127) $\displaystyle\langle T_{xx}\rangle$ $\displaystyle=\frac{1}{8\pi
G_{3}}\,\Big{(}-2f_{2}+\frac{1-2\alpha}{16}\phi_{0}^{4}-\frac{1}{2}\phi_{0}\phi_{1}\Big{)}$
(128)

These one-point functions respect the anticipated Ward identity.

$\displaystyle\langle
T^{i}_{~{}i}\rangle=\phi_{0}\,\langle\mathcal{O}_{\phi}\rangle$ (129)

The fact that the trace of the energy-momentum tensor is non-zero in general
is due to the breaking of conformal symmetry in the presence of a dimensionful
source.

##  Appendix C Evaluation of typical QNEC2 integrals for large intervals

We follow here the discussion in section 2.6 and evaluate the integrals in the
limit of large intervals, $\ell\gg 1$, for various background geometries.

We start with domain wall solutions, where we can set the null deformation
parameter to zero, $\lambda=0$, since the dual QFT state is boost invariant.
As stated in the main text, for Case 0 we parametrize the turning point as
$\rho_{\ast}=\ln\delta$ with small positive $\delta$, where $\rho$ refers to
the domain wall radial coordinate (50). The spatial integral (27) simplifies
to

$\ell\,e^{A_{\ast}}=2\int\limits_{\ln\delta}^{\infty}\operatorname{d}\\!\rho\,\Big{(}\frac{1}{\sqrt{1-y}}-\sqrt{1-y}\Big{)}$
(130)

where $y$ is defined in (78). The renormalized area integral (29) yields

${\cal
A}_{\textrm{\tiny{ren}}}(\ell)=-2\rho_{\ast}+2\int\limits_{\ln\delta}^{\infty}\operatorname{d}\\!\rho\,\Big{(}\frac{1}{\sqrt{1-y}}-1\Big{)}\,.$
(131)

The two integrals above can be converted into compact integrals

$2\int\limits_{\rho_{\ast}}^{\infty}\operatorname{d}\\!\rho\,f(y)=\int\limits_{0}^{1}\operatorname{d}\\!y\,\frac{f(y)}{y\,\frac{\operatorname{d}\\!A(\rho(y))}{\operatorname{d}\\!\rho}}$
(132)

which require the knowledge of $\rho$ as function of $y$. For Case 0 this
relation reads

$\rho=\rho_{\ast}-\frac{1}{8}\,e^{-\rho_{\ast}}-\frac{1}{2}\,\ln
y+W\big{(}\tfrac{1}{8}\,\sqrt{y}\,\exp(-\rho_{\ast}+\tfrac{1}{8}\,e^{-\rho_{\ast}})\big{)}$
(133)

with the Lambert-$W$ function. Its expansion $W(\exp(x))=x-\ln x+\frac{\ln
x}{x}+{\cal O}(\tfrac{\ln^{2}x}{x^{2}})$ permits to evaluate the two integrals
above perturbatively in $\delta$.

$\displaystyle
2\int\limits_{\ln\delta}^{\infty}\operatorname{d}\\!\rho\,\Big{(}\frac{1}{\sqrt{1-y}}-\sqrt{1-y}\Big{)}=16\delta-128\ln(2)\,\delta^{2}+{\cal
O}(\delta^{3})$ (134a) $\displaystyle
2\int\limits_{\ln\delta}^{\infty}\operatorname{d}\\!\rho\,\Big{(}\frac{1}{\sqrt{1-y}}-1\Big{)}=16\ln(2)\,\delta+\frac{16}{3}\,\big{(}\pi^{2}-24\ln
2-12\ln^{2}2\big{)}\,\delta^{2}+{\cal O}(\delta^{3})$ (134b)

The first one yields a result for the small parameter $\delta$ in terms of the
interval $\ell$,

$\delta=\frac{1}{8\ln\frac{\ell}{16}}+\dots$ (135)

where the ellipsis denotes subleading terms. The second one together with the
term $-2\rho_{\ast}$ in (131), when multiplied by $\frac{c}{6}$, yields the
result for renormalized EE displayed in the main text (80).

