Disorder in AdS3/CFT2

Moritz Dorband1,2*, Daniel Grumiller3, René Meyer1,2, Suting Zhao1,2

1 Institute for Theoretical Physics and Astrophysics,
Julius–Maximilians–Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

2 Würzburg-Dresden Cluster of Excellence ct.qmat

3 Institute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8-10/136,
A-1040 Vienna, Austria

*<EMAIL_ADDRESS>
August 27, 2024

###### Abstract

We perturbatively study marginally relevant quenched disorder in AdS3/CFT2 to
second order in the disorder strength. Using the Chern–Simons formulation of
AdS3 gravity for the Poincaré patch, we introduce disorder via the chemical
potentials. We discuss the bulk and boundary properties resulting from the
disorder averaged metric. The disorder generates a small mass and angular
momentum. In the bulk and the boundary, we find unphysical features due to the
disorder average. Motivated by these features, we propose a
Poincaré–Lindstedt-inspired resummation method. We discuss how this method
enables us to remove all of the unphysical features.

###### Contents

  1. 1 Introduction
  2. 2 Disorder in SL(2,R) Chern–Simons gravity
    1. 2.1 Disorder setup
    2. 2.2 Averaged geometry
  3. 3 Properties of the holographic averaged state
    1. 3.1 Energy-momentum tensor and trace anomaly
    2. 3.2 Quantum null energy condition
  4. 4 Poincaré–Lindstedt inspired resummation
  5. 5 Conclusions and outlook
  6. A Expanded metric components
  7. B Disorder with relative phase shift
  8. C Details of the QNEC calculation

##  1 Introduction

Strongly interacting quantum systems are of interest in both high energy
physics and condensed matter physics. While in the description of condensed
matter systems idealisations such as the perfect crystal are often useful to
understand aspects of the physics of the electron system, taking into account
the effect of impurities and disorder implemented by lattice defects is
important to fully describe the physics as measured in experiments. In
particular, the presence of random disorder completely changes the low energy
transport properties of otherwise free quantum particles, a phenomenon
nowadays known as Anderson localisation [1]. Anderson localisation states that
in disordered lower dimensional systems, disorder leads to a complete
termination of diffusive processes, making a metal a perfect insulator in the
thermodynamic limit. While this was shown for non-interacting systems in [1],
it is an open question how interactions will affect the localisation
transition, which is a contemporary open problem in theoretical physics known
as many–body localisation (for reviews see [2, 3]).

While interactions can be included perturbatively into an already disordered
system, an interesting question is whether many–body localisation persists in
the non-perturbative regime, in particular, if the interacting but non-
disordered phase would be gapless. A promising approach to describe strongly
interacting systems, and hence to answer this question, is provided by the
AdS/CFT correspondence (a.k.a. gauge/gravity duality or simply holography)
[4]. The AdS/CFT correspondence relates strongly interacting quantum field
theories (QFTs) to weakly interacting gravitational theories in asymptotically
Anti–de Sitter (AdS) spacetimes. In the limit of large central charges, the
generating functional of the dual strongly interacting quantum field theory is
defined on the asymptotic boundary of the AdS spacetime via the semiclassical
gravitational path integral with fixed boundary conditions [5, 6]. In
particular, each field in the bulk encodes both source and response
(expectation value) of an operator in the dual boundary QFT. In this work, we
will disorder the boundary sources by drawing them randomly from a Gaussian
ensemble, solve the gravity equations of motion for the bulk spacetime in each
disorder realisation, and analyse the resulting disorder averaged bulk
geometry. We work in the realm of the AdS3/CFT2 correspondence, and in
particular use the $\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})$
Chern–Simons formulation [7, 8] of three-dimensional Einstein–Hilbert gravity
with a negative cosmological constant.

The possibility of introducing quenched disorder in AdS/CFT was first
adumbrated in [9] (see [10] for a review). Momentum–relaxing impurities have
been introduced in AdS/CFT even earlier in [11], and the replica trick has
first been used in [12] to calculate the effect of disorder in AdS/CFT. Refs.
[13, 14, 15, 16] studied a disordered scalar in AdS spacetime. In [13] the
effects of disorder on a holographic superconductor were studied by
introducing a random chemical potential on the boundary. In [14], a scalar
field coupled to three-dimensional AdS gravity was considered, where the
source dual to the scalar operator was disordered. The scalar source was
disordered along the spatial boundary dimension, and several different
ensemble realisations for the disorder were discussed in [14]. Disorder was
finally introduced in [13, 14] by an ansatz using superimposed plane waves
with equally distributed random phases. By solving Einstein’s equations to
second order in the disorder strength, the backreaction of the scalar field on
the pure AdS background was determined. The resulting metric, averaged over
disorder realisations, showed a logarithmically divergent behaviour in the
infrared (IR) region in the bulk. The logarithmic behavior can be traced back
to the disorder being marginally relevant in the sense of Harris’ criterion
[17]. The Poincaré–Lindstedt resummation method was then employed to remove
this divergence. The resummed metric admitted an IR Lifshitz scaling fixed
point, with the dynamical exponent departing from the relativistic value
quadratically in the disorder strength. The quartic correction in the disorder
strength was also calculated analytically. The analytic results were confirmed
by a numerical analysis for non-perturbative disorder strength. In [15], this
analysis was extended to finite temperature, i.e., to a planar AdS black hole
background, both in $D=3$ and $D=4$. A scaling of the low–temperature entropy
density with temperature, with the exponent fixed by the Lifshitz scaling
exponent found at zero temperature [14], was constructed both analytically and
numerically. In [16], extremal planar AdS black holes, and in particular,
their near horizon behaviour, in $D=4$ Einstein–Maxwell theory was studied.
Furthermore, disorder in a p-wave holographic superconductor has been
investigated in [18], and a disorder-induced holographic metal-insulator
transition has been found in [19].

Einstein–Maxwell theory in $D=4$ with a disordered boundary chemical potential
for the U(1) charge density was investigated in [20]. AdS gravity supplemented
with a U(1) gauge field in the bulk describes a boundary state at finite U(1)
charge density at the boundary condition, with the chemical potential being
the constant mode in the time component of the bulk chemical potential. In
AdS/CFT, the full U(1) gauge field in the bulk is holographically dual to a
global conserved Noether current in the boundary QFT. By computing the
backreaction of the disordered vector potential on the metric to second order
in the disorder strength, the effect of disorder on the bulk spacetime
geometry was determined in an analogous way as in [14]. From this metric, a
positive second-order correction to the conductivity of the boundary theory
was found. This result is consistent with [21], where a lower bound for the
boundary conductivity of disordered Einstein–Maxwell AdS gravity was derived,
implying that any source of disorder will raise the average value of the
boundary conductivity in Einstein–Maxwell AdS gravity.

In this work, we use the AdS3/CFT2 correspondence to study holographically
disordered strongly interacting two-dimensional conformal field theories (2$D$
CFTs). We employ the first–order formulation of three-dimensional Einstein
gravity with a negative cosmological constant as
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ Chern–Simons theory
[7, 8]. Choosing asymptotic Brown–Hennaux boundary conditions [22], the dual
QFT is a 2$D$ CFT at large central charge, with symmetry given by two copies
of the Virasoro symmetry algebra [22]. As described in more detail below, we
introduce quenched static disorder by disordering the sources encoded in the
asymptotic behavior of the bulk Chern–Simons gauge fields, i.e., the chemical
potentials coupling to the left- and right-moving Chern–Simons currents. A
crucial advantage of this approach is the exact solvability of the bulk
Chern–Simons theory [23], providing an explicit expression for the bulk
solution for each disorder realization, and hence allowing us to derive our
results analytically to second order in the disorder strength.

Having introduced disorder into the system, we calculate the disorder averaged
bulk metric to quadratic order in the disorder strength. We then analyse both
bulk and boundary properties of the resulting geometry. We find that there
appear curvature singularities in the IR region, implying a breakdown of the
perturbation expansion in the disorder strength. Furthermore, our averaged
metric does not satisfy the AdS3 vacuum Einstein equations any longer for non-
vanishing disorder strength. The averaged metric is moreover found to contain
closed timelike curves. In the region containing closed timelike curves, a
breakdown of the semiclassical approximation for a probe string is found, one
of the three criteria for a “bad singularity” proposed in [24]. In the field
theory dual, we find that the energy-momentum tensor has a trace anomaly,
which however cannot be related to the Ricci curvature of the boundary metric
via the usual trace anomaly equation. Finally, depending on the relative
strengths of the disorder parameters for left and right moving sectors as well
as on the length of the entangling interval, we find that the quantum null
energy condition (QNEC), as well as the null energy condition (NEC), can be
violated.

To remedy all these issues, we perform a resummation inspired by the
Poincaré–Lindstedt technique used in [16]. With this method, we restore finite
curvature throughout the entire bulk. The resulting resummed averaged metric
satisfies the AdS3 vacuum Einstein equations, up to second order in the
disorder strength. The resummed metric furthermore does no longer contain
closed timelike curves and yields a well-defined semiclassical approximation
for a string probing the interior of the averaged spacetime. The dual theory’s
energy-momentum tensor is now traceless, following the usual relation between
the boundary Ricci curvature and the trace anomaly. Also, the QNEC condition
for the resummed metric is not only satisfied but saturated. This single
resummation ansatz solving all of these problems at once suggests this is the
correct bulk procedure to obtain a reasonable holographic disorder–averaged
state.

