could result in a useful model for finite-temperature lattice results in
$(2+1)$ dimensions. Furthermore, we want to keep the metric ansatz simple so
we keep the transverse and internal space symmetries, only modifying temporal
and holographic directions. This is a top-down holographic model, so the
metric includes a compact internal space; we keep the ten-dimensional Newton
constant explicit below. The temperature in this family of geometries,
assuming again $a(z_{h})=1$, is

$\displaystyle T=\frac{5}{4\pi}\frac{\sqrt{z_{p}}}{z_{h}^{3/2}}\ .$ (3.16)

The integrals for the strip width and entanglement entropy can be derived in
the same way as in the $(3+1)$-dimensional case. The relevant integrals are

$\displaystyle T\ell(z_{*})$
$\displaystyle=\frac{5}{2\pi}\frac{1}{z_{h}^{3/2}z_{*}^{7/2}}\int_{0}^{z_{*}}\frac{a(z)}{\sqrt{b(z)}}\frac{z^{4}}{\sqrt{1-(z/z_{*})^{7}}}\mathrm{d}z$
(3.17) $\displaystyle\frac{1}{TV}S_{EE}(z_{*})$
$\displaystyle=\frac{16\pi^{4}z_{p}^{17/2}}{75G_{N_{\textrm{\scriptsize
c}}}^{(10)}\sqrt{z_{h}}}\Bigg{[}\frac{z_{h}^{2}}{\epsilon^{2}}-\frac{z_{h}^{2}}{z_{*}^{2}}$
$\displaystyle\hskip
56.9055pt+2z_{h}^{2}\int_{0}^{z_{*}}\frac{1}{z^{3}}\mathopen{}\mathclose{{}\left(\frac{a(z)}{\sqrt{b(z)}}\frac{1}{\sqrt{1-(z/z_{*})^{7}}}-1}\right)\mathrm{d}z\Bigg{]}\
.$ (3.18)

Like before, the derivative of $S_{EE}(\ell)$ is a simple function, free of
integral expressions:

$\displaystyle\frac{4G_{N_{\textrm{\scriptsize
c}}}^{(4)}}{V}\frac{\mathrm{d}S_{EE}}{\mathrm{d}\ell}=\frac{z_{p}^{7/2}}{z_{*}^{7/2}}\
,$ (3.19)

where if interested in this relation in terms of ten-dimensional Newton
constant one uses ${\rm{Vol}}(S^{6})=16\pi^{3}/15$ for the volume of the
internal space.

We can work out the UV limit of $S_{EE}$ by studying (3.17) and (3.18) in the
limit $z_{*}\ll z_{0}$. The result is [34],

$\displaystyle\frac{1}{TV}S_{EE}(\ell)=\frac{16\pi^{4}z_{p}^{17/2}}{75G_{N_{\textrm{\scriptsize
c}}}^{(10)}\sqrt{z_{h}}}\mathopen{}\mathclose{{}\left[\frac{z_{h}^{2}}{\epsilon^{2}}-\frac{3\pi^{7/6}}{7^{1/3}98}\frac{\Gamma\mathopen{}\mathclose{{}\left(\frac{5}{7}}\right)^{7/3}}{\Gamma\mathopen{}\mathclose{{}\left(\frac{17}{14}}\right)^{7/3}}\frac{z_{h}^{2}}{z_{p}^{2/3}\ell^{4/3}}}\right]\
,\quad T\ell\ll 1\ .$ (3.20)

We note that in order to avoid flowing to the weakly coupled Yang-Mills
regime, we need to work with finite UV cutoff $\epsilon$, however, we are only
interested in terms depending on $\ell$ in this work.

On the other hand, in the large width limit

$\displaystyle\frac{1}{TV}S_{EE}(\ell)=\frac{16\pi^{4}z_{p}^{17/2}}{75G_{N_{\textrm{\scriptsize
c}}}^{(10)}\sqrt{z_{h}}}\mathopen{}\mathclose{{}\left(\frac{z_{h}^{2}}{\epsilon^{2}}+\frac{\sqrt{z_{p}}}{z_{h}^{3/2}}\ell+\frac{2\sqrt{2}\Gamma\mathopen{}\mathclose{{}\left(-\frac{2}{5}}\right)}{5\Gamma\mathopen{}\mathclose{{}\left(\frac{1}{10}}\right)}}\right)\
,\quad T\ell\gg 1\ .$ (3.21)

Here one should also be wary that the strict IR limit is probably not
trustworthy at face value, but better captured using 3D superconformal field
theory via M2-branes [32, 47]. Our focus is on the middle term in the bracket
in (3.21) (or actually that in (3.19)) that arises solely from black hole
horizon and corresponds to the thermal behavior. We believe that it
universally describes the thermal entropy of the deconfining phase in the
field theory. Indeed, below we show evidence for the leading $S_{EE}\sim
T^{7/3}\ell$ (as follows from plugging (3.16) in (3.21)) behavior as extracted
from lattice computations.

###  3.2 Bulk reconstruction

We now have expressions for the holographic entanglement entropy in both
$(3+1)$ and $(2+1)$ dimensions. In this work, however, we only consider the
reconstruction in $(2+1)$ dimensions since this is the case for which we have
sufficient data from the lattice. The formulas require as input the bulk
metric and output entropies. Since we have also computed the derivative of
$S_{EE}(\ell)$ with lattice simulations, it is interesting to ask whether it
is possible to find such a bulk geometry for which the holographic predictions
would agree with the lattice results. Schemes like this where the holographic
bulk geometry is inferred from QFT quantities is called bulk reconstruction.
We will use statistical methods for reconstructing the bulk from lattice data
[30]. The lattice data consists of a set of $M$ measurements of
$\mathrm{d}S_{EE}/\mathrm{d}\ell$ and corresponding uncertainties, $\sigma$,
for different widths $\ell$:

$\displaystyle\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\frac{1}{T^{2}V}\frac{\mathrm{d}S_{EE}}{\mathrm{d}\ell}}\right)_{i}\
,\ \ell_{i}\ ,\ \sigma_{i}}\right\\}\ ,\quad
i\in\mathopen{}\mathclose{{}\left\\{1,\ldots,M}\right\\}\ .$ (3.22)

The bulk geometry contains only one free function, $a(z)$, which we
parametrize as

$\displaystyle
a(z)=1+\sum_{i=1}^{N_{\text{basis}}}a_{i}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{z}{z_{h}}}\right)^{i}-\mathopen{}\mathclose{{}\left(\frac{z}{z_{h}}}\right)^{N_{\text{basis}}+1}}\right]\
,$ (3.23)

with $a_{i}\in\mathbb{R}$ $\forall
i\in\mathopen{}\mathclose{{}\left\\{1,\ldots,M}\right\\}$. The expressivity of
the ansatz is controlled by the number of free parameters, $N_{\text{basis}}$.
Furthermore, the ansatz is built so that $a(0)=a(z_{h})=1$ by construction,
which keeps the geometry asymptotically conformally $AdS$ and the horizon at a
constant temperature. We note that if we were interested in very low
temperatures and the confining phases of the theory, a metric ansatz with a
collapsing cycle at the IR end of the geometry (a ‘cigar’) would seem more
appropriate.

Our statistical model then has the parameters

$\displaystyle\mathopen{}\mathclose{{}\left\\{a_{1},a_{2},\ldots,a_{N_{\text{basis}}},c}\right\\}\
,$ (3.24)

where
$c\equiv(64\pi^{5}z_{p}^{17/2})/(375\sqrt{z_{h}}G_{N_{\textrm{\scriptsize
c}}}^{(10)})$ is the multiplicative constant on the right hand side of (3.19).
We will link strip widths $\ell_{i}$ to entropy derivatives by assuming a
likelihood

$\displaystyle\mathopen{}\mathclose{{}\left(\frac{1}{T^{2}V}\frac{\mathrm{d}S_{EE}}{\mathrm{d}\ell}}\right)_{i}\Bigg{|}\mathopen{}\mathclose{{}\left\\{\vec{a},c}\right\\}\sim\mathcal{N}\mathopen{}\mathclose{{}\left(c\mathopen{}\mathclose{{}\left(\frac{z_{h}}{z_{*}(\vec{a},\ell_{i})}}\right)^{7/2},\sigma_{i}}\right)\
,$ (3.25)

with $\mathcal{N}\mathopen{}\mathclose{{}\left(\mu,\sigma}\right)$ being the
normal distribution with mean $\mu$ and standard deviation $\sigma$. We will
use weakly informative normal priors for $\vec{a}$ and $c$ with standard
deviation 5 and 1, respectively, around their maximum likelihood estimates.
The priors represent our beliefs about the bulk geometries that might
reproduce the experimental results obtained from the lattice simulations: the
standard deviations are wide so that we prefer values closer to values which
maximize (3.25) but still want the posterior parameter estimates to be
primarily informed by experimental data instead of our priors. We combine the
priors with the information from our experimental data by bringing in their
likelihood (3.25) to form a posterior distribution for the model parameters.
The posterior distribution is of the form

$\displaystyle
p\mathopen{}\mathclose{{}\left(\vec{a},c\Bigg{|}\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\frac{1}{T^{2}V}\frac{\mathrm{d}S_{EE}}{\mathrm{d}\ell}}\right)_{i},\ell_{i},\sigma_{i}}\right\\}}\right)$
$\displaystyle\propto\prod_{i=1}^{M}\frac{1}{\sigma_{i}}\exp\mathopen{}\mathclose{{}\left\\{-\frac{1}{2\sigma_{i}^{2}}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{1}{T^{2}V}\frac{\mathrm{d}S_{EE}}{\mathrm{d}\ell}}\right)_{i}-c\mathopen{}\mathclose{{}\left(\frac{z_{h}}{z_{*}(\vec{a},\ell_{i})}}\right)^{7/2}}\right)^{2}}\right\\}$
$\displaystyle\times\exp\mathopen{}\mathclose{{}\left\\{-\frac{1}{2\cdot
1^{2}}\mathopen{}\mathclose{{}\left(c-c_{MLE}}\right)^{2}}\right\\}\times\prod_{i=1}^{N_{\text{basis}}}\exp\mathopen{}\mathclose{{}\left\\{-\frac{1}{2\cdot
5^{2}}\mathopen{}\mathclose{{}\left(a_{i}-a_{i,MLE}}\right)^{2}}\right\\}\ .$
(3.26)

Finally, we sample this posterior distribution of $\vec{a}$ and $c$ with
Hamiltonian Monte Carlo (HMC) which is a type of Markov chain Monte Carlo
method. More specifically we employ a HMC variant called No U-Turn Sampler
which reduces the number of sampling related parameters that need hand-tuning
[51], making it a good method for sampling high-dimensional distributions like
(3.26). Furthermore, this sampling approach reduces the autocorrelation
between samples which reduces the number of samples needed to obtain a good
estimate of the posterior distribution. It is important to note that for a
given strip width $\ell_{i}$ there might be multiple turning points $z_{*}$
corresponding to it. If this is the case, we select the turning point
corresponding to the true global minimum of entanglement entropy in accordance
with the Ryu-Takayanagi prescription [20].

The posterior parameter distributions can then be used to find the
distribution of bulk metrics consistent with the lattice entanglement entropy
data. In a sense this is like using the AdS/CFT duality “in reverse”. Then,
having access to the bulk gravity model, we could use the AdS/CFT dictionary
in the usual direction to derive other quantities of interest, for example,
quark-antiquark potentials.

##  4 Entanglement and thermal entropy

In this section we would like to show that

$S_{EE}\mathopen{}\mathclose{{}\left(\ell,T}\right)\approx
S_{\mathrm{th},A\mathopen{}\mathclose{{}\left(\ell}\right)}\mathopen{}\mathclose{{}\left(T}\right)\qquad\text{if}\quad\ell\,T\gg
1\ ,$ (4.1)

which tell us that at sufficiently large width $\ell$ of the entangling region
$A$, the entanglement entropy,
$S_{EE}\mathopen{}\mathclose{{}\left(\ell,T}\right)$, of $A$ reduces to the
thermal entropy, $S_{\mathrm{th},A}\mathopen{}\mathclose{{}\left(T}\right)$,
in $A$. Note that $S_{\mathrm{th},A}\mathopen{}\mathclose{{}\left(T}\right)$
is an extensive quantity and grows therefore linearly with the volume of
$A\mathopen{}\mathclose{{}\left(\ell}\right)$, i.e. linearly with $\ell$. A
more straightforward way to write this is, to show that the derivative of
$S_{EE}$ with respect to $\ell$, becomes $\ell$-independent for large $\ell$:

$\lim_{\ell\gg
T^{-1}}\frac{1}{\mathopen{}\mathclose{{}\left|A_{\perp\ell}}\right|}\frac{\partial
S_{EE}\mathopen{}\mathclose{{}\left(\ell,T}\right)}{\partial\ell}=s_{\mathrm{th},A}\mathopen{}\mathclose{{}\left(T}\right)\
,$ (4.2)

where $\mathopen{}\mathclose{{}\left|A_{\perp\ell}}\right|$ is the
$\ell$-independent area of a cross section of the entangling region $A$
perpendicular to $\ell$, and $s_{\mathrm{th},A}$ is the thermal entropy
density for region $A$.