Case I is qualitatively different from Case 0 since there is a finite lower
bound on the radial coordinate, $\ln\alpha$. For large intervals the turning
point is close to this lower bound, so as explained in the main text we
parametrize it as $\rho_{\ast}=(1-\delta)\,\ln\alpha$, again with small and
positive $\delta$. Both integrals (134) are now order ${\cal O}(\delta^{2})$,
so that the leading contribution to area comes from the term $-2\rho_{\ast}$.
Plugging $A_{\ast}=1/(16\alpha\ln(\alpha)\,\delta)$ into equation (130) allows
to express $\delta$ in terms of $\ell$.

$\delta=-\frac{1}{16\alpha\ln(\alpha)\,\ln\ell}+\dots$ (136)

The renormalized area

${\cal
A}_{\textrm{\tiny{ren}}}=-2\rho_{\ast}+\dots=-2\ln\alpha-\frac{1}{8\alpha\,\ln\ell}+\dots$
(137)

multiplied by $\frac{c}{6}$ then leads to the result (83) for EE, where we
have displayed one additional subleading term.

Case II can be treated most easily in the limit of large $|\alpha|$. Like in
Case 0 the radial coordinate is unbounded, so we parametrize again
$\rho_{\ast}=\ln\delta$ with small positive $\delta$. In the large $|\alpha|$
limit the function $A(\rho)$ is linear in the radial coordinate,

$A(\rho)=\Big{(}1-\frac{1}{16\alpha}\Big{)}\,\rho+\frac{1+\ln(-\alpha)}{16\alpha}+{\cal
O}(\alpha^{-3})$ (138)

which means that the measure factor in the right hand side of (132) is rather
trivial and yet significant, since it provides the ratio between IR and UV
values of the central charge. The simplicity of the measure factor allows to
perform all integrals in closed form and yields the result (84) in the main
text. It turns out that the same result is true at small $|\alpha|$.

Finally, we consider the three relevant integrals for the black brane in the
large $\ell$ limit, where the backreaction calculation discussed in section
6.3 applies.

The deformed entangling region integral [$A_{\ast}=A(z_{\ast})$ and
$S_{\ast}=S(z_{\ast})$]

$\frac{\ell+\lambda}{2}=\int\limits_{0}^{(\ell+\lambda)/2}\operatorname{d}\\!x=\int\limits_{z_{\ast}}^{0}\frac{\operatorname{d}\\!z}{\sqrt{\Lambda^{2}S^{2}(z)-S(z)A(z)+\frac{S^{2}(z)A(z)}{z^{2}}\,z^{2}_{\ast}\,\Big{(}\frac{1}{S_{\ast}}-\frac{\Lambda^{2}}{A_{\ast}}\Big{)}}}$
(139)

the time-shift integral

$\frac{\lambda}{2}=\int\limits_{0}^{\lambda/2}\operatorname{d}\\!t=\Lambda\,\int\limits_{z_{\ast}}^{0}\frac{\operatorname{d}\\!z}{A(z)\,\sqrt{\Lambda^{2}-\frac{A(z)}{S(z)}+\frac{A(z)}{z^{2}}\,z^{2}_{\ast}\,\Big{(}\frac{1}{S_{\ast}}-\frac{\Lambda^{2}}{A_{\ast}}\Big{)}}}\,.$
(140)

and the renormalized area integral ($z_{\textrm{\tiny{cut}}}$ tends to $+0$
when the cutoff is removed)

${\cal
A}_{\textrm{\tiny{ren}}}=2z_{\ast}\sqrt{\frac{1}{S_{\ast}}-\frac{\Lambda^{2}}{A_{\ast}}}\,\int\limits_{z_{\ast}}^{z_{\textrm{\tiny{cut}}}}\frac{\operatorname{d}\\!z}{z^{2}\,\sqrt{\Lambda^{2}-\frac{A(z)}{S(z)}+\frac{A(z)}{z^{2}}\,z^{2}_{\ast}\,\Big{(}\frac{1}{S_{\ast}}-\frac{\Lambda^{2}}{A_{\ast}}\Big{)}}}-2\ln
z_{\textrm{\tiny{cut}}}$ (141)

for large $\ell$ all lead to essentially two types of integral kernels.