This paper is organized as follows: in section 2 we briefly review the
Chern–Simons formulation of three-dimensional gravity in AdS and then
introduce the disorder implemented in section 2.1. We discuss our results from
the bulk perspective in section 2.2 and from the boundary perspective in
section 3. Afterwards, we present a Poincaré–Lindstedt inspired resummation of
these results in section 4. We conclude and give an outlook towards future
work in section 5.

##  2 Disorder in SL(2,R) Chern–Simons gravity

Classical Einstein–Hilbert gravity with negative cosmological constant in
three spacetime dimensions can equivalently be formulated as a Chern–Simons
gauge theory [7, 8]. We use this formalism to conveniently introduce disorder
into the system, while also ensuring asymptotic AdS behaviour. Before
discussing the disorder setup, we will therefore briefly review the
Chern–Simons formulation of AdS3 gravity.

The three-dimensional Einstein–Hilbert action on the classical level can
equivalently be expressed as the difference of two Chern–Simons actions [7, 8]

$\displaystyle S_{\text{EH}}[A,\bar{A}]=\frac{k}{4\pi}\int_{\cal
M}\tr(A\wedge\text{d}A+\frac{2}{3}A\wedge A\wedge
A-\bar{A}\wedge\text{d}\bar{A}-\frac{2}{3}\bar{A}\wedge\bar{A}\wedge\bar{A})\,,$
(1)

where the Chern–Simons level111We set the AdS radius to one.
$k=\frac{1}{4G}=\frac{c}{6}$ is related to Newtons constant $G$ and the
Brown–Henneaux central charge $c$. The gauge fields $A$ and $\bar{A}$ are
linear combinations of the dreibein $e_{\leavevmode\nobreak\ \mu}^{a}$ and the
dualised spin connection $\omega_{\leavevmode\nobreak\
\mu}^{a}=\frac{1}{2}\epsilon^{abc}\omega_{bc\,\mu}$. Furthermore, the gauge
fields are charged under the isometry group of the spacetime under
consideration. In this paper, we consider AdS space which in $D=3$ has the
isometry group
$\text{SO}(2,2)\simeq\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})$.
Explicitly, the gauge fields are realised as

$\displaystyle A=A^{a}T_{a}=(\omega^{a}+e^{a})T_{a}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{A}=\bar{A}^{a}T_{a}=(\omega^{a}-e^{a})T_{a}\,,$ (2)

where $T_{a}$ are generators of $\mathfrak{sl}(2,\mathbb{R})$, satisfying
$\commutator{T_{a}}{T_{b}}=(a-b)T_{a+b}$. For explicit calculations, we work
with the fundamental representation for $\mathfrak{sl}(2,\mathbb{R})$ (see
appendix A).

We consider manifolds whose topology is a solid cylinder, with radial
coordinate $\rho\in\mathbb{R}$, time $t\in\mathbb{R}$ and angular coordinate
$\phi\in[0,2\pi]$. To ensure asymptotic AdS behaviour, we impose
Brown–Henneaux boundary conditions on the gauge fields as described in [23]
(see also [25, 26] for reviews). To do so we first write the fields in radial
gauge

$\displaystyle
A=b^{-1}(\text{d}+a)b\,,\quad\quad\quad\bar{A}=b(\text{d}+\bar{a})b^{-1}\,.$
(3)

The functions $a$ and $\bar{a}$ depend only on the boundary coordinates and
capture all of the state dependence. The radial dependence is captured by the
group element $b$ only, which we choose as

$\displaystyle b=\exp(\rho T_{0})\,.$ (4)

Brown–Henneaux boundary conditions are then achieved by imposing the following
form on the state dependent functions $a$ and $\bar{a}$

$\displaystyle a_{t}$ $\displaystyle=\mu T_{+}-\partial_{\phi}\mu
T_{0}+\left(-\frac{2\pi}{k}\mu{\cal
L}+\frac{1}{2}\partial_{\phi}^{2}\mu\right)T_{-},$ (5) $\displaystyle
a_{\phi}$ $\displaystyle=T_{+}-\frac{2\pi}{k}{\cal L}T_{-},$ (6)
$\displaystyle\bar{a}_{t}$ $\displaystyle=\left(\frac{2\pi}{k}\bar{{\cal
L}}\bar{\mu}+\frac{1}{2}\partial_{\phi}^{2}\bar{\mu}\right)T_{+}+\partial_{\phi}\bar{\mu}T_{0}+\bar{\mu}T_{-},$
(7) $\displaystyle\bar{a}_{\phi}$ $\displaystyle=\frac{2\pi}{k}\bar{{\cal
L}}T_{+}+T_{-},$ (8)

where the gauge-flatness conditions on $A$ and $\bar{A}$ following from (1),

$\text{d}A+A\wedge A=0=\text{d}\bar{A}+\bar{A}\wedge\bar{A}$ (9)

have been used. Both barred and non-barred sector contain two free functions
each, the chemical potentials $\mu,\bar{\mu}$ and their corresponding
conserved currents ${\cal L},\bar{{\cal L}}$, respectively. As shown in [23],
a well-defined variational principle only allows the latter ones to vary and
generate the Virasoro algebras for the barred and non-barred sectors.

By the flatness conditions (9), the conserved currents are determined
completely by their associated chemical potentials

$\displaystyle\partial_{t}{\cal L}$ $\displaystyle=\mu\partial_{\phi}{\cal
L}+2{\cal L}\partial_{\phi}\mu-\frac{k}{4\pi}\partial_{\phi}^{3}\mu\,,$ (10)
$\displaystyle\partial_{t}\bar{{\cal L}}$
$\displaystyle=\bar{\mu}\partial_{\phi}\bar{{\cal L}}+2\bar{{\cal
L}}\partial_{\phi}\bar{\mu}+\frac{k}{4\pi}\partial_{\phi}^{3}\bar{\mu}\,.$
(11)

For the vanilla choice of chemical potentials, $\mu=1=-\bar{\mu}$, these
conservation equations are the usual holographic Ward identities
$\partial_{-}{\cal L}=0=\partial_{+}\bar{\cal L}$, with $x^{\pm}=t\pm\phi$.

In this work, to solve these conservation equations, we assume static chemical
potentials, $\partial_{t}\mu=0=\partial_{t}\bar{\mu}$. Then the solutions for
the currents are given by

$\displaystyle{\cal L}$
$\displaystyle=\frac{F\left[t+\int^{\phi}\frac{\text{d}u}{\mu(u)}\right]}{\mu^{2}}+\frac{k}{4\pi}\left(\frac{\partial_{\phi}^{2}\mu}{\mu}-\frac{1}{2}\frac{(\partial_{\phi}\mu)^{2}}{\mu^{2}}\right)\,,$
(12) $\displaystyle\bar{{\cal L}}$
$\displaystyle=\frac{G\left[t+\int^{\phi}\frac{\text{d}v}{\bar{\mu}(v)}\right]}{\bar{\mu}^{2}}-\frac{k}{4\pi}\left(\frac{\partial_{\phi}^{2}\bar{\mu}}{\bar{\mu}}-\frac{1}{2}\frac{(\partial_{\phi}\bar{\mu})^{2}}{\bar{\mu}^{2}}\right)\,.$
(13)

In general, $F$ and $G$ are arbitrary functions depending on the indicated
combination of $t$ and $\phi$.

From the above equations (12) and (13), the Poincaré AdS solution is obtained
by the vanilla choice, $\mu=1=-\bar{\mu}$, and furthermore $F=0=G$, such that
${\cal L}=0=\bar{{\cal L}}$. Using the well-known relation between metric and
gauge fields

$\displaystyle
g_{\mu\nu}=\frac{1}{2}\tr[\left(A-\bar{A}\right)_{\mu}\left(A-\bar{A}\right)_{\nu}\big{]}\,,$
(14)

we find the Poincaré patch metric

$\displaystyle\text{d}s^{2}=-e^{2\rho}\text{d}t^{2}+\text{d}\rho^{2}+e^{2\rho}\text{d}\phi^{2}\,.$
(15)

In the next subsection, we exploit the chemical potentials to introduce
disorder onto the background given by (15). For the remainder of this paper,
we use the holographic coordinate $\rho=-\ln z$.

###  2.1 Disorder setup

Our approach to introduce disorder into (15) is inspired by earlier work on
disorder [14]. In our case, the chemical potentials $\mu$ and $\bar{\mu}$
serve as sources for disorder, which is considered analogously to [14]:

$\displaystyle\mu$ $\displaystyle=1+\epsilon
f(\phi)=1+\frac{\epsilon}{\sqrt{N}}\sum_{n=1}^{N}\cos(\frac{n}{N}\phi+\gamma_{n})\,,$
(16) $\displaystyle\bar{\mu}$
$\displaystyle=-1+\bar{\epsilon}f(\phi)=-1+\frac{\bar{\epsilon}}{\sqrt{N}}\sum_{n=1}^{N}\cos(\frac{n}{N}\phi+\gamma_{n})\,.$
(17)

The random phases $\gamma_{n}$ are equally distributed in the interval
$[0,2\pi)$. The chemical potentials are then periodic in $2\pi N$, only.222As
a consequence, the chemical potentials are not single-valued anymore, in line
with the disorder ansatz by Hartnoll and Santos [14]. This multi-valuedness
yields the desired white noise for the two-point functions (21) below, which
would not be the case if the chemical potentials were single-valued.

Averaging an arbitrary function $H$ over these random phases is defined by
integration over all random phases $\gamma_{n}$ and taking the large $N$ limit
afterwards.