As mentioned in the introduction, in terms of density matrices, one can see
that the relation in Eq. (4.1) should hold, by noting that the entanglement
entropy in region $A$ is defined by:

$S_{EE}\mathopen{}\mathclose{{}\left(A,T}\right)=-\operatorname{tr}_{A}\mathopen{}\mathclose{{}\left(\rho_{A}\mathopen{}\mathclose{{}\left(T}\right)\,\log\mathopen{}\mathclose{{}\left(\rho_{A}\mathopen{}\mathclose{{}\left(T}\right)}\right)}\right)\
,$ (4.3)

where
$\rho_{A}\mathopen{}\mathclose{{}\left(T}\right)=\operatorname{tr}_{B}\mathopen{}\mathclose{{}\left(\rho\mathopen{}\mathclose{{}\left(T}\right)}\right)$
is the reduced density matrix for region $A$, obtained by taking the partial
trace of the full density matrix,
$\rho\mathopen{}\mathclose{{}\left(T}\right)$, with respect to a basis for the
complement, $B=A^{c}$, of region $A$. In the limiting case, where $A$
represents the whole system, and $B$ is empty, one has
$\rho_{A}\mathopen{}\mathclose{{}\left(T}\right)\to\rho\mathopen{}\mathclose{{}\left(T}\right)$
and therefore

$\lim_{B\to\emptyset}S_{EE}\mathopen{}\mathclose{{}\left(A,T}\right)=-\operatorname{tr}\mathopen{}\mathclose{{}\left(\rho\mathopen{}\mathclose{{}\left(T}\right)\,\log\mathopen{}\mathclose{{}\left(\rho\mathopen{}\mathclose{{}\left(T}\right)}\right)}\right)=S_{\mathrm{th},A}\mathopen{}\mathclose{{}\left(T}\right)\
,$ (4.4)

i.e. the expression for the entanglement entropy reduces to the expression for
the thermal entropy if the region $A$ is entangled with nothing. This makes it
plausible that Eq. (4.1) holds as soon as $A$ is much bigger than the thermal
screening radius, $r_{T}\sim T^{-1}$, as then most of $A$ will not “feel” that
$B$ has been traced out; only the part of $A$ that is within a distance
$r_{T}$ from the boundary to $B$ can experience entanglement with region $B$.

In the following, we will provide an additional argument for why Eq. (4.1)
resp. Eq. (4.2) should hold, based on a comparison of the replica trick
expression for the entanglement entropy and the expression for the thermal
entropy in terms of a Euclidean path integral. In order to have well-defined
path integral expressions, we will consider directly lattice regularized
theories. However, the provided relations should apply also when using other
valid regularization schemes.

###  4.1 Thermodynamics in terms of a Euclidean (lattice) field theory
partition function

Let us define the _lattice free energy_ ,
$F_{L}\mathopen{}\mathclose{{}\left(N_{t},V,N}\right)$ as

$F_{L}\mathopen{}\mathclose{{}\left(N_{t},V,N}\right)=-\log\mathopen{}\mathclose{{}\left(Z\mathopen{}\mathclose{{}\left(N_{t},V,N}\right)}\right)\
,$ (4.5)

where $Z\mathopen{}\mathclose{{}\left(N_{t},V,N}\right)$ is a canonical,
Euclidean lattice partition function, depending on the temporal lattice size,
$a\,N_{t}$, the spatial lattice volume, $a^{3}\,V$, and a conserved charge
$N$. We keep the dependency on the lattice spacing, $a$, implicit as we are
interested in the properties of a lattice system at fixed $a$. If the theory
under consideration does not contain any conserved charges, one can drop the
dependency on $N$; similarly, if there are multiple conserved charges, one
will have to extend the formalism to an appropriate set of charges,
$\mathopen{}\mathclose{{}\left\\{N_{i}}\right\\}_{i=1,2\ldots}$.

The pure gauge system we study in this paper does not contain any conserved
charges in which are interested, but we keep the $N$-dependency for the
moment, in order to emphasize the relation between $F_{L}$ and the usual
Helmholtz free energy $F\mathopen{}\mathclose{{}\left(T,V,N}\right)$, which is
given by

$F_{L}\mathopen{}\mathclose{{}\left(N_{t},V,N}\right)=N_{t}\,F\mathopen{}\mathclose{{}\left(T\mathopen{}\mathclose{{}\left(N_{t}}\right),V,N}\right)\
.$ (4.6)

From Eq. (4.6) it follows that the differential of $F_{L}$ in therms of the
usual thermodynamic quantities, $T=1/N_{t}$ (temperature), $p$ (pressure),
$\mu$ (chemical potential), $S$ (entropy), $V$ (spatial volume) and $N$
(conserved charge number), is given by:

$\mathrm{d}F_{L}=F\,\mathrm{d}N_{t}+N_{t}\,\mathrm{d}F=U\,\mathrm{d}N_{t}-p\,N_{t}\,\mathrm{d}V+\mu\,N_{t}\,\mathrm{d}N\
,$ (4.7)

where after the second equality sign we have used that the internal energy $U$
is related to $F$ by

$U=F+\underbrace{T\,S}_{\frac{S}{N_{t}}}=F+\frac{S}{N_{t}}\ ,$ (4.8)

and that

$\mathrm{d}F=\underbrace{-S\,\mathrm{d}T}_{\frac{S}{N_{t}^{2}}\,\mathrm{d}N_{t}}-p\,\mathrm{d}V+\mu\,\mathrm{d}N=\frac{S}{N_{t}^{2}}\,\mathrm{d}N_{t}-p\,\mathrm{d}V+\mu\,\mathrm{d}N\
.$ (4.9)

In order to express the thermal entropy $S$ in terms of $F_{L}$, we can use
Eqn. (4.8) and (4.7) to write:

$S=N_{t}\,\mathopen{}\mathclose{{}\left(U-F}\right)=N_{t}\,U-F_{L}=N_{t}\,\mathopen{}\mathclose{{}\left.\frac{\partial
F_{L}}{\partial N_{t}}}\right|_{V,N}-F_{L}\ .$ (4.10)

###  4.2 Relation between entanglement and thermal entropy

To demonstrate the relation from Eq. (4.1), we note that in order to determine
Eq. (2.13), one does not necessarily have to implement the change
$\ell\to\mathopen{}\mathclose{{}\left(\ell+1}\right)$ in the width $\ell$ of
the region $A$ by changing the temporal boundary conditions as described in
Fig. 5 along the current boundary of region $A$; one could just as well
increase $\ell$ by removing a slice of width $\Delta\ell$ from deep inside
region $B$, change the temporal boundary conditions of that slice, and insert
it again somewhere in the interior of region $A$, as illustrated in Fig. 9.

Figure 9: The figure illustrates how the width $\ell$ of region $A$ can be
increased by removing a slice (which is parallel to the interface between
regions $A$ and $B$) from deep within region $B$, changing the temporal
boundary conditions of the slice from those of region $B$ to those of region
$A$ (cf. Fig. 5), and then re-inserting the slice to the system deep inside
region $A$.

Let us now assume that the width $\ell$ of region $A$ and the width
$\mathopen{}\mathclose{{}\left(N_{s}-\ell}\right)$ of region $B$ are both
sufficiently large, so that the effect of the interfaces between the two
regions, located at $x=0$ and $x=N_{s}-\ell$, can be neglected for describing
the thermodynamic properties of the system deep inside of either of the two
regions. Similarly we assume that the change in the volumes of the two
regions, $A$ and $B$, when
$\ell\to\mathopen{}\mathclose{{}\left(\ell+1}\right)$ can be neglected. In
this case, deep inside region $B$, the system consists of two copies of a
system that can be described by the lattice free energy density

$f_{L}\mathopen{}\mathclose{{}\left(N_{t}}\right)\ ,$ (4.11)

whereas deep inside region $A$, the system consists of just one copy of a
system that is described by a lattice free energy density

$f_{L}\mathopen{}\mathclose{{}\left(2\,N_{t}}\right)\ .$ (4.12)

In terms of these lattice free energy densities, the derivative of the
entanglement entropy from Eq. (2.13) is given by

$\frac{1}{N_{s}^{\mathopen{}\mathclose{{}\left(d-2}\right)}}\mathopen{}\mathclose{{}\left.\frac{\partial
S_{EE}\mathopen{}\mathclose{{}\left(\ell^{\prime},N_{t},N_{s}}\right)}{\partial\ell^{\prime}}}\right|_{\ell^{\prime}=\ell+1/2}\,\approx\,f_{L}\mathopen{}\mathclose{{}\left(2\,N_{t}}\right)-2\,f_{L}\mathopen{}\mathclose{{}\left(N_{t}}\right)\
,$ (4.13)

where $N_{s}^{\mathopen{}\mathclose{{}\left(d-2}\right)}$ is the spatial area
(in lattice units) perpendicular to the $x$-direction, so that
$N_{s}^{\mathopen{}\mathclose{{}\left(d-2}\right)}\,\Delta\ell$ with
$\Delta\ell=1$ is the spatial volume of the slice that is moved from region
$B$ to region $A$.

Now, as discussed around Eq. (2.12), the lattice expression for $S_{EE}$
actually corresponds to the second order Rényi entropy, defined in Eq. (2.9).
This is a consequence of approximating in Eq. (2.10) the derivative with
respect to the replica number, $s$, in the limit
$\mathopen{}\mathclose{{}\left(s\to 1}\right)$ by a discrete forward
derivative:

$S_{EE}\mathopen{}\mathclose{{}\left(\ell,N_{t},N_{s}}\right)=-\lim_{s\to
1}\frac{\partial\log\operatorname{tr}\mathopen{}\mathclose{{}\left(\rho_{A}^{s}}\right)}{\partial
s}=\lim_{s\to 1}\frac{\partial
F_{c}\mathopen{}\mathclose{{}\left(\ell,s,N_{t},N_{s}}\right)}{\partial
s}-F\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)\\\ \approx
F_{c}\mathopen{}\mathclose{{}\left(\ell,2,N_{t},N_{s}}\right)-2\,F\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)\
.$ (4.14)

The expression on the right-hand side of Eq. (4.13) is therefore a consequence
of this forward derivative approximation in (4.14). If one tries to reverse-
engineer what the right-hand side of Eq. (4.13) would look like if the
discrete derivative approximation in Eq. (4.14) were not necessary, one
quickly finds as possible solution that:

$f_{L}\mathopen{}\mathclose{{}\left(2\,N_{t}}\right)-2\,f_{L}\mathopen{}\mathclose{{}\left(N_{t}}\right)\approx\mathopen{}\mathclose{{}\left.\frac{\partial
f_{L}\mathopen{}\mathclose{{}\left(s\,N_{t}}\right)}{\partial
s}}\right|_{s=1}-f_{L}\mathopen{}\mathclose{{}\left(N_{t}}\right)=N_{t}\,\frac{\partial
f_{L}\mathopen{}\mathclose{{}\left(N_{t}}\right)}{\partial
N_{t}}-f_{L}\mathopen{}\mathclose{{}\left(N_{t}}\right)\ ,$ (4.15)

in which case one would have:

$\frac{1}{N_{s}^{\mathopen{}\mathclose{{}\left(d-2}\right)}}\mathopen{}\mathclose{{}\left.\frac{\partial
S_{EE}\mathopen{}\mathclose{{}\left(\ell^{\prime},N_{t},N_{s}}\right)}{\partial\ell^{\prime}}}\right|_{\ell^{\prime}=\ell+1/2}\,=\,N_{t}\,\frac{\partial
f_{L}\mathopen{}\mathclose{{}\left(N_{t}}\right)}{\partial
N_{t}}-f_{L}\mathopen{}\mathclose{{}\left(N_{t}}\right)=s_{\mathrm{th},A}\ ,$
(4.16)

with $s_{\mathrm{th},A}$ being the density of thermal entropy,
$S_{\mathrm{th},A}$, of region $A$, as follows from Eq. (4.10).

With Eq. (4.16), one could therefore compute the thermal entropy of a quantum
field theory directly from a single lattice Monte Carlo simulation, using the
method for determining the derivative of the entanglement entropy, described
in Sec. 2.2. This is not possible by using only Eq. (4.10), as this would
involve knowledge of the lattice free energy, $F_{L}$, itself. Instead one
would have to measure for example
$\partial_{T}S\mathopen{}\mathclose{{}\left(T}\right)$, where $T=1/N_{t}$, for
multiple values of $T$, that will allow one to numerically approximate the
integral in

$S=S_{0}+\int\limits_{0}^{T}\mathrm{d}T^{\prime}\,\frac{\partial
S\mathopen{}\mathclose{{}\left(T^{\prime}}\right)}{\partial T^{\prime}}\ ,$
(4.17)

and where the integration constant,
$S_{0}=S\mathopen{}\mathclose{{}\left(T=0}\right)=0$, is fixed by the 3rd law
of thermodynamics. This latter procedure can be computationally very
expensive, as one has to simulate low temperatures, which means large $N_{t}$
and therefore large lattices.

The relation Eq. (4.16) might therefore in the future also be useful to study
the thermodynamic properties of strongly interacting field theories.

