We use holographic duality to study the entanglement entropy (EE)
of Conformal Field Theories (CFTs) in various spacetime dimensions

Article funded by SCOAP3

0$, while if $\s$ approaches $s$ from above ($\s\to s^+$) then $\Delta \a<0$. The sign of $\Delta \a$ will therefore be immediately obvious to the naked eye, as our examples will illustrate. Dividing by $V$ in eq.~\eqref{eq:eddef} is thus technically trivial but practically useful: otherwise, to obtain $\Delta \a$'s sign we would have to extract (typically by numerical fitting) a subtle correction in $1/L$ from the EE itself. In section~\ref{general}, we write the coefficient of the $1/L$ correction as an integral over bulk metric components, which typically must be performed numerically. This integral's sign gives us $\Delta \a$'s sign. ]]>

\dcrit$ \\ \hline \ref{adsrn} & $(d+1)$ AdS-RN & $T$, $\mu$ & Yes & Yes, for $d>\dcrit$ or low $T$ \\ \hline \ref{hyper} & $(d+1)$ AdS-to-HV & $\mathcal{O}$, $T$, $\mu$ & No & Yes, for some $d$, $\zeta$, $\theta$ \\ \hline \ref{translation} & AdS$_4$-Linear Axion & $\gamma \, x \, \mathcal{O}$, $T$, $\mu$ & No & Yes, for low $T$ \\ \hline \ref{soliton} & $(d+1)$ AdS Soliton & compact $x$ & No & No \\ \hline \end{tabular} ]]>

0$, consistent with the area theorem. However, if $d>\dcrit$ then $\s$ increases to a single global maximum, which by dimensional analysis is at an $L \propto 1/T$, and then $\s \to s^+$ as $L \to \infty$, so that $\Delta \a<0$, violating the area theorem. Figure~\ref{fig:schematic} depicts these two behaviors schematically. (These results have also been obtained using the \emph{exact} results for EE of a strip in AdS-SCH, i.e.\ without numerics, in ref.~\cite{JohannaNina}.) More generally, for any CFT excited state in which the FLEE applies and $s \neq 0$, these are the two simplest ways to connect $\s \propto \Ttt L$ at small $L$ to $\sigma \to s^{\pm}$ at large $L$. \begin{figure} \centering \includegraphics[width=0.5\textwidth]{schematic} ]]>