$\displaystyle\expectationvalue{H(\\{\gamma_{n}\\})}=\lim\limits_{N\to\infty}\int\limits_{0}^{2\pi}\prod_{n=1}^{N}\frac{\text{d}\gamma_{n}}{2\pi}H(\\{\gamma_{n}\\})$
(18)

The average, which we denote by $\expectationvalue{\cdot}$, is defined such
that

$\displaystyle\expectationvalue{f(\phi)}=0\,,$ (19)

leading to

$\displaystyle\expectationvalue{\mu}=1=-\expectationvalue{\bar{\mu}}\,.$ (20)

Therefore to zeroth order in the disorder strengths $\epsilon$ and
$\bar{\epsilon}$, this reproduces the vanilla choices $\mu=1=-\bar{\mu}$ made
earlier for (15). More generally, it can be shown that every odd power of
$f(\phi)$ averages to zero. Since the metric is bilinear in the gauge fields,
and thereby also in the chemical potentials, we obtain terms with non-trivial
average by employing (14).

The disorder that we introduced through random phases describes Gaussian
noise. Computing for example the two-point function of $f$ results in

$\displaystyle\expectationvalue{f(\phi)f(\theta)}=\frac{1}{2}\delta(\phi-\theta)\,.$
(21)

For the analysis of the perturbed geometry, we use $\epsilon$ and
$\bar{\epsilon}$ as perturbative handles to expand the metric components to
second order. This enables us to perform an analytic computation of the
averages.

While averaging the metric components to second order, there are four
different combinations of $f$ and its derivatives which occur. The averages of
these combinations are given by

$\displaystyle\expectationvalue{f^{2}}=\frac{1}{2}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\expectationvalue{(\partial_{\phi}f)^{2}}=\frac{1}{6}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\expectationvalue{f\partial_{\phi}^{2}f}=-\frac{1}{6}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\expectationvalue{(\partial_{\phi}^{2}f)^{2}}=\frac{1}{10}\,.$ (22)

In our work, we take the same random phases for $\mu$ and $\bar{\mu}$, i.e.,
we do not introduce random phases $\gamma_{n}$ and $\bar{\gamma}_{n}$, where
we could allow for different intervals, say $[0,2\pi)$ for $\gamma_{n}$ and
$[\alpha,2\pi+\alpha)$ for $\bar{\gamma}_{n}$. In that way, relative phases
between $\epsilon$ and $\bar{\epsilon}$ can be included. However, at the level
of our current analysis, this does not yield an immediate advantage. For more
details on the effect of relative phases, we refer the reader to appendix B.

A particular feature related to our choice above is that flipping signs of
both $\epsilon$ and $\bar{\epsilon}$ can be absorbed in $\gamma_{n}$, which is
not possible if only one of the signs is changed. We expect this behaviour to
reflect in the disorder averaged quantities we obtain in section 2.2 and later
on.

An important characterisation of disorder in any system is given by the Harris
criterion [17]. The criterion uses the mass dimension of the source
$[\epsilon]$ to quantify whether a source of disorder is relevant
($[\epsilon]>0$), marginal ($[\epsilon]=0$) or irrelevant ($[\epsilon]<0$).
For a holographic scenario, the criterion can be obtained by considering the
source term in the action of the dual field theory and using that the action
is dimensionless in natural units. In our case, we consider the coupling term
between the disordered source $\mu$ and its conserved current ${\cal L}$,

$\displaystyle\int\text{d}t\text{d}\phi\,\mu{\cal L}\,.$ (23)

If we were to consider the BTZ black hole instead of pure Poincaré patch,
linear combinations of the zero modes of the currents ${\cal L}$ and
$\bar{{\cal L}}$ would yield the black hole mass and angular momentum.
Therefore we have $[{\cal L}]=1$. In the above coupling term, this cancels the
inverse mass dimension of $\text{d}t$. Since the angular coordinate $\phi$ is
dimensionless, we find $[\mu]=0$ and correspondingly $[\epsilon]=0$. This is
consistent with the two-point function

$\displaystyle\expectationvalue{\mu(\phi)\mu(\theta)}=1+\frac{\epsilon^{2}}{2}\delta(\phi-\theta)\,.$
(24)

Since $[\epsilon]=0$, the disorder is marginal. To be more precise, we will
find that the disorder we implement is marginally relevant, since after
disordering and averaging the metric, there appear regions in the IR
$z\to\infty$ where our perturbative treatment breaks down.

Having discussed the setup and its characteristics, we are now ready to
analyse how the disorder affects the Poincaré patch geometry (15).

###  2.2 Averaged geometry

Using the setup described above, we proceed to analyse the disorder averaged
Poincaré patch. First, we compute the disorder averaged metric as follows. We
insert the chemical potentials (16) and (17) into the gauge field components
(5), (6), (7) and (8), with ${\cal L}$ and $\bar{{\cal L}}$ given by (12) and
(13), respectively, and setting $F=0=G$. The metric components then follow
from (14). Expanding to second order and averaging333For completeness, the
unaveraged components can be found in appendix A. using the above mean values
(22), we find

$\displaystyle\expectationvalue{g_{tt}}$
$\displaystyle=\left(-1+\frac{\epsilon\bar{\epsilon}}{2}\right)\frac{1}{z^{2}}+\frac{\epsilon\bar{\epsilon}}{12}\,,$
(25) $\displaystyle\expectationvalue{g_{t\phi}}$
$\displaystyle=\frac{\bar{\epsilon}^{2}-\epsilon^{2}}{24}\,,$
$\displaystyle\expectationvalue{g_{\phi\phi}}$
$\displaystyle=\frac{1}{z^{2}}+\frac{\epsilon^{2}+\bar{\epsilon}^{2}}{24}-\frac{\epsilon\bar{\epsilon}}{40}z^{2}\,,$
$\displaystyle\expectationvalue{g_{zt}}$
$\displaystyle=0=\expectationvalue{g_{z\phi}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\expectationvalue{g_{zz}}=\frac{1}{z^{2}}\,.$

As expected, to zeroth order in the disorder strengths this is the usual
Poincaré patch metric. As mentioned above, the metric components are invariant
under $(\epsilon,\bar{\epsilon})\to-(\epsilon,\bar{\epsilon})$, but not under
$(\epsilon,\bar{\epsilon})\to(\epsilon,-\bar{\epsilon})$. The averaged
geometry is in Fefferman–Graham form
$\expectationvalue{g_{zt}}=\expectationvalue{g_{z\phi}}=0$. For convenience,
in the following calculations, we rescale $t$ such that the leading order in
$\expectationvalue{g_{tt}}$ is equal to $-\frac{1}{z^{2}}$. Since we always
work perturbatively in $\epsilon$ and $\bar{\epsilon}$, this does not change
$\expectationvalue{g_{t\phi}}$.

Regarding the curvature properties of the averaged metric (25), we first note
that in the asymptotic region $z\to 0$, the metric describes a locally AdS
spacetime with Ricci scalar $R=-6$, Ricci tensor squared
$R_{\mu\nu}R^{\mu\nu}=12$ and vanishing Cotton tensor. This is consistent with
our choice of asymptotic boundary conditions. The above-mentioned rescaling of
$t$ does not interfere with this observation.

Now we analyse the geometry described by the averaged metric for general $z$.

$\displaystyle
R=-6+\frac{(\epsilon-\bar{\epsilon})^{2}}{12}z^{2}+\frac{\epsilon\bar{\epsilon}}{10}z^{4}\,.$
(26)

In the bulk a curvature singularity is present in the IR where $z\to\infty$.
Also, the Cotton tensor has a non-vanishing component

$\displaystyle C_{t\phi}=-\frac{3}{20}\epsilon\bar{\epsilon}z^{2}\,,$ (27)

with all other components equal to zero, so (25) in general differs from an
AdS spacetime. Moreover, there is a singularity of the causal structure
appearing at $g_{\phi\phi}(z_{0})=0$, with

$\displaystyle
z_{0}=\left(\frac{40}{\epsilon\bar{\epsilon}}\right)^{\frac{1}{4}}+\mathcal{O}(\sqrt{\epsilon})\,.$
(28)

In the region $z>z_{0}$, as the $\phi$ direction is compact, the spacetime has
closed timelike curves. This shows that at $z_{0}$ the spacetime ends, with
finite Ricci curvature.

To further characterise the singularity, we analyse whether it yields a well-
defined semiclassical approximation for a string [24]. If this is not the case
the supergravity approximation that we employ is not valid anymore. To test
this, we consider the string action for a small perturbation of a static
configuration $z(\phi)=z_{s}(\phi)+\delta z(\phi)$,

$\displaystyle S$
$\displaystyle=\frac{1}{2\pi\alpha^{\prime}}\int\text{d}\phi\sqrt{(z^{\prime})^{2}+g_{\phi\phi}(z)}$
$\displaystyle\approx\frac{1}{2\pi\alpha^{\prime}}\int\text{d}\phi\,\bigg{(}\sqrt{(z_{s}^{\prime})^{2}+g_{\phi\phi}(z_{s})}+\frac{1}{2}\frac{(\delta
z^{\prime})^{2}}{\sqrt{(z_{s}^{\prime})^{2}+g_{\phi\phi}(z_{s})}^{3}}\bigg{)}\,.$
(29)

If the square root vanishes, the semiclassical approximation for the
Nambu–Goto action (29) breaks down. For our averaged metric we find the real
root $z_{s}=z_{0}$ and thus the supergravity approximation breaks down in the
vicinity of $z_{0}$.

The appearance of singularities is not unexpected. They also appeared in prior
work on disorder [14] and [20]. As in these cases, the disorder we use is a
marginally relevant operator in the sense of the Harris criterion, therefore
drastic changes in the interior of the bulk, i.e. the IR region, can arise.

Having discussed the properties of the averaged metric from the bulk
perspective, in the next section we analyse the dual holographic state
utilizing holographic renormalisation [27].