##  5 Results

In this section we will discuss results that we have obtained from running
simulations on the lattice as described in Sec. 2.2. A comparison between
results obtained with our new method and corresponding literature results for
the case of four-dimensional Yang-Mills theory has been briefly exposed in
[28, 52]. In the following we will present our results on the entanglement
entropy in (2+1)-dimensional pure $\operatorname{SU}(2)$ gauge theory at high
temperatures. Along the way we recall how the holographic approach, as
described in Sec. 3, guides us, _e.g._ , in picking correct power laws for
fitting. We provide an example for how the replica trick, used to determine
the entanglement entropy, can be utilized to determine also the thermal
entropy, resorting to
$\lim_{\ell\to\infty}\partial_{\ell}\,S_{EE}\mathopen{}\mathclose{{}\left(\ell}\right)\propto
S_{\text{thermal}}$, as discussed in Sec. 4. In the final subsection we
reconstruct the bulk geometry from the entanglement entropy measurements.

###  5.1 Three-dimensional Yang-Mills theory at high temperature

We are interested in lattice results on the behavior of the derivative of the
entanglement entropy with respect to the slab-width, $\ell$, at high
temperatures ($T>T_{c}$) and large slab-width ($\ell>T^{-1}$), where $T$ is
the temperature and $T_{c}$ the critical temperature for deconfinement, which
serves as physical reference energy scale.

$\beta_{g}$ | $N_{s}$ | $a\,T_{c}$ | $N_{t}$ | $T/T_{c}$ | $\ell_{\text{min}}/a$ | $\ell_{\text{min}}\,T_{c}$ | $\ell_{\text{max}}/a$ | $\ell_{\text{max}}\,T_{c}$
---|---|---|---|---|---|---|---|---
16. | 48 | 0.0976593 | 2 | 5.11984 | 2.5 | 0.244148 | 20.5 | 2.00202
|  |  | 4 | 2.55992 |  |  |  |
|  |  | 6 | 1.70661 |  |  |  |
|  |  | 8 | 1.27996 |  |  |  |
24. | 48 | 0.0641104 | 4 | 3.89953 | 2.5 | 0.160276 | 15.5 | 0.99371
|  |  | 6 | 2.59968 |  |  |  |
|  |  | 8 | 1.94976 |  |  |  |
32. | 64 | 0.0477628 | 4 | 5.2342 | 2.5 | 0.119407 | 20.5 | 0.979136
|  |  | 6 | 3.48947 |  |  |  |
|  |  | 8 | 2.6171 |  |  |  |
|  |  | 10 | 2.09368 |  |  |  |
48. | 96 | 0.0316115 | 6 | 5.27234 | 2.5 | 0.0790287 | 29.5 | 0.932539
|  |  | 8 | 3.95426 |  |  |  |
|  |  | 10 | 3.16341 |  |  |  |
|  |  | 12 | 2.63617 |  |  |  |
64. | 128 | 0.0236208 | 8 | 5.29194 | 2.5 | 0.0590521 | 40.5 | 0.956643
|  |  | 12 | 3.52796 |  |  |  |
|  |  | 16 | 2.64597 |  |  |  |
Table 1: Simulation parameters and corresponding lattice spacing- and
$T/T_{c}$-values for $\operatorname{SU}(2)$ pure gauge theory on a
(2+1)-dimensional lattice, as used for the study of the large distance
behavior of $\mathrm{d}S_{EE}/\mathrm{d}\ell$ at high temperatures. The number
of replica is always $s=2$.

We have performed simulations at five different values of the lattice spacing,
$a$, and various different temperatures. The list of simulated setups is given
in Table 1.

Figure 10: The range of validity for the holographic entanglement entropy
formula derived in the weak gravity limit as function of the inverse gauge
coupling, $\beta_{g}$ and the slab width, $\ell$ in lattice units, using Eq.
(5.2) for $N_{\textrm{\scriptsize c}}=2$. Figure 11: Example of fitting the
lattice equivalent of $\mathrm{d}S_{EE}/\mathrm{d}\ell$ as a function of
$\ell$ for $\operatorname{SU}(2)$ in (2+1) dimensions at three different
temperatures $T\sim 1/N_{t}$ with $N_{t}$=8,12,16 for a spatial lattice size
of $V=128^{2}$. As $T/T_{c}>1$, the derivative of the entanglement entropy
with respect to $\ell$ approaches at large $\ell$ a temperature-dependent
plateau-value. The length unit $\Lambda_{cr}$ corresponds to the inverse
critical temperature, _i.e._ , $\Lambda_{cr}=1/T_{c}$. The faint vertical
lines represent the lower and upper limit of the holography-validity interval
from Eq. (5.2). Note that the data, corresponding to different $N_{t}$-values
(_i.e._ different temperatures), does not yet lay on a unique curve at short
distances, indicating that our lattice data does not resolve a sufficiently
high UV-regime at which the temperature would no longer matter. Furthermore,
we note that the short distance data lays outside the holography-validity
range and does not follow the expected pure power law $\sim x^{-7/3}$ (black
line); we had to allow for a shift, $x_{0}$, in order to fit the data (red
lines).

Note that due to our interest in the high-temperature regime, we are forced to
go to rather large values of the inverse gauge coupling, $\beta_{g}$. As
discussed in [32], this means that the criteria for the applicability of the
classical approximation for the bulk theory in which the holographic
computations in Sec. 3 are carried out, are satisfied only at relatively large
values of the slab-width $\ell$, compared to the lattice spacing $a$. The
range of validity as function of $\beta_{g}$ and $\ell$ in lattice units is
sketched in Fig. 10 for $N_{\textrm{\scriptsize c}}=2$, using the inequalities
from [32]:

$\frac{1}{N_{\textrm{\scriptsize
c}}\,g^{2}_{c}}\ll\ell_{c}\ll\frac{1}{N_{\textrm{\scriptsize
c}}^{1/5}\,g^{2}_{c}}\ ,$ (5.1)

where $g^{2}_{c}$ is the continuum theory gauge coupling. In terms of lattice
quantities, using the relations $\beta_{g}=2\,N_{\textrm{\scriptsize
c}}/g^{2}_{0}$, $g^{2}_{0}=a\,g^{2}_{c}$, and $\ell_{c}=a\,\ell$, the
inequalities take the form:

$\frac{\beta_{g}}{2\,N_{\textrm{\scriptsize
c}}^{2}}\ll\ell\ll\frac{\beta_{g}}{2\,N_{\textrm{\scriptsize c}}^{6/5}}\ .$
(5.2)

Our lattice data for the derivative of the entanglement entropy might
therefore at short distances not conform with the power law behavior, expected
from the discussion in Sec. 3.

With this in mind, we will allow for certain corrections to the expected
behavior discussed in Sec. 3 when performing fits to our lattice data. An
example is shown in Fig. 11. From holography (cf. Eq. (3.20)), we would expect
our lattice data for $\mathrm{d}S_{EE}/\mathrm{d}\ell$ as function of $\ell$
to behave in the UV like a power law $\sim\ell^{-7/3}$. As our simulations are
carried out at temperatures, $T$, which are 2-5 times the critical temperature
$T_{c}$, this power law behavior is almost completely screened by the plateau
within the range of validity for the holographic predictions from Eq. (5.2)
resp. Fig. 10.

However, our focus in this section is not on the $\ell$ dependency of
$\mathrm{d}S_{EE}/\mathrm{d}\ell$ at small $\ell$, but on its $T/T_{c}$
dependency at large $\ell$. More precisely, we aim to verify that for
$T/T_{c}\gg 1$ one has

$\lim_{\ell\to\infty}\frac{1}{\mathopen{}\mathclose{{}\left|\partial
A}\right|}\partial_{\ell}\,S_{EE}\mathopen{}\mathclose{{}\left(\ell}\right)=s_{\mathrm{th},0}\,\mathopen{}\mathclose{{}\left(T/T_{c}}\right)^{7/3}\
,$ (5.3)

which follow from
$\lim_{\ell\to\infty}\partial_{\ell}\,S_{EE}\mathopen{}\mathclose{{}\left(\ell}\right)\propto
s_{\mathrm{th},A\mathopen{}\mathclose{{}\left(\ell}\right)}$ (cf. Sec. 4), in
combination with the holographic prediction that for a
$\mathopen{}\mathclose{{}\left(2+1}\right)$-dimensional $\operatorname{SU}(N)$
gauge theory the thermal entropy behaves like
$S_{\mathrm{th}}\propto\mathopen{}\mathclose{{}\left(T/T_{c}}\right)^{7/3}$
for $T/T_{c}\gg 1$.

Figure 12: The three upper panels show $\mathrm{d}S_{EE}/\mathrm{d}\ell$ for
$\operatorname{SU}(2)$ in (2+1) dimensions, rescaled by a factor of
$\mathopen{}\mathclose{{}\left(T/T_{c}}\right)^{-7/3}=\mathopen{}\mathclose{{}\left(T\,\Lambda_{cr}}\right)^{-7/3}$,
as function of $\ell\,T$ at three different lattice spacing values,
corresponding to $\beta=32$ (top), $\beta=48$ (second from top), and
$\beta=64$ (third from top). Included is only data for $T/T_{c}>2.5$ as the
scaling
$\mathrm{d}S_{EE}/\mathrm{d}\ell\propto\mathopen{}\mathclose{{}\left(T/T_{c}}\right)^{7/3}$
is expected to hold only at large temperatures, away from the critical point.
As can be seen, for the two finest lattice spacings, corresponding to
$\beta=48,64$, the data collapses nicely. The plateau value is fitted by two
different ansatzes, as indicated (red: $y=a\,x^{-7/3}+b$, blue: $y=b$). The
bottom panel shows the continuum extrapolation of the plateau values, $b$,
obtained in the previous three panels with the two different fit ansatzes in
red and blue.

As a first test of the scaling law from Eq. (5.3), we show in the first three
panels of Fig. 12 our lattice results for
$\partial_{\ell}\,S_{EE}\mathopen{}\mathclose{{}\left(\ell}\right)$, rescaled
by $\mathopen{}\mathclose{{}\left(T/T_{c}}\right)^{-7/3}$ and as function of
$\mathopen{}\mathclose{{}\left(\ell\,T}\right)$ for $\beta_{g}=32$ (top), $48$
(second from top), and $64$ (third from top). The ratio $T/T_{c}$ as function
of $\beta_{g}$, $N_{s}$, and $N_{t}$ is determined as described in Appendix
A.2.3. For each value of $\beta_{g}$ the plots consist of data from various
temperatures. Thanks to the rescaling of
$\partial_{\ell}\,S_{EE}\mathopen{}\mathclose{{}\left(\ell}\right)$ by
$\mathopen{}\mathclose{{}\left(T/T_{c}}\right)^{-7/3}$, the plateau data
corresponding to different temperatures coincides. This is in particular true
for $\beta_{g}=48,64$, corresponding to lattice spacing values
$a\,T_{c}\approx 0.024$, $0.032$. For $\beta_{g}=32$, resp. $a\,T_{c}=0.048$
the data does not collapse as nicely anymore, and for the even lower
$\beta_{g}$-values, resp. larger lattice spacings, it would look even worse.
The bottom panel in Fig. 12 shows a linear extrapolation of the rescaled
plateau values, which provides a first rough estimate of the continuum value
of the proportionality factor, $s_{\mathrm{th},0}$ in Eq. (5.3).

In order to be able to use correction terms in the continuum extrapolation of
our lattice data for the plateau values
$\lim_{\ell\to\infty}\partial_{\ell}S_{EE}\mathopen{}\mathclose{{}\left(\ell}\right)$
and refine the estimate for $s_{\mathrm{th},0}$ obtained in the bottom panel
of Fig. 12, we extract the plateau values
$\lim_{\ell\to\infty}\partial_{\ell}S_{EE}\mathopen{}\mathclose{{}\left(\ell}\right)$
for all simulated values of $\beta_{g}$ and $T/T_{c}=T\,\Lambda_{c}$
separately by fitting the ansatz

$y=a\,\mathopen{}\mathclose{{}\left(x-x_{0}}\right)^{-7/3}+b\ $ (5.4)

to our data, where
$\mathopen{}\mathclose{{}\left(x,y}\right)\in\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\ell/\Lambda_{c},\Lambda_{c}^{2}\,N_{s}^{-1}\mathrm{d}S_{EE}/\mathrm{d}\ell}\right)\,|\,\forall\ell}\right\\}$.
The ratio $T/T_{c}$ is again determined as described in Appendix A.2.3 and
$a\,T_{c}=1/\mathopen{}\mathclose{{}\left(N_{t}\,\mathopen{}\mathclose{{}\left(T/T_{c}}\right)}\right)$.
The shift $x_{0}$ in Eq. (5.4) has been added to account for expected
deviations from the pure $\sim\ell^{-7/3}$ power law behavior of
$\mathrm{d}S_{EE}/\mathrm{d}\ell$, mentioned at the beginning of this section.
The form of the chosen correction is not important, as long as it does not
significantly affect the fit results for the parameter $b$. Alternatively, one
could perform constant fits ($y=b$) on restricted data for which $\ell>T^{-1}$
(cf. Fig. 12).