\dcrit$ then for all $T/\mu$, $\s$ resembles the upper curve in figure~\ref{fig:schematic}, with a single maximum, whose position changes as $T/\mu$ decreases, and $\s \to s^+$ as $L \to \infty$. In particular, when $d>\dcrit$ the area theorem is always violated. On the other hand, if $d<\dcrit$, then at high $T/\mu$ we recover the result of section~\ref{adssc}, where $\s$ resembles the lower curve in figure~\ref{fig:schematic}, with no maximum and $\s \to s^-$ as $L \to \infty$. However, as we lower $T/\mu$, a transition occurs at a critical value of $T/\mu$ from the lower curve in figure~\ref{fig:schematic} to the upper curve, i.e.\ a peak appears. In particular, at the critical $T/\mu$, $\Delta \a$ changes sign and the area theorem is violated. In short, for any $d$, at sufficiently low $T/\mu$, $\s$ resembles the upper curve in figure~\ref{fig:schematic}, with a single maximum, $\s \to s^+$ as $L \to \infty$, and area theorem violation. In section~\ref{hyper} we consider the model of ref.~\cite{Lucas-ml-2014sba}, namely gravity in ${\rm AdS}_{d+1}$ coupled to a real scalar field and two $\U(1)$ gauge fields, which at $T=0$ yields domain-wall solutions from ${\rm AdS}_{d+1}$ to HV geometries~\cite{Huijse-ml-2011ef}. Such solutions are dual to CFTs in which $\mu$ and $\Op$ produce an IR fixed point with HV exponent $\theta$ and Lifshitz scaling $t \to \lambda^{\zeta} t$, $\vec{x} \to \lambda \vec{x}$, with $\lambda \in \mathbb{R}^+$, spatial coordinates $\vec{x}$, and dynamical exponent\footnote{The dynamical exponent is usually called $z$, but our $z$ is the coordinate normal to the ${\rm AdS}_{d+1}$ boundary.} $\zeta$. Similarly to section~\ref{rg}, in general the FLEE does not apply in these cases, and $\mathcal{O}$'s dimension $\Delta$ controls the leading power of $L$ in $\s$ at small $L$. We consider only the three examples of ref.~\cite{Lucas-ml-2014sba}, which all have $d<\dcrit$, and find several different behaviors as $T/\mu$ decreases, including both $\s \to s^{\pm}$ as $L \to \infty$, depending on the values of $\theta$ and $\zeta$, and area law violation at $T/\mu=0$, when $\theta = d-2$~\cite{Ogawa-ml-2011bz,Huijse-ml-2011ef}. \looseness=-1 In section~\ref{translation} we consider the solution of ref.~\cite{Andrade-ml-2013gsa}, namely gravity in ${\rm AdS}_4$ coupled to a $\U(1)$ gauge field and real, massless scalar ``axion'' fields scaling as $\g x$ with constant $\g$, which at $T=0$ is dual to an RG flow from a $d=3$ UV CFT driven by $\mu$ and a marginal $\Op$ with source $\g \, x$. The FLEE does not apply in this case, and at small $L$ we find $\s$ is a linear function of $L$ with slope $\propto\Ttt$ and \emph{non-zero} intercept $\propto \g^2$. When $\g=0$ the geometry reduces to AdS-RN, and we recover the results of section~\ref{adsrn} with $d=3<\dcrit$. When $\g/\mu \neq 0$ but $T/\mu=0$ the solutions of ref.~\cite{Andrade-ml-2013gsa} are dual to a semi-local quantum liquid state, similar to AdS-RN with $T/\mu=0$, with $s \neq 0$. Indeed, as $T/\g$ decreases we find a transition similar to that of AdS-RN, from the lower curve in figure~\ref{fig:schematic} to the upper curve. Finally, in section~\ref{soliton} we consider the AdS soliton, namely ${\rm AdS}_{d+1}$ with one direction $x$ compactified into a circle, with anti-periodic boundary conditions for fermions~\cite{Witten-ml-1998zw,Horowitz-ml-1998ha}. The compact direction shrinks to zero deep in the bulk, producing a ``hard wall,'' signaling mass gap generation and confinement in the dual QFT~\cite{Witten-ml-1998zw}. The QFT also has negative Casimir energy, $\langle T_{tt} \rangle <0$~\cite{Horowitz-ml-1998ha}. The FLEE does not apply in this case, nevertheless we find $\s \propto \langle T_{tt} \rangle L$ at small $L$. We find $\s<0$ for all $L$, and in particular as $L$ increases, $\s$ decreases to a minimum and then $\s \to 0^-$ as $L \to \infty$, similar to the relativistic RG flows of section~\ref{rg}. In summary, we find area theorem violation in AdS-SCH at large $d$, AdS-RN at low $T/\mu$, some models with HV geometries, and the model of ref.~\cite{Andrade-ml-2013gsa} at small $T/\g$. What do these all have in common? One obvious answer is: an IR fixed point that is not a $d$-dimensional CFT like the UV fixed point. In particular, the solutions of section~\ref{hyper} describe HV IR fixed points at $T/\mu=0$, while the other cases describe $(0+1)$-dimensional IR fixed points, meaning invariance under rescaling of $t$ but not $\vec{x}$~\cite{Iqbal-ml-2011in,Iqbal-ml-2011ae,Emparan-ml-2013moa,Emparan-ml-2013xia}, which can be interpreted as HV in the limit $\zeta \to \infty$ with $-\theta/\zeta$ fixed~\cite{Hartnoll-ml-2012wm}. More precisely, in AdS-SCH when $d\to \infty$, in the near-horizon region $t$ and the holographic radial coordinate, $z$, form the $\SL(2,\mathbb{R})/\U(1)$ group manifold, while $\vec{x}$ forms $\mathbb{R}^{d-1}$~\cite{Emparan-ml-2013xia}. In AdS-RN at $T/\mu=0$ or the model of ref.~\cite{Andrade-ml-2013gsa} at $T/\g=0$, in the near-horizon region $t$ and $z$ form ${\rm AdS}_2$ while the $\vec{x}$ form $\mathbb{R}^{d-1}$. As a result, in each near-horizon region, linearized fluctuations of fields transform covariantly under rescalings that act on $t$ but not $\vec{x}$~\cite{Emparan-ml-2013moa,Emparan-ml-2013xia}. Strictly speaking, such non-relativistic scale invariance occurs only for a limiting value of some parameter: $d = \infty$, $T/\mu=0$, etc. However, in our examples area theorem violation occurs at intermediate values of these parameters, as we dial them towards the limits. In other words, area theorem violation first occurs while the non-relativistic scale invariance is \emph{nascent}, i.e.\ not yet exact, and hence signals the emergence of non-relativistic massless degrees of freedom. ]]>

\ell$ they exhibit exponential decay~\cite{Iqbal-ml-2011in}. (In extremal AdS-RN, $\ell \propto 1/\mu$.) If the two systems have different operators, then we cannot compare $\ell$ precisely. If we instead define $\ell$ from the maximum in $\s$, then we can. Turning the holographic duality around, $\s$ can also help characterize geometries. For example, in a solution such as extremal AdS-RN, $\s$'s global maximum could provide a precise division between near- and far-horizon regions. We can also use $\s$ to characterize scaling geometries deep in the bulk or near a horizon, even away from the strict limit in which the geometry is scale invariant. Imagine for instance that we did not know the AdS-RN solution at $T/\mu=0$ (as often occurs when numerically solving for a metric). Area theorem violation would occur at finite $T/\mu$, not just at $T/\mu=0$, already suggesting that the extremal near-horizon geometry may have scale invariance, but cannot be ${\rm AdS}_{d+1}$. In sum, $\s$ is clearly useful for ``fingerprinting'' states of QFTs, holographic or otherwise. We therefore believe $\s$ deserves further exploration in future research. ]]>