##  3 Properties of the holographic averaged state

In the previous section, we showed how the averaged metric is obtained,
followed by a discussion of its curvature properties. Here, we study the
effects of the disorder on the boundary field theory using the holographic
dictionary. In particular, we calculate the boundary energy-momentum tensor
and the QNEC.

###  3.1 Energy-momentum tensor and trace anomaly

Here, we compute the energy-momentum tensor of the dual theory on the
boundary. Since the disorder averaged metric (25) is in Fefferman–Graham
gauge, we can do so by use of the standard result [27]

$\displaystyle\langle\langle
T_{ij}^{\text{ren}}\rangle\rangle=\frac{c}{12\pi}\left(\gamma_{ij}^{(2)}-\tr\big(\gamma^{(2)}\big{missing})\gamma_{ij}^{(0)}\right)\,,$
(30)

where $\gamma_{ij}$ is the metric induced on the boundary, expanded as

$\displaystyle\gamma_{ij}(z)=\gamma_{ij}^{(0)}\frac{1}{z^{2}}+\gamma_{ij}^{(2)}+...\,.$
(31)

The trace is understood in terms of $\gamma_{ij}^{(0)}$ as
$\tr\big(\gamma^{(2)}\big{missing})=\gamma^{(0)\,ij}\gamma_{ji}^{(2)}$. We
denote the expectation value as $\langle\langle\cdot\rangle\rangle$ to
distinguish it from disorder averages.

Applying (30) to the averaged metric in (25), we find the energy-momentum
tensor

$\displaystyle\langle\langle T_{ij}^{\text{ren}}\rangle\rangle$
$\displaystyle=\frac{c}{288\pi}\begin{pmatrix}2\epsilon\bar{\epsilon}&\bar{\epsilon}^{2}-\epsilon^{2}\\\
\bar{\epsilon}^{2}-\epsilon^{2}&\epsilon^{2}+\bar{\epsilon}^{2})\end{pmatrix}_{ij}\,.$
(32)

It is clear from this result that by including disorder, we introduce a small
amount of energy into the system. Provided that the disorder strengths
$\epsilon$ and $\bar{\epsilon}$ are different, we also introduce angular
momentum.

The trace of the energy-momentum tensor does not vanish in general,

$\displaystyle\tr(\langle\langle T_{ij}^{\text{ren}}\rangle\rangle)$
$\displaystyle=-\frac{c(\epsilon-\bar{\epsilon})^{2}}{288\pi}\,.$ (33)

However, since the disorder averaged metric induced on the boundary
$\langle\gamma_{ij}^{(0)}\rangle=\text{diag}(-1,1)$ is Ricci flat, the general
result for the trace anomaly in curved backgrounds

$\displaystyle\tr(\langle\langle
T_{ij}^{\text{ren}}\rangle\rangle)=\frac{c}{24\pi}R[\langle\gamma^{(0)}\rangle]\,,$
(34)

cannot be fulfilled for general disorder strengths $\epsilon$,
$\bar{\epsilon}$. For equal strengths, both sides vanish. Even though
$\tr\big(\langle\langle T_{ij}^{\text{ren}}\rangle\rangle\big{missing})\propto
R[\langle\gamma^{(0)}\rangle]$ is fulfilled for $\epsilon=\bar{\epsilon}$, the
bulk curvature (26) is still divergent in the IR, so these two features are
not in one-to-one correspondence. Nevertheless, as we will show in section 4,
there is a procedure to restore finite bulk curvature and absence of the trace
anomaly for general $\epsilon$ and $\bar{\epsilon}$. Before doing so, we
consider one additional holographic probe of our geometry, namely the QNEC.

###  3.2 Quantum null energy condition

QNEC locally constrains the null projections of the expectation value of the
energy-momentum tensor [28, 29, 30]. For QFT2 with a CFT2 UV-fixed point it
reads [28] (see also [31])

$2\pi\,\langle\langle T_{ij}\rangle\rangle k^{i}k^{j}\geq
S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\,.$ (35)

The right-hand side contains first and second variations of entanglement
entropy (EE) explained below as well as the UV central charge $c$.

The variations of EE work as follows. One of the endpoints of the entangling
interval is chosen arbitrarily, while the other must coincide with the
spacetime point where the left-hand side of the QNEC inequality (35) is
evaluated. Then the entangling region is deformed into the null direction
$k^{i}$, with a deformation parameter $\lambda$. As a consequence of this
lightlike deformation, EE depends on this parameter. Primes denote derivatives
with respect to $\lambda$, and after taking these derivatives $\lambda$ is set
to zero. For more details on QNEC in two dimensions and explicit examples see
[32].

We investigate now QNEC for our averaged field theory, to verify whether or
not QNEC holds/saturates/is violated. If the latter happened we had a nogo
result showing that the averaged field theory cannot be a unitary relativistic
QFT.

For states dual to three-dimensional vacuum Einstein solution QNEC saturates
[33, 34], which in our context implies that QNEC saturates for each of our
realisations. Therefore, also the averaged QNEC inequality trivially
saturates,

$\big{\langle}2\pi\,\langle\langle
T_{ij}\rangle\rangle\,k^{i}k^{j}\big{\rangle}=\big{\langle}S^{\prime\prime}+\frac{6}{c}\,\big{(}S^{\prime}\big{)}^{2}\big{\rangle}\,.$
(36)

The Cauchy–Schwarz inequality then implies a QNEC inequality for the averaged
EE

$\big{\langle}2\pi\,\langle\langle
T_{ij}\rangle\rangle\,k^{i}k^{j}\big{\rangle}\geq\langle
S\rangle^{\prime\prime}+\frac{6}{c}\,\big{(}\langle
S\rangle^{\prime}\big{)}^{2}\,.$ (37)

However, in our work, we have to differentiate between averaged quantities and
quantities computed from the averaged metric. Therefore, in the following, we
calculate both sides of (35) explicitly for the averaged geometry (25).

We start our analysis by calculating the right-hand side of (35) for the
averaged metric (25). The calculation follows the above description of
introducing a lightlike deformation $k^{i}$ parametrised by a small $\lambda$.
In the following, we choose the lightlike direction $k^{i}=(1,1)$. The details
of the calculation are included in appendix C, following the method described
in [32]. From the computation using (25), we find

$\displaystyle
S^{\prime\prime}\big{|}_{\lambda=0}+\frac{6}{c}(S^{\prime})^{2}\big{|}_{\lambda=0}=\frac{c}{3}\left[-\epsilon^{2}+\left(2-\frac{39}{25}l^{2}\right)\epsilon\bar{\epsilon}+11\bar{\epsilon}^{2}\right]\,,$
(38)

where $l$ is the size of the undeformed entangling interval. The fact that
this length appears proves that QNEC is not saturated, since $l$ cannot appear
in the left-hand side of (35). Computing this explicitly from (32), we find

$\displaystyle 2\pi\langle\langle T_{ij}^{\mathrm{ren}}\rangle\rangle
k^{i}k^{j}=c\left(-\epsilon^{2}+2\epsilon\bar{\epsilon}+3\bar{\epsilon}^{2}\right)\,.$
(39)

Comparing with (38), we find that whether QNEC holds or not depends on the
length $l$ and on the ratio of the disorder strengths
$a=\frac{\bar{\epsilon}}{\epsilon}$. QNEC is satisfied within a symmetric
region around $a=1$ for sufficiently large $l$. When the disorder strengths
are equal, QNEC always holds and even saturates for $l=0$. Considering also
the corresponding NEC condition, given by (39) via $\langle\langle
T_{ij}^{\mathrm{ren}}\rangle\rangle k^{i}k^{j}\geq 0$, we find that there are
regions in parameter space where NEC is violated while QNEC holds, as well as
the contrary. We summarise our results in figure 1.

Figure 1: Results of the QNEC/NEC analysis for (25). All combinations of
either QNEC or NEC being satisfied/violated exist for the averaged metric in
(25).

We point out for the averaged geometry in (25), the EE shows unphysical
behaviour in that it becomes negative for sufficiently large $l$; the explicit
result follows from (82) in appendix C for $\lambda=0$,

$\displaystyle
S=\frac{c}{3}\ln\frac{l}{z_{\mathrm{cut}}}+\frac{c}{864}\left[(\epsilon^{2}+\bar{\epsilon}^{2})l^{2}-\frac{3\epsilon\bar{\epsilon}l^{4}}{25}\right]\,.$
(40)

The possibility for negative $S$ is related to the singular behaviour of (25)
in the IR region. In particular, the turning point $z_{s}$ of the
Ryu–Takayanagi geodesic comes close to the causal singularity $z_{0}$ obtained
in (28), where our approximation breaks down. This feature of negative EE
again shows that (25) includes unphysical behaviour. Motivated by the
unphysical features associated with the averaged metric and its dual
holographic quantities, in the next section we discuss a resummation method
inspired by the Poincaré–Lindstedt resummation performed in earlier works on
disorder [14, 20].

##  4 Poincaré–Lindstedt inspired resummation

As discussed in the previous section, the averaged metric has several
unphysical properties. In this section, we resolve these deficiencies by a
resummation method that we explain below.

In earlier works on disorder in holography such as [14, 20], it was observed
that the averaged metric components contain secular terms, leading to
Lifshiftz scaling IR fixed points. In order to regulate the averaged
solutions, the so-called Poincaré–Lindstedt resummation technique was
employed. Inspired by this resummation, in the following we discuss an
approach to modify the averaged Poincaré patch metric (25).