Figure 13: Plateau values of $\mathrm{d}S_{EE}/\mathrm{d}\ell$ at large $\ell$
for $\operatorname{SU}(2)$ in (2+1) dimensions for various temperatures
$T/T_{c}>1$. The different colors represent data for different lattice spacing
values, $a\,T_{c}$ (in units of the critical temperature, $T_{c}$), and the
corresponding fits to their temperature dependency. The black line shows the
continuum extrapolation ($\beta\to\infty$).

The so obtained plateau values, $b$, are plotted in Fig. 13 as function of the
reduced temperature $\tau=T/T_{c}-1$. Superimposed to the data is for each
value of $\beta_{g}$ separately a fit of the form

$y=a_{0}\,\mathopen{}\mathclose{{}\left(1+x}\right)^{7/3}\,\mathopen{}\mathclose{{}\left(1+a_{1}\mathopen{}\mathclose{{}\left(1+x}\right)^{-1}}\right)\
,$ (5.5)

with
$\mathopen{}\mathclose{{}\left(x,y}\right)\in\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\tau,b\mathopen{}\mathclose{{}\left(\tau}\right)}\right)\,|\,\forall\tau}\right\\}$.
As before, this fit ansatz is motivated by the identification of the plateau
value with the thermal entropy density. The latter should, according to
holography, for $T\gg\ell^{-1}$ behave like
$s_{\mathrm{thermal}}\propto\mathopen{}\mathclose{{}\left(T/T_{c}}\right)^{7/3}=\mathopen{}\mathclose{{}\left(1+\tau}\right)^{7/3}$,
and we include a leading correction term, whose relative magnitude is
parametrized by $a_{1}$. The fits are performed on data for which
$T/T_{c}>1.9$, as lower temperatures would require higher order corrections.
The only exception is the case of $\beta_{g}=16$, where $T/T_{c}>1.7$ is used
as lower bound in order to have a minimum of three data points to fit the two
parameters $a_{0}$ and $a_{1}$. However, as can be seen, the quality of fit
for $\beta_{g}=16$ is poor. This is on the one hand due to the extended lower
bound of the fit-range which, as mentioned before, would require higher order
corrections, and on the other hand due to the fact that the lattice is too
coarse at $\beta_{g}=16$, so that $N_{t}=2$ for the highest temperature,
$T/T_{c}\approx 5.12$. We therefore exclude the $\beta_{g}=16$ data from
further analysis.

The data from the remaining $\beta_{g}$-values is used to perform a continuum
extrapolation $\mathopen{}\mathclose{{}\left(\beta_{g}\to\infty}\right)$ in
two different ways (black and cyan bands).

The first extrapolation (black) is obtained using the fitted functions and
corresponding error bands to obtain values and errors for $b$ at fixed
$T/T_{c}$ but different lattice spacing values, $a\,T_{c}$, and perform a fit:

$y=b_{0}+b_{1}\,x\ ,$ (5.6)

with
$\mathopen{}\mathclose{{}\left(x,y}\right)\in\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(a\,T_{c},b\mathopen{}\mathclose{{}\left(a\,T,T/T_{c}}\right)}\right)\,|\,\forall\,a\,T_{c}}\right\\}$.
This is repeated for different values of
$T/T_{c}\in\mathopen{}\mathclose{{}\left\\{2.0,2.2,\ldots,5.0}\right\\}$ while
keeping track of the values of $b_{0}$ as function of $\tau=T/T_{c}-1$. A few
examples of such fits are shown in Fig. 14. To the so obtained data we then
fit again Eq. (5.5) with
$\mathopen{}\mathclose{{}\left(x,y}\right)\in\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\tau,b_{0}\mathopen{}\mathclose{{}\left(\tau}\right)}\right)\,|\,\forall\tau\in\mathopen{}\mathclose{{}\left\\{1.0,1.2,\ldots,4.0}\right\\}}\right\\}$,
which yields the black curve and corresponding error band from Fig. 13.

Figure 14: Example of fits used to perform continuum extrapolations of the
plateau value, $b$, of the entanglement entropy, as determined from the fit
results shown in Fig. 13 for $\beta_{g}=24,32,48,64$.

The second continuum extrapolation in Fig. 13 (cyan) is obtained by
considering directly the lattice spacing-dependency of the fit results for the
parameters $a_{0}$ and $a_{1}$ for $\beta_{g}=24,32,48,64$. As shown in Fig.
15, we perform quadratic fits to the data for $a_{0}$ and $a_{1}$ as functions
of $a\,T_{c}$. The so obtained fit results for the constants $a_{0,0}$ and
$a_{1,0}$ and their errors are then used to draw the cyan curve and error band
in Fig. 13. As can be seen, the two continuum extrapolations are in good
agreement.

Figure 15: Continuum extrapolation of the parameters $a_{0}$ and $a_{1}$,
obtained from the fits in Fig. 13 for $\beta_{g}=24,32,48,64$.

Finally, let us also perform a quick check of whether in Eq. (5.3) a power law
different from $\mathopen{}\mathclose{{}\left(T/T_{c}}\right)^{7/3}$ would be
consistent with the data. To this end, we fit to the plateau data from Fig. 13
for each value of $\beta_{g}$ separately the function form

$y=c_{0}\,\mathopen{}\mathclose{{}\left(1+x}\right)^{c_{1}}\ ,$ (5.7)

where
$\mathopen{}\mathclose{{}\left(x,y}\right)\in\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\tau,b\mathopen{}\mathclose{{}\left(\tau}\right)}\right)|\forall\tau}\right\\}$
as before in the fits with Eq. (5.5). As our data sets cover for some
$\beta_{g}$ values of only three different reduced temperature values, $\tau$,
we do not include in Eq. (5.7) the leading corrections from Eq. (5.5) to keep
the number of fit parameters at $2$. As can be seen in Fig. 16, which
summarizes the results of the fits of the form Eq.(5.7) to the data, without
the leading correction term, $\sim a_{1}$, in the fit function the achievable
quality of fit is generally worse than in Fig. 13. But, with increasing
$\beta_{g}$ the fit quality improves in Fig. 16 as the correction parameter
$a_{1}$ in the corresponding fits in Fig. 13 becomes smaller in magnitude with
increasing $\beta_{g}$. As $\beta_{g}$ increases, the fitted exponent $c_{1}$
in Fig. 16 approaches the expected value $7/3\approx 2.33$ and the value of
the amplitude parameter, $c_{0}$, approaches the value of $a_{0}$ from Fig.
13.

Figure 16: Plateau values of $\mathrm{d}S_{EE}/\mathrm{d}\ell$ at large $\ell$
for $\operatorname{SU}(2)$ in (2+1) dimensions for various temperatures
$T/T_{c}>1$. The different colors represent data for different lattice spacing
values, $a\,T_{c}$ (in units of the critical temperature, $T_{c}$), and the
corresponding fits to their temperature dependency. The black line shows the
continuum extrapolation ($\beta\to\infty$).

In Fig. 17 we perform continuum extrapolations for the $c_{0}$ and $c_{1}$
parameters, based on the fit results for different $\beta_{g}$, shown in Fig.
16. These extrapolations are analogous to the continuum extrapolations
performed in Fig. 15 for the $a_{0}$ and $a_{1}$ parameters for different
$\beta_{g}$ values listed in Fig. 13. However, as in Fig. 16, the quality of
fit for the $\beta_{g}=24$ data is much worse than was the case in Fig. 13, we
show in Fig. 17 in addition to the extrapolations based on quadratic fits
(blue) to the data for $\beta_{g}=24,32,48,64$ also extrapolations based on
linear fits to the data for $\beta_{g}=32,48,64$. Both extrapolation methods
yield within errorbars consistent continuum values for $c_{0}$ and $c_{1}$ and
the results for $c_{1}$ are consistent with $7/3\approx 2.33$.

Figure 17: Continuum extrapolation of the parameters $c_{0}$ and $c_{1}$, base
on linear (red) and quadratic (blue) fits to the $c_{0}$ and $c_{1}$
parameters shown in Fig. 16 for the different $\beta_{g}$ values. The linear
fits use the data from $\beta_{g}=32,48,64$ while the quadratic fits use also
$\beta_{g}=24$.

###  5.2 Reconstructed bulk geometries

In Section 3 we have discussed our statistical model which we use to connect
the metric tensor of the holographic dual to entanglement entropy values. We
will now use data of the derivative of entanglement entropy with respect to
the slab width obtained from the lattice to find which metric fits these
observations. More specifically, the data is for $\operatorname{SU}(2)$
lattice gauge theory on a $V=128^{2}$ lattice at temperature $T/T_{c}=2.646$.
Since the gauge theory is in $(2+1)$ dimensions, the appropriate holographic
background is the asymptotically D2-brane geometry.

Figure 18: Left: Maximum likelihood fit (red dashed line) of the derivative of
entanglement entropy in $\mathopen{}\mathclose{{}\left(2+1}\right)$
dimensions. Black dots are a subset of the interpolated lattice data for
$\mathrm{d}S_{EE}/\mathrm{d}\ell$. Right: The metric function reconstructed
from lattice data in $\mathopen{}\mathclose{{}\left(2+1}\right)$ dimensions.
Red dashed line is the maximum likelihood estimate of the data. The black
dashed line is the median in the distribution of $g_{zz}=a(z)^{2}/b(z)$. The
dashed gray lines represent the $50\%$ and $95\%$ central confidence intervals
of the metric. Both figures use $N_{basis}=3$. The utilized lattice data is
the same as in Fig. 13.

We are sampling the posterior distribution of the parameters which define the
metric tensor to find metrics which would fit the lattice data well under the
assumption that the holographic model represents the gauge theory well. The
results are shown in Fig. 18. There is a pronounced bump in the metric towards
the UV-region of the geometry. It would be interesting to analyze the
ramifications for this feature on some derived quantities. We have also
plotted the predictions for $\mathrm{d}S_{EE}/\mathrm{d}\ell$ using the
maximum likelihood holographic geometry and find good agreement with the
lattice gauge theory data.

##  6 Discussion and outlook

We demonstrated that holography has entered a new phase, precision science.
Gravity duals of realistic field theories can be reconstructed from underlying
data that is _always_ bound to be imprecise, a fact which subsequently feeds
in likelihood estimates on any holographic computations. Systematically
reducing the statistical uncertainties on the extracted quantities from the
lattice leads to an increasingly accurate metric of the dual geometry. The
current work is a proof-of-concept and we did not yet investigate much which
operators drive the renormalization group flow away from the UV or what
predictions the bulk reconstruction yields, other than the computation of the
thermal entropy per usual identification with the Bekenstein-Hawking entropy
of the dual black hole.

Given the formidable route to establishing the dual quantum geometry at finite
numbers of colors, we advocate that there seems to be a (naive) way to
quantify the trustworthiness of classical dual geometry. The holographic
entanglement c-function has a sudden jump across the energy scale marked by
the deconfinement-confinement transition. This can be associated with the
degrees of freedom scaling as ${\cal{O}}(N_{\textrm{\scriptsize c}}^{2})$ and
${\cal{O}}(N_{\textrm{\scriptsize c}}^{0})$ in the respective phases. This
jump is likely an artifact of the $N_{\textrm{\scriptsize c}}\to\infty$ limit,
and is less drastic for finite values of $N_{\textrm{\scriptsize c}}$, as
first illustrated in [8] for $N_{\textrm{\scriptsize c}}=3$ and later in [9]
for $N_{\textrm{\scriptsize c}}=2,3,4$. Nevertheless, if one sticks to the
deconfining phase, the reconstructed dual metrics behave smoothly as a
function of $N_{\textrm{\scriptsize c}}$, giving one in principle means to
interpolate the emerging classical geometries. There are of course many
complications along the way to making this precise, especially if the
entanglement entropy is being used as a tool to reconstruct the metric, since
the RT formula is expected to be modified in the presence of quantum
corrections [53]. It is clearly interesting to study further the entanglement
entropies for higher rank Yang-Mills theories. We hope to report on
entanglement c-functions in (3+1) dimensions for $N_{\textrm{\scriptsize
c}}>4$ in the future.

Other possible avenues for further research include the shape dependence on
the entangling region. In principle, parts of this information is already
encoded in our link updates and one could not only extract, say, the corner
contribution [54], but globally extend the entangling region to spherical
shapes. In addition to this, aspects of multipartite entanglement measures are
clearly interesting targets. Simplest extension of the present work being to
that of two strips (slabs) and questions related with mutual information,
entanglement negativity [55], or purification [56]. In particular, it would be
interesting to analyze if in the last case scenario one finds non-monotonous
behavior as a function of the energy scale as suggested by the holographic
dual quantity [57] known as the entanglement wedge cross section [58].