$\displaystyle I_{1}[h(y);\,\Delta]$
$\displaystyle=\int\limits_{0}^{1}\operatorname{d}\\!y\,\frac{h(y)}{\sqrt{1-y}(1-y+\Delta\,y)^{3/2}}$
(142a) $\displaystyle I_{2}[h(y);\,\Delta]$
$\displaystyle=\int\limits_{0}^{1}\operatorname{d}\\!y\,\frac{h(y)}{\sqrt{1-y}(1-y+\Delta\,y)^{5/2}}$
(142b)

Here $h(y)$ is a function that is continuous in the interval $[0,1]$ and
Taylor expandable around $y=1$, while $\Delta=2\delta-\delta^{2}$ is a small
parameter. The integration variable $y$ is related to our original radial
coordinate $z$ by the simple coordinate transformation
$y=z^{2}/(1-\delta)^{2}$.

We evaluate such integrals perturbatively in $\Delta$, displaying one more
order than we need.

$\displaystyle I_{1}[h(y);\,\Delta]$
$\displaystyle=\frac{2h(1)}{\Delta}+h^{\prime}(1)\,\ln\frac{\Delta}{4}+2h^{\prime}(1)+{\cal
I}_{1}[h(y)]+{\cal O}(\Delta\,\ln\Delta)$ (143) $\displaystyle
I_{2}[h(y);\,\Delta]$
$\displaystyle=\frac{4h(1)}{3\Delta^{2}}+\frac{2h(1)-2h^{\prime}(1)}{3\Delta}-\frac{1}{2}h^{\prime\prime}(1)\ln\frac{\Delta}{4}-\frac{4}{3}h^{\prime\prime}(1)+{\cal
I}_{2}[h(y)]+{\cal O}(\Delta\ln\Delta)$ (144)

The ${\cal O}(1)$ integrals are defined as ($h^{(m)}$ denotes the
$m^{\textrm{th}}$ derivative of $h$)

${\cal
I}_{n}[h(y)]=\int\limits_{0}^{1}\operatorname{d}\\!y\,\frac{h(y)-\sum_{m=0}^{n}\frac{(y-1)^{m}}{m!}\,h^{(m)}(1)}{(1-y)^{n+1}}\,.$
(145)

In the rest of the appendix we apply these formulas to the three integrals
mentioned above.

The time-shift integral (140) yields

$\lambda=\frac{\Lambda}{\delta}\,\Big{(}1+\frac{\epsilon^{2}}{4}\,\big{(}3A_{2}^{\prime}(1)+2S_{2}(1)+S_{2}^{\prime}(1)\big{)}\Big{)}\,\big{(}1+{\cal
O}(\delta\,\ln\delta)+{\cal O}(\epsilon^{3})\big{)}$ (146)

where we assumed $A_{2}(1)=0$ since we keep the horizon fixed at $z=1$. In
this appendix prime denotes derivative with respect to $y$ (and not with
respect to $z$, as it does in the main text). Moreover, we set
$A_{2}^{\prime}(1)=0$ since we also keep the temperature fixed. Solving (146)
for $\Lambda$ shows that it scales linearly in $\lambda$ and, to leading
order, linearly in $\delta$. The entangling region interval (139) allows to
solve $\delta$ in terms of $\ell$, $\lambda$ and $\epsilon$.

$\delta=2e^{-\ell}\,\Big{(}1+\epsilon^{2}\ell\,f-\lambda\,\big{(}1+\epsilon^{2}\,(\ell-1)\,f\big{)}+\frac{3}{4}\,\lambda^{2}\,\big{(}1+\epsilon^{2}(\ell-2)\,f\big{)}\Big{)}+\dots$
(147)

with

$f:=\frac{S_{2}^{\prime}(1)}{4}-\frac{S_{2}(1)}{2}\,.$ (148)

We deduce from (147) that each power of $\delta$ is suppressed at large $\ell$
by an instanton-like factor. Note that the suppression by $e^{-\ell}$ is
compatible with general expectations and precisely agrees with the exponent
derived for holographic EE in Fischler:2012ca . Finally, the area integral
(141) yields the result (103) stated in the main text.