The idea for the resummation in our case is to change the averaged metric such
that the curvature divergence is absent. To do so, we make an ansatz for the
resummed metric, satisfying the following conditions:

  * •

Since the divergence of the curvature is only in $z$, also the modifying
functions should only depend on $z$.

  * •

The divergent terms in (26) can be traced back to $\expectationvalue{g_{tt}}$
and $\expectationvalue{g_{\phi\phi}}$. Therefore only those components are
modified.

  * •

In the asymptotic region, $z\to 0$, the averaged metric is well behaved.
Therefore, we demand the modifying functions to approach one for $z\to 0$.

Since the modifying functions only depend on $z$ while the disorder is
associated with the $\phi$ direction, this modification commutes with the
disorder average (18). The resummation should be understood as a change of
ansatz on the metric level which, since it does not depend on the random
phases and therefore commutes with the averaging procedure, is valid both
before and after averaging. For suitable choices, this change of ansatz will
in particular cure the issues in the averaged metric derived in sections 2.2
and 3.1. We elaborate on the implementation of this ansatz in the Chern–Simons
formulation in section 5.

The above conditions lead us to the modified ansatz for the averaged metric

$\displaystyle\text{d}s^{2}=\frac{\expectationvalue{g_{tt}}}{\alpha(z)}\text{d}t^{2}+\frac{\text{d}z^{2}}{z^{2}}+2\expectationvalue{g_{t\phi}}\text{d}t\text{d}\phi+\frac{\expectationvalue{g_{\phi\phi}}}{\beta(z)}\text{d}\phi^{2},$
(41)

where $\alpha$ and $\beta$ are functions of $z$ and $\epsilon$,
$\bar{\epsilon}$. To satisfy the above conditions, we make a general ansatz

$\displaystyle\alpha(z)=1+\sum_{n=1}^{\infty}a_{n}(\epsilon,\bar{\epsilon})z^{n},\quad\beta(z)=1+\sum_{n=1}^{\infty}b_{n}(\epsilon,\bar{\epsilon})z^{n},$
(42)

where the coefficients $a_{n}$, $b_{n}$ can be expanded444This expansion may
be generalised to higher order than quadratic straightforwardly. in $\epsilon$
and $\bar{\epsilon}$,

$\displaystyle
a_{n}(\epsilon,\bar{\epsilon})=a_{1n}\epsilon^{2}+a_{2n}\epsilon\bar{\epsilon}+a_{3n}\bar{\epsilon}^{2},\quad
b_{n}(\epsilon,\bar{\epsilon})=b_{1n}\epsilon^{2}+b_{2n}\epsilon\bar{\epsilon}+b_{3n}\bar{\epsilon}^{2}.$
(43)

In the following, the undetermined coefficients $a_{ij}$ and $b_{ij}$ will be
fixed by recalculating the Ricci scalar for (41) to
$\mathcal{O}(\epsilon^{2})$ and demand the diverging terms to vanish. When
expanding (41) to second order, the terms of $\mathcal{O}(z^{n})$ in
$\alpha(z)$ and $\beta(z)$, i.e., the coefficients
$a_{n}(\epsilon,\bar{\epsilon})$ and $b_{n}(\epsilon,\bar{\epsilon})$,
contribute at $\mathcal{O}(z^{n-2})$ in the metric. Therefore, we expect that
only $a_{2}$, $b_{2}$, $a_{4}$ and $b_{4}$ need to be non-zero.

Computing the Ricci scalar for (41) to $\mathcal{O}(\epsilon^{2})$ and
demanding the result to be non-singular yields the following conditions:

$\displaystyle b_{14}=b_{34}$
$\displaystyle=a_{14}=a_{34}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle a_{24}+b_{24}$
$\displaystyle=-\frac{1}{40},$ (44) $\displaystyle a_{12}+b_{12}$
$\displaystyle=\frac{1}{24}=a_{32}+b_{32},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle a_{22}+b_{22}$
$\displaystyle=-\frac{1}{12}\,.$

As expected, all other coefficients have to vanish. Using the above relations
to express $a_{ij}$ in terms of $b_{ij}$, the resummed metric (41) expanded to
$\mathcal{O}(\epsilon^{2})$ is given by

$\displaystyle\text{d}s^{2}$
$\displaystyle=\left[-\frac{1}{z^{2}}+\left(\frac{1}{24}-b_{12}\right)\epsilon^{2}-b_{22}\epsilon\bar{\epsilon}+\left(\frac{1}{24}-b_{32}\right)\bar{\epsilon}^{2}-\left(\frac{1}{40}+b_{24}\right)z^{2}\epsilon\bar{\epsilon}\right]\text{d}t^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\left[\frac{1}{z^{2}}+\left(\frac{1}{24}-b_{12}\right)\epsilon^{2}-b_{22}\epsilon\bar{\epsilon}+\left(\frac{1}{24}-b_{32}\right)\bar{\epsilon}^{2}-\left(\frac{1}{40}+b_{24}\right)z^{2}\epsilon\bar{\epsilon}\right]\text{d}\phi^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{\text{d}z^{2}}{z^{2}}+2\left(\frac{\bar{\epsilon}^{2}-\epsilon^{2}}{24}\right)\text{d}t\text{d}\phi.$
(45)

By construction, the leading order terms in $z$ do not change. Hence, the
averaged boundary metric is still given by two-dimensional Minkowski space.

The remaining undetermined parameters will be fixed in the following, but
before we summarise what is achieved already by this resummation. To lowest
order in $\epsilon$ the scalar curvature is now finite by construction, $R=-6$
for the entire range of $z$. Calculating the Ricci tensor for (45), we find
that its trace-free part vanishes only provided that we set
$b_{24}=-\frac{1}{40}$. This condition can also be found from related
calculations: the three-dimensional Einstein equations with negative
cosmological constant are satisfied only for this value of $b_{24}$. In
addition, the Cotton tensor for the resummed metric does not vanish in
general,

$\displaystyle
C_{\mu\nu}=-(1+40b_{24})\epsilon\bar{\epsilon}z^{2}\begin{pmatrix}\frac{9}{80}&0&\frac{3}{20}\\\
0&0&0\\\ \frac{3}{20}&0&\frac{3}{16}\end{pmatrix}_{\mu\nu},$ (46)

but only when fixing $b_{24}=-\frac{1}{40}$. That is, removing the
$\mathcal{O}(z^{2})$ terms in (45) ensures that (45) is locally an AdS
spacetime everywhere.555We find the same result if we had inserted the
coefficients $b_{ij}$ in terms of $a_{ij}$. In this case, the terms $\propto
z^{2}$ in the metric are proportional to an overall factor of $a_{24}$, which
then has to be set to zero in order to satisfy Einsteins equations. This is
consistent with the constraint $a_{24}+b_{24}=-\frac{1}{40}$ and
$b_{24}=-\frac{1}{40}$.

While the above features concern the bulk, the resummation also effects the
dual holographic state. Since the terms of $\mathcal{O}(z^{0})$ in
$\expectationvalue{g_{tt}}$ and $\expectationvalue{g_{\phi\phi}}$ change, the
boundary energy-momentum tensor is now given by

$\displaystyle\langle\langle T_{tt}^{\mathrm{ren}}\rangle\rangle$
$\displaystyle=\frac{c}{288\pi}\left[(1-24b_{12})\epsilon^{2}-24b_{22}\epsilon\bar{\epsilon}+(1-24b_{32})\bar{\epsilon}^{2}\right],$
(47) $\displaystyle\langle\langle T_{t\phi}^{\mathrm{ren}}\rangle\rangle$
$\displaystyle=\frac{c(\bar{\epsilon}^{2}-\epsilon^{2})}{288\pi},$
$\displaystyle\langle\langle T_{\phi\phi}^{\mathrm{ren}}\rangle\rangle$
$\displaystyle=\langle\langle T_{tt}^{\mathrm{ren}}\rangle\rangle\,.$

A particularly interesting feature is that the trace vanishes, since the
diagonal terms in (47) are equal. Therefore, by the resummation method, we
obtain a metric with dual energy-momentum tensor that satisfies the usual
trace anomaly condition (34).

As apparent from the result above, the energy-momentum tensor now depends on
the three remaining coefficients, $b_{12}$, $b_{22}$, and $b_{32}$. They enter
in the diagonal terms, which can be thought of as a small mass induced by the
disorder. Correspondingly, the off-diagonal terms are understood as small
angular momentum.