Entanglement measures are by far not the only data with which one can
reconstruct the bulk geometry; work in this direction include [59, 60, 61, 62,
30, 63]. Other quantities that have been investigated in the literature
include operators [64, 65], four-point correlators [66], differential entropy
[67, 68], fidelity [69], complexity [70], light-cone cuts [71, 72], chiral
condensate [73, 74], conductivity [75], hadron spectra [76], and magnetization
[77] with varying degree of assumptions about the dual classical bulk gravity.
To all of these cases one could incorporate the Hamiltonian Monte Carlo
approach discussed in Sec. 5.2 to also infer the maximum likelihood estimates
of the reconstructed duals. In addition to these examples, one can also use
the Wilson/Polyakov loops to infer the bulk metric [30, 78], though the
fundamental string breaking at finite temperature limits the most
straightforward application and only part of the geometry is visible to the
boundary. In an upcoming work [79] the authors overcome this limitation and
reconstruct the full geometry. Besides the murky status of Debye screening at
strong coupling as revealed in Appendix A, the ability to use Polyakov loop
data also for the reconstruction will further motivate putting extra effort in
a more systematic study of quark-anti-quark potentials already in the pure
Yang-Mills theory in any dimension. Having simultaneous access to bulk metric
both in the string and the Einstein frame by reconstruction from $q\bar{q}$
potential and entanglement entropy data, respectively, would enable us to make
a direct comparison with the beta function of
$\operatorname{SU}(N_{\textrm{\scriptsize c}})$ Yang-Mills theory at strong
coupling.

Finally, to address the deconfinement phase transition and lower temperatures
in general in the bulk gravity side, one seeks for an alternative ansatz for
the metric other than those (3.3), (3.14) where a black hole is assumed to
reside at the bottom of the geometry.

##### Acknowledgments

We would like to thank Jacopo Ghiglieri, Marco Panero, and Javier G. Subils
for useful discussions. N. J. is supported in part by the Academy of Finland
grant. no 1322307, and A. S. and K. R. by the Academy of Finland grants
319066, 320123 and 345070. T. R. is supported in part by the Swiss National
Science Foundation (SNSF) through the grants no. 200021_175761 and
TMPFP2_210064. The authors wish to acknowledge CSC - IT Center for Science,
Finland, for generous computational resources.

##  Appendix A Quark-anti-quark potential

In this appendix we slightly extend the discussion in the main text to cover
also the Polyakov loop which give rise to a potential between quarks and anti-
quarks. This is to illustrate that the D2-brane background, which lead to the
rather unconventional power law behavior for the entanglement entropy
$S_{EE}\propto\ell^{-4/3}$, predicts also an unexpected power law behavior for
the quark-anti-quark potential at small separation $\propto L^{-2/3}$. We will
see that the lattice data that we will generate seems to conform with this and
thus lends more support on the usage of the gravity dual of (2+1)-dimensional
super Yang-Mills theory at largish energy scales. In addition to this
matching, we learned from EE studies that at finite temperature, the black
hole present in the D2-background also explains the behavior of the
entanglement entropy at large subregions sizes. It is therefore not completely
unreasonable to ask if the large separation behavior of the
$q\bar{q}$-potential be also within grasp via D2-brane black holes in this
regime. Indeed, our frugal results from the lattice seem to conform with
$\propto L^{-10/3}$ behavior at large separation, extractable from the
D2-brane background. If this result persists under further scrutiny, then the
holographic approach predicts that the analytic continuation of the lattice
results to real time cf. [39] would result in an imaginary part of the
potential linear in the $q\bar{q}$ separation $L$.

###  A.1 Static quark potential from thermal D2-brane background

To derive these $q\bar{q}$-potential results we start with the metric (3.14)
with $a(z)=1$ and
$b(z)=1-\mathopen{}\mathclose{{}\left(\frac{z}{z_{0}}}\right)^{5}$. This is a
black hole geometry with a horizon at $z=z_{h}$. The potential is found by
studying a string whose endpoints are at the boundary, separated by a distance
$L$ in the $x_{1}$-direction. The action of such a string is

$\displaystyle
S_{NG}=\frac{1}{2\pi\alpha^{\prime}}\int\sqrt{\mathop{\mathrm{det}}\nolimits{g_{2}}}\
,$ (A.1)

where $g_{2}$ is the metric induced on the string worldsheet. The action can
be written in terms of the profile function $x_{1}=x_{1}(z)$ which determines
the shape of the hanging string in the bulk as

$\displaystyle S_{NG}=\frac{\tau
z_{p}^{2}}{2\pi\alpha^{\prime}}\int\frac{1}{z^{2}}\sqrt{1+\mathopen{}\mathclose{{}\left(\frac{z_{p}}{z}}\right)b(z)x_{1}^{\prime}(z)^{2}}\
\mathrm{d}z\ ,$ (A.2)

where $\tau$ is the time interval spanned by the string worldsheet. The
$q\bar{q}$-potential can be found by studying the equation of motion for the
string profile $x_{1}(z)$. The Euler-Lagrange equation is

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathopen{}\mathclose{{}\left(\frac{b(z)x_{1}^{\prime}(z)}{z^{3}\sqrt{1+z_{p}b(z)x_{1}^{\prime}(z)^{2}/z}}}\right)=0\
.$ (A.3)

The solution with the boundary conditions $x_{1}(z=0)=L/2$ and
$x_{1}^{\prime}(z\to z_{*})\to-\infty$ is

$\displaystyle
x_{1}(z)=\frac{L}{2}-\frac{z^{4}}{4z_{p}^{1/2}z_{*}^{5/2}}\sqrt{1-z_{*}^{5}/z_{h}^{5}}\
F_{1}\mathopen{}\mathclose{{}\left(\frac{4}{5};\frac{1}{2},\frac{1}{2};\frac{9}{5};\frac{z^{5}}{z_{h}^{5}},\frac{z^{5}}{z_{*}^{5}}}\right)\
,$ (A.4)

where $F_{1}$ is the Appell hypergeometric function. The constants $L$ and
$z_{*}$ are the quark separation and string turning point, respectively. They
are connected by the equation $x_{1}(z=z_{*})=L/2$, that is

$\displaystyle
L(z_{*})=\frac{\sqrt{\pi}\Gamma\mathopen{}\mathclose{{}\left(\frac{9}{5}}\right)z_{*}^{3/2}}{2\Gamma\mathopen{}\mathclose{{}\left(\frac{13}{10}}\right)z_{p}^{1/2}}\sqrt{1-\frac{z_{*}^{5}}{z_{h}^{5}}}\
{}_{2}F_{1}\mathopen{}\mathclose{{}\left(\frac{1}{2},\frac{4}{5};\frac{13}{10};\frac{z_{*}^{5}}{z_{h}^{5}}}\right)\
.$ (A.5)

The string action can be evaluated by plugging $x_{1}^{\prime}(z)$ from (A.4)
into (A.2). One finds

$\displaystyle\frac{2\pi\alpha^{\prime}}{\tau
z_{p}^{2}}S_{NG}=\frac{2}{\epsilon}-\frac{\sqrt{\pi}\Gamma\mathopen{}\mathclose{{}\left(\frac{4}{5}}\right)}{\Gamma\mathopen{}\mathclose{{}\left(\frac{3}{10}}\right)z_{*}}\
{}_{2}F_{1}\mathopen{}\mathclose{{}\left(-\frac{1}{2},-\frac{1}{5};\frac{3}{10};\frac{z_{*}^{5}}{z_{h}^{5}}}\right)\
,$ (A.6)

where $\epsilon$ is the usual UV-cutoff to regulate the otherwise divergent
action. Now we can find the UV-behavior of the potential by studying (A.5) and
(A.6) in the limit $z_{*}\to 0$

$\displaystyle L$
$\displaystyle=\frac{\sqrt{\pi}\Gamma\mathopen{}\mathclose{{}\left(\frac{9}{5}}\right)}{2\sqrt{z_{p}}\Gamma\mathopen{}\mathclose{{}\left(\frac{13}{10}}\right)}z_{*}^{3/2}$
(A.7) $\displaystyle\frac{2\pi\alpha^{\prime}}{\tau z_{p}^{2}}S_{NG}$
$\displaystyle=\frac{2}{\epsilon}-\frac{2\sqrt{\pi}\Gamma\mathopen{}\mathclose{{}\left(\frac{4}{5}}\right)}{z_{*}\Gamma\mathopen{}\mathclose{{}\left(\frac{3}{10}}\right)}\
.$ (A.8)

Together, these formulas imply that when $z_{*}$ is small the action behaves
as

$\displaystyle\frac{2\pi\alpha^{\prime}}{\tau z_{p}^{2}}S_{NG}$
$\displaystyle=\frac{2}{\epsilon}-\frac{2^{1/3}4\pi^{5/6}\Gamma\mathopen{}\mathclose{{}\left(\frac{4}{5}}\right)^{5/3}}{3^{2/3}\Gamma\mathopen{}\mathclose{{}\left(\frac{3}{10}}\right)^{5/3}z_{p}^{1/3}L^{2/3}}\
.$ (A.9)

The $q\bar{q}$-potential is defined

$\displaystyle
V(L)=-\mathopen{}\mathclose{{}\left(S_{NG}(L)-\operatorname{Re}\mathopen{}\mathclose{{}\left[S_{NG}(L\to\infty)}\right]}\right)/\tau\
.$ (A.10)

There are different ways to do the regularization, often the potential is
computed from the difference of the string action and the action of two
straight disconnected strings hanging from the boundary. This time we choose a
different method, namely we find it useful to regulate so that the real part
of the potential goes to zero at large separations. We will shortly show that
the latter term in (A.10) is a constant. Therefore, the potential indeed is
$V(L)\propto L^{-2/3}$ at small separations.

In order to study the potential at large separations we need to analyze the
string profile carefully. Usually the large separation limit is somewhat
trivial as often the string is assumed to break at some critical separation
making the potential constant at larger separations. It is possible, however,
to extract non-trivial IR-behavior of the potential by analytically continuing
$L(z_{*})$ and $S_{NG}(z_{*})$ to the complex $z_{*}$-plane [80]. The quark
separation $L$ is always non-negative and real but the turning point $z_{*}$
need not be. In Fig. 19 we show a curve along which $L$ is real and non-
negative, corresponding to complex string configurations. The curve represents
the solutions to (A.5) when solving for $z_{*}$ and setting $L$ to fixed non-
negative real values. At small separations $z_{*}$ is real as usual. The
separation at which the string would break into the disconnected configuration
happens in this region. Shortly after that separation the turning point
becomes complex. Also the conjugate of this curve solves the equation but we
choose to focus on the $\operatorname{Im}[z_{*}]\geq 0$ branch.

Figure 19: The curve in the complex $z_{*}$-plane which corresponds to
positive real quark separations in (A.5). At quark separation $L=0$ the
corresponding turning point is $z_{*}=0$ and $L$ increases as one moves to the
right along the curve. At small separations $z_{*}$ is real but moves to the
complex plane shortly after the separation where the string would break.

The large separation limit can be worked out by first inverting (A.5) at large
$L$. We find

$\displaystyle\frac{z_{*}}{z_{h}}=$
$\displaystyle\frac{5^{2/3}\Gamma\mathopen{}\mathclose{{}\left(\frac{4}{5}}\right)^{4/2}}{2^{14/15}\pi^{2/3}\Gamma\mathopen{}\mathclose{{}\left(\frac{3}{5}}\right)^{2/3}}\mathopen{}\mathclose{{}\left(\frac{z_{p}L^{2}}{z_{h}^{3}}}\right)^{1/3}$
$\displaystyle+\frac{4(-1)^{3/10}2^{1/3}(5-\sqrt{5})(5\pi)^{1/6}\Gamma\mathopen{}\mathclose{{}\left(\frac{4}{5}}\right)^{5/3}}{9\Gamma\mathopen{}\mathclose{{}\left(\frac{3}{10}}\right)^{5/3}}\mathopen{}\mathclose{{}\left(\frac{z_{h}^{3}}{z_{p}L^{2}}}\right)^{1/6}+\mathcal{O}\mathopen{}\mathclose{{}\left(L^{-4/3}}\right)\
.$ (A.11)

This can be used together with (A.6) to see that the action at large
separations behaves as

$\displaystyle\frac{2\pi\alpha^{\prime}}{\tau
z_{p}^{2}}S_{NG}(L\to\infty)=\frac{2}{\epsilon}-i\frac{z_{p}^{1/2}L}{z_{h}^{3/2}}+\frac{2(-1)^{4/5}}{\sqrt{\pi}}\Gamma\mathopen{}\mathclose{{}\left(\frac{7}{10}}\right)\Gamma\mathopen{}\mathclose{{}\left(\frac{4}{5}}\right)+\mathcal{O}\mathopen{}\mathclose{{}\left(L^{-1}}\right)\
.$ (A.12)

In the definition of the $q\bar{q}$-potential, we can now see that the real
part of the IR-action will cancel the $1/\epsilon$-divergence present in
$S_{NG}(L)$. The constant term in (A.12) contributes to the counter term in
(A.10) and makes the real part of $V(L)$ vanish at large separations. We can
also immediately see that
$\operatorname{Im}[V(L\to\infty)]=-iz_{p}^{1/2}L/z_{h}^{3/2}+\mathcal{O}(1)$.
Further, analyzing subleading terms of (A.6) we can find that
$\operatorname{Re}[V(L\to\infty)]$ behaves as $L^{-10/3}$ like we claimed in
the beginning of this section.