## References

  * (1) S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602, hep-th/0603001.
  * (2) D. Harlow, “Jerusalem Lectures on Black Holes and Quantum Information,” Rev. Mod. Phys. 88 (2016) 015002, 1409.1231.
  * (3) P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and Z. Yang, “Holographic duality from random tensor networks,” JHEP 11 (2016) 009, 1601.01694.
  * (4) M. Van Raamsdonk, “Lectures on Gravity and Entanglement,” in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 297–351.  2017\.  1609.00026.
  * (5) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, “The entropy of Hawking radiation,” 2006.06872.
  * (6) R. Bousso, Z. Fisher, S. Leichenauer, and A. C. Wall, “Quantum focusing conjecture,” Phys. Rev. D93 (2016), no. 6, 064044, 1506.02669.
  * (7) R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer, and A. C. Wall, “Proof of the Quantum Null Energy Condition,” Phys. Rev. D93 (2016), no. 2, 024017, 1509.02542.
  * (8) T. A. Malik and R. Lopez-Mobilia, “Proof of the quantum null energy condition for free fermionic field theories,” Phys. Rev. D 101 (2020), no. 6, 066028, 1910.07594.
  * (9) J. Koeller and S. Leichenauer, “Holographic Proof of the Quantum Null Energy Condition,” Phys. Rev. D94 (2016), no. 2, 024026, 1512.06109.
  * (10) S. Balakrishnan, T. Faulkner, Z. U. Khandker, and H. Wang, “A General Proof of the Quantum Null Energy Condition,” JHEP 09 (2019) 020, 1706.09432.
  * (11) F. Ceyhan and T. Faulkner, “Recovering the QNEC from the ANEC,” Commun. Math. Phys. 377 (2020), no. 2, 999–1045, 1812.04683.
  * (12) D. Grumiller, P. Parekh, and M. Riegler, “Local quantum energy conditions in non-lorentz-invariant quantum field theories,” Phys. Rev. Lett. 123 (Sep, 2019) 121602.
  * (13) A. C. Wall, “Testing the Generalized Second Law in 1+1 dimensional Conformal Vacua: An Argument for the Causal Horizon,” Phys. Rev. D85 (2012) 024015, 1105.3520.
  * (14) S. S. Gubser and A. Nellore, “Mimicking the QCD equation of state with a dual black hole,” Phys. Rev. D78 (2008) 086007, 0804.0434.
  * (15) R. A. Janik, J. Jankowski, and H. Soltanpanahi, “Nonequilibrium Dynamics and Phase Transitions in Holographic Models,” Phys. Rev. Lett. 117 (2016), no. 9, 091603, 1512.06871.
  * (16) M. Attems, J. Casalderrey-Solana, D. Mateos, I. Papadimitriou, D. Santos-Oliván, C. F. Sopuerta, M. Triana, and M. Zilhão, “Thermodynamics, transport and relaxation in non-conformal theories,” JHEP 10 (2016) 155, 1603.01254.
  * (17) R. A. Janik, J. Jankowski, and H. Soltanpanahi, “Quasinormal modes and the phase structure of strongly coupled matter,” JHEP 06 (2016) 047, 1603.05950.
  * (18) M. Attems, Y. Bea, J. Casalderrey-Solana, D. Mateos, M. Triana, and M. Zilhao, “Phase Transitions, Inhomogeneous Horizons and Second-Order Hydrodynamics,” JHEP 06 (2017) 129, 1703.02948.
  * (19) R. A. Janik, J. Jankowski, and H. Soltanpanahi, “Real-Time dynamics and phase separation in a holographic first order phase transition,” Phys. Rev. Lett. 119 (2017), no. 26, 261601, 1704.05387.
  * (20) J. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys. 104 (1986) 207–226.
  * (21) M. Banados, “Three-dimensional quantum geometry and black holes,” AIP Conf. Proc. 484 (1999), no. 1, 147–169, hep-th/9901148.
  * (22) M. M. Roberts, “Time evolution of entanglement entropy from a pulse,” JHEP 12 (2012) 027, 1204.1982.
  * (23) M. Sheikh-Jabbari and H. Yavartanoo, “Excitation entanglement entropy in two dimensional conformal field theories,” Phys. Rev. D 94 (2016), no. 12, 126006, 1605.00341.
  * (24) C. Holzhey, F. Larsen, and F. Wilczek, “Geometric and renormalized entropy in conformal field theory,” Nucl. Phys. B 424 (1994) 443–467, hep-th/9403108.
  * (25) P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J. Stat. Mech. 0406 (2004) P06002, hep-th/0405152.