In the following, we argue how to fix the open coefficients such that the
least additional energy is injected to the system. In particular, we demand
that the mass and angular momentum in the energy-momentum tensor are
consistent with the BPS bound

$\displaystyle M^{2}-J^{2}\geq 0\,.$ (48)

We start our argumentation by considering the limits where either $\epsilon=0$
or $\bar{\epsilon}=0$. Then, from (47), up to common constant coefficients, we
find

$\displaystyle\mathrm{for}\quad\bar{\epsilon}=0:$ $\displaystyle\quad
M\propto(1-24b_{12})\epsilon^{2},\quad J\propto-\epsilon^{2}$ (49)
$\displaystyle\mathrm{and\leavevmode\nobreak\ for}\quad\epsilon=0:$
$\displaystyle\quad M\propto(1-24b_{32})\bar{\epsilon}^{2},\quad
J\propto\bar{\epsilon}^{2}\,.$ (50)

Inserting these values into (48), we obtain conditions on either $b_{12}$ or
$b_{32}$,

$\displaystyle(1-24b_{12})^{2}-1$ $\displaystyle\geq
0\quad\to\quad\begin{cases}b_{12}\geq\frac{1}{12}\\\ b_{12}\leq 0\end{cases}$
(51) $\displaystyle\mathrm{and}\quad(1-24b_{32})^{2}-1$ $\displaystyle\geq
0\quad\to\quad\begin{cases}b_{32}\geq\frac{1}{12}\\\ b_{32}\leq 0\end{cases}$
(52)

The upper conditions on $b_{12}$ and $b_{32}$ yield negative contributions to
the mass. Therefore, we exclude them by further requiring $M\geq 0$. All
values $b_{12}<0$ and $b_{32}<0$ yield a positive contribution to the mass. In
order to introduce as little change by the resummation parameters as possible,
we set both of them to zero, consistent with the above bounds. The expression
for $M$ simplifies to

$\displaystyle
M\propto\epsilon^{2}-24\epsilon\bar{\epsilon}b_{22}+\bar{\epsilon}^{2}\,.$
(53)

For this mass, we analyse again the BPS bound for general $\epsilon$ and
$\bar{\epsilon}$. We find that (48) is satisfied when

$\displaystyle a\leq 0\land b_{22}\geq 0\quad\mathrm{or}\quad a\geq 0\land
b_{22}\leq 0,$ (54)

where $a=\frac{\bar{\epsilon}}{\epsilon}$ is the ratio between the disorder
parameters. The only choice for $b_{22}$ such that (48) holds for a generic
ratio $a$ is $b_{22}=0$. Therefore, with the parameters inserted in (47), we
obtain

$\displaystyle M\propto 1+a^{2}\quad\mathrm{and}\quad J\propto-1+a^{2},$ (55)

such that

$\displaystyle\langle\langle
T_{ij}^{\mathrm{ren}}\rangle\rangle=\frac{c\epsilon^{2}}{288\pi}\begin{pmatrix}1+a^{2}&-1+a^{2}\\\
-1+a^{2}&1+a^{2}\end{pmatrix}\,.$ (56)

A particularly nice observation related to this choice of resummation
parameters is that $\expectationvalue{g_{\phi\phi}}$ cannot have any real
roots and does not change sign, since it consists of a sum of manifestly
positive terms. Therefore, also the closed timelike curves are removed by the
resummation. Moreover, also the semiclassical approximation is well-defined
for a string (29) on the resummed metric.

Although these are promising results, we point out that the above description
has certain differences from the method used in [14]. The most pertinent is
that the Poincaré–Lindstedt method was employed in [14] before averaging while
we choose to change the effective metric after averaging. Furthermore, while
we encounter IR-divergences in curvature invariants, in their case only the
metric components are IR-divergent while curvature remains finite.

Having discussed some basic geometric quantities as well as boundary
properties of the resummed metric, in the remainder of this section we analyse
the effect of the resummation on QNEC.

Since the resummation leads to an averaged geometry which is a solution to the
AdS vacuum Einstein equations, we also expect a change in the result for QNEC.
Doing the calculation for (45), we find that the right-hand side of (35) is
given by

$\displaystyle
S^{\prime\prime}\big{|}_{\lambda=0}+\frac{6}{c}\left(S^{\prime}\right)^{2}\big{|}_{\lambda=0}=\frac{c}{3}\left[-b_{12}\epsilon^{2}-b_{22}\epsilon\bar{\epsilon}+\left(\frac{1}{12}-b_{32}\right)\bar{\epsilon}^{2}\right]\,.$
(57)

The energy-momentum tensor after resummation is given in (47). Projecting onto
the lightlike direction $k^{i}(1,1)$ yields

$\displaystyle 2\pi\langle\langle T_{ij}^{\mathrm{ren}}\rangle\rangle
k^{i}k^{j}=\frac{c}{3}\left[-b_{12}\epsilon^{2}-b_{22}\epsilon\bar{\epsilon}+\left(\frac{1}{12}-b_{32}\right)\bar{\epsilon}^{2}\right],$
(58)

so (35) saturates independently of $a=\frac{\bar{\epsilon}}{\epsilon}$, $l$,
or the remaining parameters of the resummation. This result is consistent with
the fact that the resummed geometry is a solution to the vacuum Einstein
equations and thereby a Bañados geometry, since for all Bañados geometries
QNEC saturates [32].

In particular, this means that also the values $b_{12}=b_{22}=b_{32}=0$ are
allowed, for which we argued above. In this case, also NEC given by (58) via
$\langle\langle T_{ij}^{\mathrm{ren}}\rangle\rangle k^{i}k^{j}\geq 0$ is
satisfied.

As a final comment, after resummation the EE is positive and receives only
positive perturbative corrections. Explicitly, it is given by (82) for
$\lambda=0$ without the term $\propto\epsilon\bar{\epsilon}l^{4}$,

$\displaystyle
S=\frac{c}{3}\ln\frac{l}{z_{\mathrm{cut}}}+\frac{c(\epsilon^{2}+\bar{\epsilon}^{2})l^{2}}{864}\,.$
(59)

##  5 Conclusions and outlook

In this paper, we studied the effect of random disorder on the Poincaré patch
solution of the three-dimensional Einstein equations for the AdS vacuum.
Starting from the Chern–Simons formulation of AdS3 gravity with Brown–Henneaux
boundary conditions, we used the left- and right-moving chemical potentials
$\mu$ and $\bar{\mu}$ to introduce disorder. Inspired by earlier work on
disorder in holography, we chose a convenient ansatz for the disorder
potential and worked perturbatively to second order in the disorder strengths
$\epsilon$ and $\bar{\epsilon}$. This enabled us to perform an analytic
average. Before discussing our findings, it should be noted that this choice
of implementing the disorder is not unique. An alternative approach would be
to directly disorder the boundary metric on which the dual 2$D$ CFT lives. It
is not obvious whether both approaches lead to the same results.

Having obtained the averaged metric, we first studied its bulk properties. As
expected from our choice of boundary conditions, in the ultraviolet region
$z\to 0$ we maintained asymptotic AdS behaviour, i.e., the Ricci scalar was
constant and equal to $-6$. Furthermore, the trace-free Ricci tensor as well
as the Cotton tensor vanished asymptotically. Deeper in the bulk, we found
that this is no longer true. All of the quantities mentioned before receive
corrections quadratic in $\epsilon$ and $\bar{\epsilon}$. These corrections
depended on the holographic coordinate $z$. This resulted in a curvature
divergence and a corresponding naked singularity in the IR region
$z\to\infty$, implying a breakdown of our perturbative treatment. Such
divergent behaviour was not unexpected, since similar effects appeared also in
earlier work on disorder [14, 20]. Moreover, by the Harris criterion, the
disorder we used is marginally relevant. Therefore, large changes in the IR
region can happen. We further found that the averaged metric did not yield a
well-defined semiclassical approximation for a string, which is related to the
fact that the averaged metric component $\expectationvalue{g_{\phi\phi}}$ has
real roots $z_{0}$. This also implied that the averaged metric had a
singularity in the causal structure at $z_{0}$. Since the $\phi$-direction is
compact, the region $z>z_{0}$ includes closed timelike curves, indicating that
the spacetime actually ends at $z_{0}$ with finite Ricci curvature.

We also studied the holographic state dual to the averaged metric. Computing
the energy-momentum tensor, we found a non-vanishing trace. Since the metric
induced on the boundary is just two-dimensional Minkowski space, the relation
between trace anomaly and boundary curvature was not fulfilled. We also
analysed the entanglement properties of the averaged geometry. We found that
the EE can become negative for sufficiently large length of the entangling
region $l$ in the boundary theory, related to the breakdown of perturbation
theory in the bulk IR region. Furthermore, we computed QNEC for the averaged
metric. It depended on the length $l$ and the ratio of the disorder strengths
$a$ whether or not it was fulfilled. Moreover, we also analysed NEC and found
that it can be violated as well, see Figure 1.

In earlier works on disorder, to regulate diverging behaviour, the so-called
Poincaré–Lindstedt method was employed. Inspired by this, we use an analogous
method to cure the above-mentioned deficiencies of the averaged geometry.
Although we modify only the averaged geometry, the averaging procedure and our
resummation method commute, since the resummation functions do not depend on
the random phases sourcing the disorder. We rescaled the averaged metric
components by the resummation functions containing undetermined coefficients.
By computing quantities such as the Ricci curvature and the trace-free Ricci
tensor, we obtain constraints on these coefficients. By our method, we were
able to restore finite curvature throughout the entire bulk. The resummed
metric was a solution to the AdS3 vacuum Einstein equations with vanishing
Cotton tensor. Furthermore, the closed timelike curves were removed and the
semiclassical approximation for a string could be defined. In the dual theory,
the energy-momentum tensor on the boundary was traceless, such that the
conformal anomaly equation was satisfied. The EE was manifestly positive and
QNEC saturated. Three of the resummation coefficients entered the boundary
energy-momentum tensor of the resummed metric. We fixed them to zero by
requiring consistency with the BPS bound for generic values of $\epsilon$ and
$\bar{\epsilon}$.

Our analysis conducted in this work enables numerous future research projects.
In the following, we give a list of questions and issues that we have not
answered or studied in this work, but which might be interesting to study in
the future.

#### Effective Theory.

Before averaging the metric, the dynamics of the theory is determined by the
Einstein–Hilbert action. After averaging, this is no longer true. The question
arises if one can find an effective action where the averaged metric follows
as solution to the equations of motion. We made two such attempts by
considering topologically massive gravity (TMG) and Einstein-dilaton gravity.
For TMG, the Chern–Simons term yielding the non-vanishing graviton mass has a
coupling constant $\mu$. Imposing the Hamiltonian constraint, Eq. (4) in [35],
we found that the resulting $\mu$ does not have a constant asymptotic piece.
Therefore, (25) cannot be a solution to TMG [35]. To check Einstein-dilaton
gravity, we followed the steps discussed in [36]. To calculate the dilaton
field from the metric, the averaged metric components need to satisfy certain
convexity conditions. Unfortunately, the averaged metric does not satisfy
these conditions. Finding a bulk effective theory would significantly improve
the understanding of the disorder averaging in AdS/CFT, and in particular
allow to study the dual field theory directly.