###  A.2 Static quark potential from the lattice

The lattice gauge theory formalism was introduced by K. G. Wilson in 1974 to
show that quarks are confined in QCD [44]. The static quark potential was
therefore from the beginning of primary interest in lattice Monte Carlo
studies of $\operatorname{SU}(N_{\textrm{\scriptsize c}})$ gauge theories [81]
and various observables have been employed to study its properties. In
particular Wilson loops, which are traces over parallel transporters along
closed loops, and Polyakov loops, which are Wilson loops that wrap around a
compact direction (usually the Euclidean time direction), played an important
role in understanding the non-perturbative properties of QCD, as, _e.g._ ,
showing that $\operatorname{SU}(2)$ pure gauge theory is both, confining and
asymptotically free [82], and that the asymptotically free and the confining
region of parameter space are separated from each other by a deconfinement
transition [83].

With increasing computer performance and the development of improved
simulation techniques [84] these non-perturbative studies of the properties of
the static quark potential have been extended first to $\operatorname{SU}(3)$
[85, 86, 87] and then to higher $N_{\textrm{\scriptsize c}}$ and spacetime
dimensions different from (3+1) [88]. Results at $T=0$ in (2+1) dimensions can
for $N_{\textrm{\scriptsize c}}=2$, for example, be found in [89, 90], for
$N_{\textrm{\scriptsize c}}=3$ in [91], and for $N_{\textrm{\scriptsize c}}=5$
in [92]. Recall that the static quark potential can also be computed
perturbatively [37].

At $T=0$ the determination of the static quark potential using Polyakov loops
suffers from a bad signal to noise ratio as the magnitude of Polyakov loop
correlation function drops exponentially with $1/T$. At $T=0$ the static quark
potential is therefore usually determined from Wilson loops [86]. As we are in
the present work interested not in the zero but in high temperature properties
of $\operatorname{SU}(N_{\textrm{\scriptsize c}})$ gauge theory, we will use
Polyakov loop correlators to determine the static quark potential.

####  A.2.1 Static quarks and Polyakov loops

In this section we illustrate the relation between Polyakov loops and static
quarks. Starting point is Wilson’s lattice fermion action

$S_{F}\mathopen{}\mathclose{{}\left[U,\bar{\psi},\psi}\right]=\sum\limits_{x,y}\bar{\psi}\mathopen{}\mathclose{{}\left(x}\right)\,D_{x,y}\mathopen{}\mathclose{{}\left[U}\right]\,\psi\mathopen{}\mathclose{{}\left(y}\right)\
,$ (A.13)

with $\psi\mathopen{}\mathclose{{}\left(x}\right)$ and
$\bar{\psi}\mathopen{}\mathclose{{}\left(x}\right)$ being Euclidean Dirac
spinors appropriate for $d$ dimensions and
$D_{x,y}\mathopen{}\mathclose{{}\left[U}\right]$ is the corresponding Wilson-
Dirac operator,

$D_{x,y}\mathopen{}\mathclose{{}\left[U}\right]\,=\,\delta_{xy}\mathbbm{1}-\kappa\underbrace{\sum\limits_{\nu=1}^{d-1}\biggl{(}\bigg{.}\delta_{x+\widehat{\nu},y}\mathopen{}\mathclose{{}\left(\mathbbm{1}-\gamma_{\nu}}\right)U_{\nu}(x)+\delta_{x-\widehat{\nu},y}\mathopen{}\mathclose{{}\left(\mathbbm{1}+\gamma_{\nu}}\right)U^{\dagger}_{\nu}(x-\widehat{\nu})\bigg{.}\biggr{)}}_{S_{xy}}\\\
-\,\kappa\underbrace{\mathopen{}\mathclose{{}\left(\delta_{x+\widehat{4},y}\mathopen{}\mathclose{{}\left(\mathbbm{1}-\gamma_{4}}\right)\,\operatorname{e}^{\mu}\,U_{4}(x)+\delta_{x-\widehat{4},y}\mathopen{}\mathclose{{}\left(\mathbbm{1}+\gamma_{4}}\right)\,\operatorname{e}^{-\mu}\,U^{\dagger}_{4}(x-\widehat{4})}\right)}_{T_{xy}}\
,$ (A.14)

with the hopping parameter,
$\kappa=1/\mathopen{}\mathclose{{}\left(2\mathopen{}\mathclose{{}\left(m_{0}+d}\right)}\right)$,
which is related to the bare quark mass, $m_{0}=a\,m_{c}$, and
$\mathopen{}\mathclose{{}\left\\{\gamma_{\nu}}\right\\}_{\nu=1,\ldots,d}$ are
Euclidean Dirac matrices in $d$ dimensions. We have added a quark chemical
potential, $\mu$, to Eq. (A.14) in order to be able to keep track of whether a
terms correspond to a quark or an anti-quark in what follows.

In the Euclidean path integral,

$Z=\int\mathcal{D}\bigl{[}U,\psi,\bar{\psi}\bigr{]}\,\operatorname{e}^{-S_{F}\mathopen{}\mathclose{{}\left[U,\bar{\psi},\psi}\right]-S_{G}\mathopen{}\mathclose{{}\left[U}\right]}\
,$ (A.15)

the integration over the spinor fields
$\psi\mathopen{}\mathclose{{}\left(x}\right)$ and
$\bar{\psi}\mathopen{}\mathclose{{}\left(x}\right)$ is a multi-dimensional
Grassmann integral which upon evaluation yields a so-called fermion
determinant:

$Z_{f}\mathopen{}\mathclose{{}\left[U}\right]=\int\mathcal{D}\bigl{[}\psi,\bar{\psi}\bigr{]}\,\operatorname{e}^{-S_{F}\mathopen{}\mathclose{{}\left[U,\bar{\psi},\psi}\right]}\,=\,\operatorname{Det}\mathopen{}\mathclose{{}\left(D\mathopen{}\mathclose{{}\left[U}\right]}\right)\
,$ (A.16)

in terms of which:

$Z=\int\mathcal{D}\bigl{[}U\bigr{]}\,\operatorname{Det}\mathopen{}\mathclose{{}\left(D\mathopen{}\mathclose{{}\left[U}\right]}\right)\,\operatorname{e}^{-S_{G}\mathopen{}\mathclose{{}\left[U}\right]}\
.$ (A.17)

Now, if we are interested in static quarks, _i.e._ , in quarks that do not
perform spatial hops but simply move along straight lines around the time
direction, we can in Eq. (A.14) drop the spatial hopping terms, $S_{x\,y}$,
and keep only the temporal hopping terms, $T_{x\,y}$. In this case, the
fermion determinant in Eq. (A.16) simplifies significantly. If we stick for
the moment to $d=4$, it takes the form:

$\operatorname{Det}\mathopen{}\mathclose{{}\left(D\mathopen{}\mathclose{{}\left[U}\right]}\right)=\operatorname{Det}\mathopen{}\mathclose{{}\left(\mathbbm{1}-\kappa\,T}\right)=\\\
\prod\limits_{\bar{x}}\bigl{(}\big{.}\mathop{\mathrm{det}}\nolimits^{2}_{c}\bigl{(}\big{.}\mathbbm{1}+\mathopen{}\mathclose{{}\left(2\,\kappa}\right)^{N_{t}}\,\operatorname{e}^{\mu\,N_{t}}\,P\mathopen{}\mathclose{{}\left(\bar{x}}\right)\big{.}\bigr{)}\mathop{\mathrm{det}}\nolimits^{2}_{c}\bigl{(}\big{.}\mathbbm{1}+\mathopen{}\mathclose{{}\left(2\,\kappa}\right)^{N_{t}}\,\operatorname{e}^{-\mu\,N_{t}}\,P^{\dagger}\mathopen{}\mathclose{{}\left(\bar{x}}\right)\big{.}\bigr{)}\big{.}\bigr{)}\
,$ (A.18)

where the subscript $c$ of the $\operatorname{det_{c}}$-operator is meant to
indicate that it acts only in color-space (_i.e._ , that the operand is a
matrix with only color indices) and the product runs over all spatial
locations $\bar{x}$.

As can be seen from Eqn. (A.18) and (A.17), the static quarks couple to the
gauge field through the Polyakov and anti-Polyakov loops. The Polyakov loop,
$P\mathopen{}\mathclose{{}\left(\bar{x}}\right)$ at spatial location
$\bar{x}$, is given by the ordered product of all temporal link-variables (cf.
Eq. (2.4)) over $\bar{x}$, to form a parallel transporter that connects the
site $x=\mathopen{}\mathclose{{}\left(\bar{x},t=0}\right)$ with itself by
wrapping once on a straight temporal trajectory around the periodic time-
direction:

$P\mathopen{}\mathclose{{}\left(\bar{x}}\right)=U_{d}\mathopen{}\mathclose{{}\left(\bar{x},0}\right)\,U_{d}\mathopen{}\mathclose{{}\left(\bar{x},1}\right)\,\ldots\,U_{d}\mathopen{}\mathclose{{}\left(\bar{x},N_{t}-1}\right)\
.$ (A.19)

Its hermitian conjugate,
$P^{\dagger}\mathopen{}\mathclose{{}\left(\bar{x}}\right)$, does the same
thing but in opposite direction, as illustrated in Fig. 20.

Figure 20: Illustration of a pair of anti-Polyakov loop, $P^{\dagger}$, and
Polyakov loop, $P$, separated by a spatial distance $L$. Both loops are
wrapping around the periodic time extent of the lattice, but in opposite
directions.

Now, as we forced the fermions in Eq. (A.18) to be static, it is reasonable to
require them as well to be heavy, in which case $\kappa$ becomes very small
and the color-determinants in Eq. (A.18) can be written as

$\mathop{\mathrm{det}}\nolimits_{c}\bigl{(}\big{.}\mathbbm{1}+\mathopen{}\mathclose{{}\left(2\,\kappa}\right)^{N_{t}}\operatorname{e}^{\mu\,N_{t}}P\mathopen{}\mathclose{{}\left(\bar{x}}\right)\big{.}\bigr{)}=1+\mathopen{}\mathclose{{}\left(2\,\kappa}\right)^{N_{t}}\operatorname{e}^{\mu\,N_{t}}\operatorname{tr}\bigl{(}\big{.}P\mathopen{}\mathclose{{}\left(\bar{x}}\right)\big{.}\bigr{)}+\mathcal{O}\mathopen{}\mathclose{{}\left((2\,\kappa)^{2\,N_{t}}}\right)\
,$ (A.20a) resp.
$\mathop{\mathrm{det}}\nolimits_{c}\bigl{(}\big{.}\mathbbm{1}+\mathopen{}\mathclose{{}\left(2\,\kappa}\right)^{N_{t}}\operatorname{e}^{-\mu\,N_{t}}P^{\dagger}\mathopen{}\mathclose{{}\left(\bar{x}}\right)\big{.}\bigr{)}=1+\mathopen{}\mathclose{{}\left(2\,\kappa}\right)^{N_{t}}\operatorname{e}^{-\mu\,N_{t}}\operatorname{tr}\bigl{(}\big{.}P^{\dagger}\mathopen{}\mathclose{{}\left(\bar{x}}\right)\big{.}\bigr{)}+\mathcal{O}\mathopen{}\mathclose{{}\left((2\,\kappa)^{2\,N_{t}}}\right)\
.$ (A.20b)

As can be seen from Eqn. (A.20) by counting the powers of
$\operatorname{e}^{\mu}$ resp. $\operatorname{e}^{-\mu}$: having a static
quark or anti-quark winding around the time direction over a spatial site
$\bar{x}$ is accompanied by a Polyakov loop resp. anti-Polyakov loop located
on that site.

####  A.2.2 Static quark potential from Polyakov loops

We can now define the static quark potential,
$V\mathopen{}\mathclose{{}\left(L,T,\beta_{g}}\right)$, in terms of the
spatial correlation function between a traced Polyakov loop, $P$, and a traced
anti-Polyakov loop, $P^{\dagger}$ (cf. Fig. 20), using the relation [93],

$\operatorname{e}^{-V\mathopen{}\mathclose{{}\left(L,T,\beta_{g}}\right)/T}=\frac{1}{N_{\textrm{\scriptsize
c}}^{2}}\mathopen{}\mathclose{{}\left\langle\operatorname{tr}\bigl{(}\big{.}P^{\dagger}\mathopen{}\mathclose{{}\left(\bar{x}}\right)\big{.}\bigr{)}\operatorname{tr}\bigl{(}\big{.}P\mathopen{}\mathclose{{}\left(\bar{y}}\right)\big{.}\bigr{)}}\right\rangle_{\beta_{g}}\
,$ (A.21)

with $L=\mathopen{}\mathclose{{}\left|\bar{y}-\bar{x}}\right|$ and $T=1/N_{t}$
(in lattice units). Note that Eq. (A.21) is particularly useful at finite
temperatures, _i.e._ , when $N_{t}$ is not too large. As the formula
indicates, at low temperatures, the magnitude of the correlation function
decays quickly with $N_{t}=1/T$ and develops a bad signal to noise ratio. If
one is interested in the static quark potential at low or zero temperatures,
one therefore typically uses a different approach which is based on so-called
Wilson loops [94].