#### Poincaré–Lindstedt inspired resummation.

While it is clear how our resummation method works on the metric level, it is
currently unclear how to implement it in the Chern–Simons formulation. This
may require to go beyond the radial gauge (3)-(4).

Another question regarding the resummation method is tied to the effective
theory. Above we discussed in great detail how the resummation regulates the
averaged geometry. If we had an effective theory at hand, it will be
interesting to study how it affects the resummation.

#### Disordering different background geometries.

For simplicity, in the current work we restricted our focus on the Poincaré
patch metric. On top of this, we introduced disorder. From the first order
formulation of AdS3 gravity that we introduced in section 2, it is
straightforward to extend to other cases. Simple examples are the BTZ black
hole, global AdS3, or conical defect geometries. These examples are obtained
simply by fixing the functions $F$ and $G$ in (12) and (13) to be non-
vanishing constants. It will be interesting to see if there arise
qualitatively different features after averaging the corresponding disordered
metrics.

#### Beyond perturbation theory.

Throughout the entire paper, we worked perturbatively in the disorder
strengths $\epsilon$ and $\bar{\epsilon}$. While this enables us to perform
the averages analytically, it begs the question to what extent the
perturbative treatment affects the result. One might expect that in a non-
perturbative treatment, some of the problematic features are absent. Since the
non-perturbative calculation involves averages of inverse powers of $\mu$ and
$\bar{\mu}$, an analytic answer to this question is hard to obtain. However, a
numeric analysis will be very useful to properly address these questions.

#### Different boundary conditions.

In our procedure, we average over descendant geometries of Bañados type, which
include a conical singularity in the IR. It is known that by relaxing the
Brown–Henneaux boundary conditions to e.g. near horizon boundary conditions,
this conical singular behaviour can be removed [37, 38]. It is reasonable to
assume that different boundary conditions might lead to an averaged metric
with regular curvature. To answer this question, but also to study disorder in
2$D$ CFTs with extended symmetries [23], it is interesting to study disorder
averages with different boundary conditions.

## Acknowledgements

We thank Souvik Banerjee, Johanna Erdmenger, Simeon Hellerman, Ioannis
Papadimitriou and Shahin Sheikh-Jabbari for useful discussions.

M. D., R. M. and S. Z. are grateful for the kind hospitality during their stay
at TU Wien where a part of this project was realised.

#### Funding information

M. D., R. M. and S. Z. acknowledge support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy through the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id
390858490). Their work is furthermore supported via project-id 258499086 - SFB
1170 ‘ToCoTronics’. D. G. was supported by the Austrian Science Fund (FWF),
projects P 30822, P 32581 and P 33789.

##  Appendix A Expanded metric components

In this appendix, we give the unaveraged metric components, following from
inserting (16) and (17) into (5), (6), (7) and (8), with ${\cal L}$ and
${\cal\bar{L}}$ given by (12) and (13), and employing (14) with the group
element $b=\exp(\rho T_{0})$. We use the SL(2,$\mathbb{R}$) generators
represented as

$\displaystyle T_{+}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix},\quad
T_{0}=\frac{1}{2}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\quad\mathrm{and}\quad
T_{-}=\begin{pmatrix}0&-1\\\ 0&0\end{pmatrix}$ (60)

such that the trace of two generators is given by

$\displaystyle\tr(T_{a}T_{b})=\begin{pmatrix}0&0&-1\\\ 0&\frac{1}{2}&0\\\
-1&0&0\end{pmatrix}$ (61)

where we ordered $(+,0,-)$.

Converting to the holographic coordiante $z=\exp(-\rho)$, we find the
following metric components:

$\displaystyle g_{tt}$
$\displaystyle=\left(-1+(\bar{\epsilon}-\epsilon)f+\epsilon\bar{\epsilon}f^{2}\right)\frac{1}{z^{2}}+\frac{\epsilon\bar{\epsilon}}{2}f^{\prime\,2}$
(62a) $\displaystyle g_{tz}$
$\displaystyle=-\frac{(\bar{\epsilon}+\epsilon)f^{\prime}}{2z}$ (62b)
$\displaystyle g_{t\phi}$
$\displaystyle=\frac{\epsilon+\bar{\epsilon}}{2}f\frac{1}{z^{2}}+\frac{\bar{\epsilon}^{2}-\epsilon^{2}}{4}f^{\prime\,2}+\frac{\epsilon+\bar{\epsilon}}{4}f^{\prime\prime}$
(62c) $\displaystyle g_{zz}$ $\displaystyle=\frac{1}{z^{2}}$ (62d)
$\displaystyle g_{z\phi}$ $\displaystyle=0$ (62e) $\displaystyle g_{\phi\phi}$
$\displaystyle=\frac{1}{z^{2}}-\frac{\bar{\epsilon}^{2}+\epsilon^{2}}{2}ff^{\prime\prime}-\frac{\bar{\epsilon}^{2}+\epsilon^{2}}{4}f^{\prime\,2}+\frac{\epsilon-\bar{\epsilon}}{2}f^{\prime\prime}-\frac{\epsilon\bar{\epsilon}}{4}f^{\prime\prime\,2}z^{2}.$
(62f)

##  Appendix B Disorder with relative phase shift

In this appendix, we discuss the effect of a more general ansatz for the
disorder functions. As mentioned in the main text, we could allow for random
phases $\gamma_{n}$ and $\bar{\gamma}_{n}$ with a relative phase shift in the
intervals that they are drawn from, $[0,2\pi)$ and $[\alpha,2\pi+\alpha)$,
respectively. Compared to (16) and (17), this relative phase shift in the
intervals results in a relative phase shift in the cosines,

$\displaystyle
f(\phi)=\frac{1}{\sqrt{N}}\sum_{n=1}^{N}\cos(\frac{n}{N}\phi+\gamma_{n})\quad\mathrm{and}\quad\bar{f}(\phi)=\frac{1}{\sqrt{N}}\sum_{n=1}^{N}\cos(\frac{n}{N}\phi+\gamma_{n}+\alpha).$
(63)

Using this ansatz, we first observe that all of the one point functions still
vanish,

$\displaystyle\expectationvalue{f}=\expectationvalue{\bar{f}}=0.$ (64)

Moreover, all disorder averages without mixing of $f$ and $\bar{f}$ are left
invariant,

$\displaystyle\expectationvalue{f^{2}}=\expectationvalue{\bar{f}^{2}}=\frac{1}{2},\quad\expectationvalue{(f^{\prime})^{2}}=\expectationvalue{(\bar{f}^{\prime})^{2}}=\frac{1}{6},$
(65)
$\displaystyle\expectationvalue{ff^{\prime}}=\expectationvalue{\bar{f}\bar{f}^{\prime}}=0,\quad\expectationvalue{f\bar{f}^{\prime\prime}}=\expectationvalue{f^{\prime\prime}\bar{f}}=-\frac{1}{6}.$
(66)

All of the mixed averages receive a correction depending on this phase,

$\displaystyle\expectationvalue{f^{\prime}\bar{f}^{\prime}}=\frac{\cos\alpha}{6},\quad\expectationvalue{f\bar{f}^{\prime\prime}}=\expectationvalue{f^{\prime\prime}\bar{f}}=-\frac{\cos\alpha}{6},\quad\expectationvalue{f^{\prime}\bar{f}^{\prime\prime}}=-\expectationvalue{f^{\prime\prime}\bar{f}^{\prime}}=-\frac{\sin\alpha}{8},$
(67)
$\displaystyle\expectationvalue{f\bar{f}}=\frac{\cos\alpha}{2},\quad\expectationvalue{f^{\prime\prime}\bar{f}^{\prime\prime}}=\frac{\cos\alpha}{10},\quad\expectationvalue{f\bar{f}^{\prime}}=-\expectationvalue{f^{\prime}\bar{f}}=-\frac{\sin\alpha}{4}.$
(68)

Consistently, $\alpha\to 0$ reduces back to the averages discussed in the main
text.

Although many of the averages are modified, the resulting averaged metric and
Ricci curvature do not differ much compared to (25) and (26),

$\displaystyle\big{\langle}g^{(\alpha)}\big{\rangle}$
$\displaystyle=\begin{pmatrix}\left(-1+\frac{\epsilon\bar{\epsilon}}{2}\cos\alpha\right)\frac{1}{z^{2}}+\frac{\epsilon\bar{\epsilon}}{12}\cos\alpha&0&\frac{\bar{\epsilon}^{2}-\epsilon^{2}}{24}\\\
0&\frac{1}{z^{2}}&0\\\
\frac{\bar{\epsilon}^{2}-\epsilon^{2}}{24}&0&\frac{1}{z^{2}}+\frac{\epsilon^{2}+\bar{\epsilon}^{2}}{24}-\frac{\epsilon\bar{\epsilon}}{40}z^{2}\cos\alpha\end{pmatrix}$
(69) $\displaystyle\mathrm{and}\quad R^{(\alpha)}$
$\displaystyle=-6+\frac{\epsilon^{2}-2\epsilon\bar{\epsilon}\cos\alpha+\bar{\epsilon^{2}}}{12}z^{2}+\frac{\epsilon\bar{\epsilon}}{10}z^{4}\cos\alpha.$
(70)

As expected from the changes in the averages given above, the phase shift only
appears in terms $\propto\epsilon\bar{\epsilon}$. A particular interesting
value might be $\alpha=\frac{\pi}{2}$: in this case, the $\mathcal{O}(z^{2})$
term in $\big{\langle}g_{\phi\phi}^{(\alpha)}\big{\rangle}$ vanishes and
$\big{\langle}g_{tt}^{(\alpha)}\big{\rangle}=-\frac{1}{z^{2}}$. However, the
curvature still contains a divergent piece $\propto z^{2}$. To summarise this
brief analysis, including a relative phase shift does not yield an obvious
advantage compared to our analysis in the main text.