The potential $V\mathopen{}\mathclose{{}\left(L,T}\right)$ in Eq. (A.21) has
an explicit temperature-dependency, which is particularly prominent for
$T/T_{c}>1$, where the potential approaches for $L\gg T^{-1}$ a temperature-
dependent plateau value. We will subtract this plateau value and consider the
subtracted potential, the binding energy (in units of the critical temperature
scale),

$V_{s}\mathopen{}\mathclose{{}\left(L,T,\beta_{g}}\right)=\frac{V\mathopen{}\mathclose{{}\left(L,T,\beta_{g}}\right)-V\mathopen{}\mathclose{{}\left(\infty,T,\beta_{g}}\right)}{T_{c}}\\\
=-\frac{T}{T_{c}}\log\mathopen{}\mathclose{{}\left(\frac{\mathopen{}\mathclose{{}\left\langle\operatorname{tr}\bigl{(}\big{.}P^{\dagger}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}\operatorname{tr}\bigl{(}\big{.}P\mathopen{}\mathclose{{}\left(y}\right)\big{.}\bigr{)}}\right\rangle_{\beta_{g}}}{\big{|}\mathopen{}\mathclose{{}\left\langle\operatorname{tr}\bigl{(}\big{.}P\big{.}\bigr{)}}\right\rangle_{\beta_{g}}\big{|}^{2}}}\right)\
,$ (A.22)

instead of $V\mathopen{}\mathclose{{}\left(L,T,\beta_{g}}\right)$ itself. Fig.
21 shows Monte Carlo results for Eq. (A.22) as function of $L$ in the case of
$\operatorname{SU}(2)$ in $\mathopen{}\mathclose{{}\left(2+1}\right)$
dimensions at different temperatures and for two different lattice spacings,
_i.e._ , different values of $\beta_{g}$.

$a\,T_{c}=0.032$

$a\,T_{c}=0.048$

Figure 21: Subtracted static quark potential from Eq. (A.22) for
$\operatorname{SU}(2)$ at finite temperature in (2+1) dimensions, plotted as
function of distance $L$ for five different temperatures $T>T_{c}$. The two
panels corresponds to different lattice spacings, $a\,T_{c}=0.032$ (left) and
$a\,T_{c}=0.048$ (right). Error bars are smaller than the plot markers.

####  A.2.3 Setting the scale

We use the critical temperature, $T_{c}$, as reference energy scale. To be
able to do so, we need to determine the ratio $T/T_{c}$ as function of the
inverse lattice coupling $\beta_{g}$ and the spatial and temporal lattice
sizes $N_{s}$, $N_{t}$. We therefore want a function
$C_{R}\mathopen{}\mathclose{{}\left(\beta_{g},N_{t},N_{s}}\right)=T/T_{c}$,
which can be obtained along the lines described in [95, Sec. 4.3-4.4]. Using
the expression given in [95] for the pseudo-critical lattice coupling as
function of $N_{t}$ and $N_{s}$,

$\beta_{g,c}\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)=N_{t}\,\mathopen{}\mathclose{{}\left(4\,c_{r}-d_{2}\mathopen{}\mathclose{{}\left(\frac{N_{t}}{N_{s}}}\right)^{\rho}}\right)-\mathopen{}\mathclose{{}\left(4\,c_{1}+d_{1}\,\mathopen{}\mathclose{{}\left(\frac{N_{t}}{N_{s}}}\right)^{\rho}}\right)\\\
-\frac{1}{N_{t}}\mathopen{}\mathclose{{}\left(4\,\frac{c_{2}}{c_{r}}+d_{0}\,\mathopen{}\mathclose{{}\left(\frac{N_{t}}{N_{s}}}\right)^{\rho}}\right)\
,$ (A.23)

as well as the function

$C\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)=N_{t}\,\mathopen{}\mathclose{{}\left(4\,c_{r}-d_{2}\mathopen{}\mathclose{{}\left(\frac{N_{t}}{N_{s}}}\right)^{\rho}}\right)+\frac{1}{N_{t}}\mathopen{}\mathclose{{}\left(4\,\frac{c_{2}}{c_{r}}+d_{0}\,\mathopen{}\mathclose{{}\left(\frac{N_{t}}{N_{s}}}\right)^{\rho}}\right)\
,$ (A.24)

where [95]

$c_{1}=-0.176(5)\ ,\ c_{2}=0.0675(5)\ ,\
c_{r}=\frac{T_{c}}{g_{c}^{2}}=0.3757(5)\ ,$ (A.25)

and

$d_{0}=-0.51(9)\ ,\ d_{1}=0\ ,\ d_{2}=0.134(4)\ ,\ \text{and}\ \rho=2.61(9)\
,$ (A.26)

one can define a preliminary

$C_{R,c}\mathopen{}\mathclose{{}\left(\beta_{g},N_{t},N_{s}}\right)=1+\frac{\beta_{g}-\beta_{g,c}\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)}{C\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)}\
.$ (A.27)

The subscript $c$ in Eq. (A.27) indicates that $T/T_{c}\approx
C_{R,c}\mathopen{}\mathclose{{}\left(\beta_{g},N_{t},N_{s}}\right)$ is only
valid if $\beta_{g}$ is sufficiently close to its pseudo-critical value
$\beta_{g,c}\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)$.

To obtain an expression for
$C_{R}\mathopen{}\mathclose{{}\left(\beta_{g},N_{t},N_{s}}\right)$ that is
valid for arbitrary values of $\beta_{g}$, $N_{s}$ and $N_{t}$, We note that
$\mathopen{}\mathclose{{}\left(a\,T_{c}}\right)$ depends only on $\beta_{g}$
and $N_{s}$, and the lattice spacing $a$ itself is controlled merely by
$\beta_{g}$. At constant $\beta_{g}$ and $N_{s}$ (which implies also constant
$a\mathopen{}\mathclose{{}\left(\beta_{g}}\right)$), we therefore have [95]:

$\frac{T\mathopen{}\mathclose{{}\left(\beta_{g},N_{t},N_{s}}\right)}{T_{c}\mathopen{}\mathclose{{}\left(\beta_{g},N_{s}}\right)}=\frac{T\mathopen{}\mathclose{{}\left(\beta_{g},N_{t},N_{s}}\right)}{T\mathopen{}\mathclose{{}\left(\beta_{g},N_{t}^{\prime},N_{s}}\right)}\frac{T\mathopen{}\mathclose{{}\left(\beta_{g},N_{t}^{\prime},N_{s}}\right)}{T_{c}\mathopen{}\mathclose{{}\left(\beta_{g},N_{s}}\right)}=\frac{N_{t}^{\prime}}{N_{t}}\,\frac{T\mathopen{}\mathclose{{}\left(\beta_{g},N_{t}^{\prime},N_{s}}\right)}{T_{c}\mathopen{}\mathclose{{}\left(\beta_{g},N_{s}}\right)}\
,$ (A.28)

which allows us to define the function
$C_{R}\mathopen{}\mathclose{{}\left(\beta_{g},N_{t},N_{s}}\right)=T/T_{c}$ for
arbitrary values of $\beta_{g}$, $N_{s}$ and $N_{t}$ by:

$C_{R}\mathopen{}\mathclose{{}\left(\beta_{g},N_{t},N_{s}}\right)=\frac{N_{t}^{\prime}\mathopen{}\mathclose{{}\left(\beta_{g},N_{s}}\right)}{N_{t}}\,C_{R,0}\mathopen{}\mathclose{{}\left(\beta_{g},N_{t}^{\prime}\mathopen{}\mathclose{{}\left(\beta_{g},N_{s}}\right),N_{s}}\right)\
,$ (A.29)

with

$N_{t}^{\prime}\mathopen{}\mathclose{{}\left(\beta_{g},N_{s}}\right)=\mathopen{}\mathclose{{}\left.\lfloor
N_{t}\rfloor}\right|_{N_{t}:\beta_{g,c}\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)=\beta_{g}}\
,$ (A.30)

_i.e._ , $N_{t}^{\prime}\mathopen{}\mathclose{{}\left(\beta_{g},N_{s}}\right)$
is the value of $N_{t}\in\mathbb{N}$ for which
$\mathopen{}\mathclose{{}\left(\beta_{g}-\beta_{g,c}\mathopen{}\mathclose{{}\left(N_{t},N_{s}}\right)}\right)$
assumes its smallest, positive value for given $\beta_{g}$ and $N_{s}$.

####  A.2.4 Data and interpolation to arbitrary temperatures

In Tab. 2 we summarize the simulation parameters for which we have produced
data. As can be seen, although the data for different lattice spacings covers
a similar temperature range, the simulated temperatures are in general
different. In order to be able to directly compare the static quark potentials
obtained at different lattice spacings, we will have to interpolate between
the simulated temperature values.

$\beta_{g}$ | $N_{s}$ | $a\,T_{c}$ | $N_{t}$ | $T/T_{c}$
---|---|---|---|---
16. | 48 | 0.0976593 | 2 | 5.11984
|  |  | 4 | 2.55992
|  |  | 6 | 1.70661
|  |  | 8 | 1.27996
32. | 64 | 0.0477628 | 4 | 5.2342
|  |  | 6 | 3.48947
|  |  | 8 | 2.6171
|  |  | 10 | 2.09368
|  |  | 12 | 1.74473
48. | 96 | 0.0316115 | 8 | 3.95426
|  |  | 10 | 3.16341
|  |  | 12 | 2.63617
|  |  | 14 | 2.25958
|  |  | 16 | 1.97713
Table 2: Simulation parameters and corresponding lattice spacing- and
$T/T_{c}$-values for $\operatorname{SU}(2)$ pure gauge theory on a
(2+1)-dimensional lattice.

We perform this interpolation as follows: we pick one of the simulated
$\beta_{g}$ and a value of $L$, and fit for this choice the form

$y=\alpha\,x^{\gamma}+\delta\ $ (A.31)

to the data
$\mathopen{}\mathclose{{}\left(x,y}\right)\in\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(T/T_{c},V_{s}\mathopen{}\mathclose{{}\left(L,T,\beta_{g}}\right)}\right)}\right\\}_{T}$,
with $V_{s}\mathopen{}\mathclose{{}\left(L,T,\beta_{g}}\right)$ from Eq.
(A.22) considered as function of $T$. This fit yields a function

$\tilde{V}_{s,\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}\mathopen{}\mathclose{{}\left(T}\right)=\alpha_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}\,\mathopen{}\mathclose{{}\left(\frac{T}{T_{c}}}\right)^{\gamma_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}}+\delta_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}\
,$ (A.32)

which allows us to obtain for the given values of $\beta_{g}$ and $L$ the
subtracted potential at arbitrary $T$. This procedure is then repeated for all
available $L$ and $\beta_{g}$. The error in the value for the subtracted
potential, given by Eq. (A.32) at arbitrary $T$, is obtained by determining
the 65% confidence band for Eq. (A.32). Some fit examples are shown in Fig. 22
and Fig. 23 provides an overview over the fitted parameter values at different
lattice spacings $a$ and different distances $L$.

Figure 22: Two examples of fits of the form (A.31) to the subtracted static
quark potential from Eq. (A.22) for $\operatorname{SU}(2)$ in (2+1)
dimensions, considered as function of $T/T_{c}$ at fixed
$\mathopen{}\mathclose{{}\left(\beta_{g},L}\right)$, resp.
$\mathopen{}\mathclose{{}\left(a\,T_{c},L\,T_{c}}\right)$. The error bands
correspond to the 65% confidence band.

Figure 23: The panels show the values of
$\alpha_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}$ (left),
$\gamma_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}$ (middle), and
$\delta_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}$ (right), used to
parametrize Eq. (A.32). The parameters are plotted as functions of $L\,T_{c}$
for three different lattice spacings, corresponding to $\beta_{g}=16$ (black
circles), $32$ (blue triangles), and $48$ (red diamonds).

Note that the chosen form of the fit function, Eq. (A.31), is essentially a
power law with a small additive correction, $\delta$, to account for the fact
that we subtract in Eq. (A.22) effectively the log of
$\mathopen{}\mathclose{{}\left\langle|P|}\right\rangle^{2}_{\beta_{g}}$
instead of $|\mathopen{}\mathclose{{}\left\langle
P}\right\rangle_{\beta_{g}}|^{2}$, in order to avoid cancellations between
contributions to the Polyakov loop observable coming from the two degenerate
vacua that exist in the deconfined phase for $\operatorname{SU}(2)$, where
$\operatorname{Re}\mathopen{}\mathclose{{}\left(P}\right)$ is either positive
or negative. Provided that the ansatz (A.31) works well and assuming that the
parameter $\delta$ indeed just accounts for the systematic error caused by
using $\mathopen{}\mathclose{{}\left\langle|P|}\right\rangle^{2}_{\beta_{g}}$
instead of $|\mathopen{}\mathclose{{}\left\langle
P}\right\rangle_{\beta_{g}}|^{2}$ in (A.22), we could in principle subtract
$\delta_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}$ from Eq. (A.32)
in order to get rid of this systematic error. Instead of Eq. (A.32) we would
therefore use,

$\tilde{V}_{s,\mathrm{imp},\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}\mathopen{}\mathclose{{}\left(T}\right)=\alpha_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}\,\mathopen{}\mathclose{{}\left(\frac{T}{T_{c}}}\right)^{\gamma_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}}\
,$ (A.33)

with the same values of
$\alpha_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}$ and
$\gamma_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}$. However, as our
spatial lattices are relatively large,
$\delta_{\mathopen{}\mathclose{{}\left(L,\beta_{g}}\right)}$ is typically very
small (cf. Fig. 23, right) and whether one uses Eq. (A.33) or sticks with Eq.
(A.32), does not give rise to significantly (within error bars) different
results.