##  Appendix C Details of the QNEC calculation

In this appendix, we explain in detail how the results for the QNEC conditions
are obtained following the method discussed in section 2.6 in [32]. We will
first explain the idea and give explicit expressions afterwards.

The calculation results from the following idea: the EE $S$ is computed by a
static geodesic of length $l$, connecting two points in the boundary CFT. To
obtain QNEC, we need to know how $S$ changes under small null deformations of
the entangling region. For concreteness, this means we change one of the
endpoints by a vector of the form $k=\lambda(1,1)$, where $\lambda$ is assumed
to be small compared to $l$. Therefore, and also due to the fact that we will
need only second and first derivatives of $S$ evaluated at $\lambda=0$, we can
always work perturbatively in $\lambda$. For our disordered metric, we will
also work perturbatively in $\epsilon$. The calculation works as follows: we
choose the spatial boundary coordinate $\phi$ as affine parameter to denote
the geodesic Lagrangian ${\cal L}$ in terms of $\dot{t}$, $\dot{z}$ and $z$,
where a dot refers to a derivative w.r.t $\phi$. This makes use of the
existence of the Killing vectors $\partial_{t}$ and $\partial_{\phi}$, i.e.
the metric components depend only on the radial coordinate $z$. The area, and
thereby the EE, is then given by the integral of ${\cal L}$ over $\phi$. It is
however more convenient to reexpress this as an integral over $z$. To do so,
we make use of the Noether charges of $\partial_{\phi}$ and $\partial_{t}$,
$Q_{1}$ and $Q_{2}$ respectively, where for calculational convenience we
evaluate $Q_{1}$ at the turning point $z_{\ast}$ of the geodesic. It is then
possible to express $\dot{t}$ and $\dot{z}$ as functions $h_{t}$ and $h_{z}$
depending on $z$, $z_{\ast}$ and $\Lambda$, where $\Lambda$ is a combination
of the two charges. For example, one may take $\Lambda=Q_{2}/Q_{1}$, which
turns out to be very convenient in the case of diagonal metrics. Furthermore,
$\Lambda$ has to vanish when the null deformation is turned of. Therefore,
$\Lambda$ and $\lambda$ are linearly related to lowest order, such that
expansions in $\Lambda$ are also possible. The null deformation changes the
extent of the entangling region by $\lambda$ both in the spatial and temporal
direction. However using $h_{t}$ and $h_{z}$, the change of size of the
entangling interval can be calculated in terms of $z_{\ast}$ and $\Lambda$,
such that these two parameters can be expressed in terms of $l$ and $\lambda$.
Finally, to calculate the EE the integral can be written as an integral over
$z$, using again $h_{z}$, where $z_{\ast}$ and $\Lambda$ are expressed in
terms of $l$ and $\lambda$. Having obtained this result, it is straightforward
to calculate the QNEC combination in (35).

After this more general discussion, we present in detail the calculation for
the metric in (25). The geodesic Lagrangian

$\displaystyle{\cal
L}(\dot{t},\dot{z},z)=\sqrt{\expectationvalue{g_{\phi\phi}}+2\expectationvalue{g_{t\phi}}\dot{t}+\expectationvalue{g_{tt}}\dot{t}^{2}+\frac{\dot{z}^{2}}{z^{2}}},$
(71)

has the Noether charges

$\displaystyle Q_{1}$ $\displaystyle=\dot{z}\frac{\partial{\cal
L}}{\partial\dot{z}}+\dot{t}\frac{\partial{\cal L}}{\partial\dot{t}}-{\cal
L}=-\frac{\expectationvalue{g_{\phi\phi}}+\expectationvalue{g_{t\phi}}\dot{t}}{{\cal
L}},$ (72) $\displaystyle Q_{2}$ $\displaystyle=\frac{\partial{\cal
L}}{\partial\dot{t}}=\frac{\expectationvalue{g_{t\phi}}+\expectationvalue{g_{tt}}\dot{t}}{{\cal
L}}\,.$ (73)

Evaluating $Q_{1}$ at $z_{\ast}$ and solving for $\dot{t}$ and $\dot{z}$, the
solutions are of the form

$\displaystyle\dot{t}=\Lambda h_{t}(z,z_{\ast},\Lambda)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \dot{z}=h_{z}(z,z_{\ast},\Lambda).$
(74)

Inserting the metric components (25), we find to $\mathcal{O}(\epsilon^{2})$

$\displaystyle h_{t}$
$\displaystyle=\Lambda\left[1+\frac{z^{2}\epsilon^{2}}{24\Lambda}\left(-(1-\Lambda+\Lambda^{2})+\left(2-\frac{3}{5}z^{2}\right)a\Lambda+(1+\Lambda+\Lambda^{2})a^{2}\right)\right],$
(75) $\displaystyle h_{z}$
$\displaystyle=-\frac{\sqrt{(z_{\ast}^{2}-z^{2})(1-\Lambda^{2})}}{z}-\frac{\epsilon^{2}}{240z}\sqrt{\frac{z_{\ast}^{2}-z^{2}}{1-\Lambda^{2}}}\bigg{[}5(2\Lambda-1)z_{\ast}^{2}+5(1-2\Lambda^{2}+2\Lambda^{3})z^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\big{(}3z_{\ast}^{4}-10z_{\ast}^{2}\Lambda^{2}+z^{2}(3z_{\ast}^{2}-10\Lambda^{2})+z^{4}(6\Lambda^{2}-3)\big{)}a$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\big{(}5(1+2\Lambda)z_{\ast}^{2}+5(-1+2\Lambda^{2}+2\Lambda^{3})z^{2}\big{)}a^{2}\bigg{]},$
(76)

where $\Lambda=\frac{Q_{2}}{Q_{1}}$ and $a=\frac{\bar{\epsilon}}{\epsilon}$ is
the ratio of the disorder strengths.

The next step is to calculate the expressions for the spatial and temporal
deformation of the entangling interval. In principle, they are given by
$\phi$\- and $t$-integrals respectively, but using the above functions, both
can be computed as a $z$-integral:

$\displaystyle\lambda$
$\displaystyle=\int\limits_{0}^{\lambda}\text{d}t=2\Lambda\int\limits_{z_{\ast}}^{0}\text{d}z\frac{h_{t}}{h_{z}},$
(77) $\displaystyle l+\lambda$
$\displaystyle=\int\limits_{0}^{l+\lambda}\text{d}\phi=2\int\limits_{z_{\ast}}^{0}\frac{\text{d}z}{h_{z}}.$
(78)

These integrals can be performed after expanding the argument to second order
in $\epsilon$. Solving for $\Lambda$ and $z_{\ast}$ yields

$\displaystyle\Lambda$
$\displaystyle=\frac{\lambda}{2z_{\ast}}+\epsilon^{2}\left[\frac{a^{2}-1}{36}z_{\ast}^{2}+\frac{-125(1+a^{2})+(99z_{\ast}^{2}-200)a}{7200}z_{\ast}\lambda+\frac{1-a^{2}}{96}\lambda^{2}\right],$
(79) $\displaystyle z_{\ast}$
$\displaystyle=\frac{l}{2}\left(1+\frac{51al^{2}-100(1+a^{2})}{57600}l^{2}\epsilon^{2}\right)+\frac{\lambda}{2}\left(1+\frac{51al^{2}-20-100a^{2}}{11520}l^{2}\epsilon^{2}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{\lambda^{2}}{4l}\left(1-\frac{300-1300a^{2}-(400-867l^{2})a}{57600}l^{2}\epsilon^{2}\right).$
(80)

Finally, the area integral can be evaluated to give a result depending only on
$l$, $\lambda$ and the radial cutoff $z_{\text{cut}}$

$\displaystyle A=2\int\limits_{z_{\ast}}^{z_{\text{cut}}}\text{d}z\frac{{\cal
L}(\Lambda h_{t},h_{z},z)}{h_{z}}\,.$ (81)

The explicit result using $l\gg z_{\text{cut}}$, expanded to second order in
$\epsilon$ and $\lambda$, is given by

$\displaystyle S(\lambda)$
$\displaystyle=\frac{c}{3}\ln\frac{l}{z_{\text{cut}}}+\frac{c\lambda}{3l}-\frac{c\lambda^{2}}{3l^{2}}+\frac{c\epsilon^{2}}{864}\Big{[}\left(1+a^{2}\right)l^{2}-\frac{3al^{4}}{25}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\left(4a^{2}-\frac{12al^{2}}{25}\right)l\lambda$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\left(-1+2a+3a^{2}-\frac{3al^{2}}{5}\right)\lambda^{2}\Big{]}.$ (82)

Taking derivatives w.r.t $\lambda$ and setting $\lambda=0$ afterwards, we find
the result given in (38). While the above $S$ is calculated making use of
$l\gg z_{\text{cut}}$ in the last step, we find the same result (38) when
using the full result from (81), without assuming large $l$. The calculation
for the resummed geometry (45) works analogously and is even considerably
shorter.

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