####  A.2.5 Scaling of deconfined potential at short and large distances

As discussed in Sec. A.1 holographic considerations suggest that the static
quark potential in the deconfined phase of a strongly coupled gauge theory in
$\mathopen{}\mathclose{{}\left(2+1}\right)$ dimensions should follow different
power laws at short and long distances, namely
$V_{\mathopen{}\mathclose{{}\left(L\,T\ll
1}\right)}\mathopen{}\mathclose{{}\left(T,L}\right)\propto L^{-2/3}$ and
$V_{\mathopen{}\mathclose{{}\left(L\,T\gg
1}\right)}\mathopen{}\mathclose{{}\left(T,L}\right)\propto L^{-10/3}$.

Figure 24: Static quark potential for $\operatorname{SU}(2)$ in (2+1)
dimensions as function of distance $L$ at finite temperature $T=2.646\,T_{c}$
(left) and $1.8\,T_{c}$ (right) for three different lattice spacings
$a=$0.098$\,T_{c}^{-1}$ (black), 0.048$\,T_{c}^{-1}$ (blue), and
0.032$\,T_{c}^{-1}$ (red). At short distances, well below $L\sim T^{-1}$, the
behavior of the potential appears compatible with an asymptotic power-law,
$\sim L^{-2/3}$ (dashed lines). The displayed fits were performed on data for
$L<0.3\,T^{-1}$ (where available). At distances above $L\sim T^{-1}$, the
potential appears to behave like $\sim L^{-10/3}$ (dotted lines). Here the
fits were obtained on data for $L>1.7\,T^{-1}$.

In Fig. 24 we fit these power law ansatzes to the short and long distance
pieces of the interpolated potential (A.32) (as function of $L\,T_{c}$) at
fixed $T/T_{c}=2.646$ (left) and fixed $T/T_{c}=1.8$ (right) for different
lattice spacing values. In order to stay away from the region around
$L\,T\approx 1$, where the log-log plots in Fig. 24 have clearly visible
curvature, we included for the short distance fits only data with $L\,T<0.3$,
and for the long distance fits only data with $L\,T>1.7$. The short distance
fit could therefore only be performed for the two smaller lattice spacing
values, as for the largest one with $a\,T_{c}=0.98$, there is only one point
satisfying the criterion $L\,T<0.3$ .

At the current stage, these fits should of course not be taken too seriously,
as the available data is of limited statistics and does also not sufficiently
cover the asymptotic short- and long-distance regimes. In both regimes, we see
at most the onset of a possible linear behavior of the subtracted potential in
the log-log plots in Fig. 24. It should also be mentioned that at short
distances, the displayed data contains also the points corresponding to
$L=1\,a$, which can be expected to suffer from relevant finite-lattice spacing
effects. At large distances one can infer that the data does not cover a
sufficiently large range to decide whether the potential really approaches a
power law and starts to follow a straight line in the log-log plots in Fig.
24, or whether its slope will continue to become even steeper with increasing
$L$.

Figure 25: Same data as in Fig. 24, but fitted with a global (_i.e._ , all
$L$-values with $L>a$) screened power-law ansatz.

To this end, we attempted to fit also a screened power law, _i.e._ , the form

$y=x^{-s}\,\operatorname{e}^{-m\,x}\ ,$ (A.34)

to the same data, where
$\mathopen{}\mathclose{{}\left(x,y}\right)\in\bigl{\\{}\big{.}\bigl{(}\big{.}L\,T_{c},\tilde{V}_{s,(L,\beta_{g})}\mathopen{}\mathclose{{}\left(T}\right)\big{.}\bigr{)}|L/a>1\
,L\,T_{c}<1.35\big{.}\bigr{\\}}$, with the interpolating potential from Eq.
(A.32). The results from these fits are summarized in Fig. 25. Note that these
are global fits in $L$, where we have excluded only the points at $L/a\leq 1$
to avoid the most-severe finite lattice spacing effects, and the points with
$L\,T_{c}>1.35$ where for the two smaller lattice spacings the finite lattice
volume would start to become visible. As can be seen, these fits worked out
quite well for the given quality of the data.

In order to confirm that the use of Eq. (A.33) instead of Eq. (A.32) to
interpolate our data for the subtracted static quark potential between the
simulated temperature values, would not significantly affect the fit results,
we show in Fig. 26 the analogous plots to those in Fig. 25, using the improved
interpolating potential form Eq. (A.33). These latter fits might appear
slightly cleaner than the ones in Fig. 25, in regards of the lattice spacing
dependency of the fitted parameters. But, the fitted parameters in Fig. 25 and
Fig. 26 are mutually compatible within error bars.

Figure 26: Same data as in Fig. 25, but using Eq. (A.33) instead of Eq. (A.32)
to obtain the values of the subtracted potential at the specified values of
$T/T_{c}$.

##  Appendix B Change of temporal boundary conditions and gauge invariance

Recall from Sec. 2.2 that when updating the boundary of region $A$ by changing
the temporal boundary conditions over spatial links, so that they change from
region $B$ to region $A$ or vice versa, we undergo transitions between
different partition functions
$\mathopen{}\mathclose{{}\left\\{Z_{i}}\right\\}_{i=0,\ldots}$, as introduced
above Eq. (2.24). We are interested in the free energy difference between
subsequent $Z_{i}$, which we determine numerically by probing the overlap
between the distributions of gauge field configurations of neighboring
$Z_{i}$.

When changing from one $Z_{i}$ to a neighboring one, it turns out that it
makes a huge difference whether the spatial link over which the temporal
boundary conditions are changed has before and after the update at least one
of its ends connect to a spatial site that is either completely in region $A$
or completely in region $B$ (cf. Fig. 27, left), or the spatial link touches
before and/or after the update at both ends a spatial site at which spatial
links from both regions meet (cf. Fig. 27, right).

Figure 27: The two panels show the spatial links corresponding to regions $A$
(red) and $B$ (blue) at two intermediate stages during the process of
interpolating between the situations depicted on the left- resp. right-hand
side of Fig. 5. The spatial link that is highlighted by a red frame in the
left-hand panel is at the moment blue and belongs therefore to region $B$. The
spatial site $x_{1}$ is touched only by spatial links belonging to region $B$,
while the spatial site $x_{2}$ is touched only by spatial links belonging to
region $A$, except for the highlighted spatial link. If the highlighted link
changes from region $B$ to region $A$, the spatial site $x_{2}$ will be
surrounded merrily by spatial links from region $A$, while the spatial site
$x_{1}$ will now also be touched by one link from region $A$. However, before
and after the change of the highlighted spatial link from region $B$ to region
$A$, one of its ends is connected to spatial site that is touched by spatial
links of only one region. In contrast: the spatial sites at both ends of the
highlighted spatial link in the right-hand panel are always touched by spatial
sites from both regions, $A$ and $B$, no matter whether the highlighted link
is still in region $B$ or changes to region $A$.

In order to illustrate why this is the case, we look at a transition between
some $Z_{i}$ and corresponding neighboring $Z_{i+1}$, which will require a
spatial link to change from region $B$ to region $A$. We recall from Fig. 5
that when a spatial link changes from region $B$ to region $A$ or vice versa,
two temporal plaquettes over that spatial link, denoted in Fig. 5 by $P_{1}$
and $P_{2}$, switch their top links, due to the change of temporal boundary
conditions. Such an update is in general gauge-dependent: if it were, _e.g._ ,
possible to use a local gauge transformation to make the to-be-swapped links
the same before doing the change of temporal boundary conditions, then the
Euclidean action would remain unchanged when changing form $Z_{i}$ to
$Z_{i+1}$. Such a gauge-dependency is okay here as we are sampling the overlap
between gauge configuration distributions of different systems with different
temporal boundary conditions and therefore different sites identified with
each other. The latter, _i.e._ , the fact that in the two systems different
sites are identified, affects locally the amount of gauge-freedom the system
has as one cannot perform independent local gauge-transformations on
identified sites. The overlap between the gauge configuration distributions of
neighboring $Z_{i}$ has therefore an entropic component coming from the change
in the amount of gauge freedom due to the identification of different numbers
of sites.

Consider as a concrete example the situation depicted on the left-hand side of
Fig. 27: if the indicated link between $x_{1}$ and $x_{2}$ belongs to region
$B$ (blue), we are in the system described by $Z_{1}$, and if the link changes
to region $A$ (red), we would be in the system described by $Z_{2}$. If the
link under consideration is currently part of region $B$, _i.e._ , we are in
the $Z_{1}$-system, the spatial site $x_{1}$ is touched only by spatial links
that belong to region $B$, and as explained in the left-hand panel of Fig. 28,
we could therefore apply independent gauge transformations over $x_{1}$ to
make the two links the same that would be swapped between the plaquettes
$P_{1}$ and $P_{2}$ from Fig. 5 when the temporal boundary conditions are
changed to those from region $A$. The same is true for the opposite move,
where the marked spatial link from the left-hand panel of Fig. 27 between the
spatial sites $x_{1}$ and $x_{2}$ is initially part of region $A$. In this
case the site $x_{2}$ is touched only by links from region $A$ and would allow
for the required local gauge transformations to make the to-be-swapped top-
links of plaquettes $P_{1}$ and $P_{2}$ identical. We can therefore define a
Metropolis update that connects a gauge configuration of $Z_{1}$ to a gauge
configuration of $Z_{2}$: if the system currently belongs to $Z_{1}$, the
update proposes to pick a local gauge transformation over $x_{1}$ that makes
the link-swap trivial, to then perform the link swap, and to finally do an
inverse gauge transformation over $x_{2}$; if the system belongs currently to
$Z_{2}$, the Metropolis update would do the same thing in the opposite
direction, _i.e._ , with initial local gauge transformation over $x_{2}$ and
inverse gauge transformation over $x_{1}$ after the swap. As the freedom in
choosing the appropriate gauge transformations is for both directions the
same, and the action does not change during the move, the corresponding
transition probability is always $1$. Although the way in which sites are
identified changes during such an update due to the change of temporal
boundary conditions, the number of identified points remains the same. Note
also that, although we can achieve that the swapped links themselves are the
same before and after such a move, the gauge field around them gets changed,
as the local gauge transformations that are applied at the beginning and end
of the move are applied at different spatial locations.

Figure 28: Left: the four sites highlighted by a red, orange, blue and green
circle are fully inside of one of the two regions $A$ (red,orange) or region
$B$ (blue,green) and therefore touched only by links that are all subject to
the same type of temporal boundary conditions: either those of region $A$, or
those of region $B$. If is therefore possible to apply independent local gauge
transformations at the green and blue, as well as the red and orange sites,
which will affect the correspondingly colored links.
Right: if one applies a local gauge transformation to the site highlighted by
a solid-lined black circle, which is touched by links from both regions, $A$
and $B$, the situation is more complicated. In region $B$, a gauge
transformation at this site changes the link $l_{1}$. From the point of view
from region $A$, however, the link $l_{1}$ is connected to the site
highlighted by the dashed black circle, on which we therefore have to apply
the same local gauge transformation as on the site with the solid black circle
in order to leave the gauge action unchanged, which will affect all the links
highlighted by dashed lines. The same conclusion is reached by focusing on the
link $l_{2}$ instead of $l_{1}$. The gauge transformation on the site with the
black circle changes in region $A$ directly the link $l_{2}$. In region $B$,
the link $l_{2}$ is connected to the site with the dashed black circle, on
which we therefore have to apply the same gauge transformation as on the site
with the solid-lined black circle.

As a second concrete example consider now the situation depicted in the right-
hand panel of Fig. 27. The spatial link emphasized there by the red frame
connects the sites $x_{1}$ and $x_{3}$. If the link belongs to region $B$, the
system is described by $Z_{7}$ and if the link belongs to region $A$ by
$Z_{8}$. Both spatial sites, $x_{1}$ and $x_{3}$, are touched by links from
both regions, $A$ and $B$. As explained in the right-hand panel of Fig. 28,
this means that it is not possible to find a local gauge transformation that
makes the links, that are swapped between $P_{1}$ and $P_{2}$ in Fig. 5 when
the temporal boundary conditions change, the same. Spatial sites that are
touched by spatial links from both regions, $A$ and $B$, are (indirectly)
subject to both temporal boundary conditions simultaneously, meaning that the
corresponding site at time $t=0$ over such a spatial site is not just
identified with either the site at the same spatial location at time $t=N_{t}$
(as inside region $B$) or time $t=2\,N_{t}$ (as inside region $A$) but with
both.

Note that in $\mathopen{}\mathclose{{}\left(1+1}\right)$ dimensions, the
situation from the second example cannot occur and one is always in the
analogous situation to the first example. In
$\mathopen{}\mathclose{{}\left(1+1}\right)$ dimensions, the boundary update of
region $A$ is always trivial and the derivative of the entanglement entropy
with respect to the width, $\ell$, of region $A$ has therefore to be
identically zero.

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