0$ and the area theorem is obeyed, while if $C(z_H)>0$ then $\Delta \a <0$ and the area theorem is violated. ]]>

0$. The NEC also requires $g'(z)\geq 0$, so any solution of eq.~\eqref{eq:superpotential} is guaranteed to obey the NEC. We also want $\mathcal{O}$ to be relevant, $\D

\dcrit$. For example, figure~\ref{fig:adsscd48} shows our numerical results for $\s/s$ as a function of $LT$ for (a) the strip and (b) the sphere in AdS-SCH with $d=4$ and $8$. In all cases we find $\s/s \propto (\Ttt/s) L$ at small $LT$, as expected. For $d=4$ and for both the strip and sphere, we find $\s/s$ increases monotonically and $\s/s \to 1^-$ as $LT \to \infty$, whereas for $d=8$, $\s/s$ rises to a global maximum at an $L$ that by dimensional analysis must be $\propto 1/T$, and then $\s/s \to 1^+$ as $LT \to \infty$. \begin{figure}[t] \centering \setlength{\tabcolsep}{0pt} \begin{tabular}{cc} \includegraphics[width=0.49\textwidth]{schwarzschild_strip_large_width} & \includegraphics[width=0.49\textwidth]{schwarzschild_sphere_large_width} \\ {\small(a)} & {\small(b)} \end{tabular} ]]>

0$ for $d=8$. \begin{figure}[t] \centering \includegraphics[width=0.5\textwidth]{adssc_cplot.pdf} ]]>

\dcrit$, we find $\s/s$ rises to a global maximum before $\s/s \to 1^+$ as $LT \to \infty$. The maximum occurs at an $LT$ on the order of $10^0$ to $10^2$. \fussy \begin{figure}[t] \centering \begin{subfigure}[b]{0.48\textwidth} \centering \includegraphics[width=\textwidth]{adssc_strip_alld} ]]>

\dcrit$, $\s/s$ rises to a global maximum before $\s/s \to 1^+$ as $LT \to \infty$.]]>

0$ when $d<\dcrit$ to $C(z_H)>0$ and $\Delta \a <0$ when $d>\dcrit$ represents area theorem violation~\cite{Ryu-ml-2006ef,Myers-ml-2012ed,Casini-ml-2012ei,Casini-ml-2016udt}. How does AdS-SCH evade the holographic proof in ref.~\cite{Myers-ml-2012ed} of the area theorem for the strip in relativistic RG flows? The proof of ref.~\cite{Myers-ml-2012ed} relies on the NEC. As mentioned below eq.~\eqref{eq:rg_flow_g_condition_1}, for relativistic RG flows the NEC implies $g'(z)\geq 0$, that is, $g(z)$ is strictly non-decreasing as $z$ increases. However, for AdS-SCH the NEC imposes no such constraint, and indeed $g(z) = 1-mz^d$ \emph{decreases} monotonically as $z$ increases, from $g(z=0)=1$ to $g(z=z_H)=0$. Apparently, as $d$ increases, eventually $g(z)$ decreases quickly enough to render $C(z_H)>0$. How does AdS-SCH evade the field theory proofs in refs.~\cite{Casini-ml-2012ei,Casini-ml-2016udt} of the area theorem for the sphere in relativistic RG flows? The proofs of refs.~\cite{Casini-ml-2012ei,Casini-ml-2016udt} relied crucially on Lorentz invariance, which non-zero $T$ clearly breaks. In fact, in the $d\to\infty$ limit AdS-SCH is dual to an RG flow from a $(d+1)$-dimensional UV CFT to a $(0+1)$-dimensional IR CFT, which is clearly only possible when Lorentz symmetry is broken. More specifically, when $d \to \infty$ the AdS-SCH near-horizon geometry becomes $\SL(2,\mathbb{R})/\U(1) \times \mathbb{R}^{d-1}$, where the latter factor represents the spatial directions $\vec{x}$~\cite{Emparan-ml-2013moa,Emparan-ml-2013xia}. After a mode decomposition on $\mathbb{R}^{d-1}$, the action in eq.~\eqref{eq:adsscaction} gives rise to a string theory with target space $\SL(2,\mathbb{R})/\U(1)$~\cite{Emparan-ml-2013xia}. Linearized fluctuations in the near-horizon region then exhibit scale invariance in $t$ and $z$ but not $\vec{x}$~\cite{Emparan-ml-2013xia,Castro-ml-2010fd}. AdS-SCH thus provides our first hint that area theorem violation can occur as we dial a parameter towards a limiting value in which an IR fixed point emerges with scaling different from the UV fixed point. We will find further examples of such behavior in the following. ]]>

\dcrit$ for various $T/\mu$. We find $\s/s \propto LT$ at small $L$ for all $T/\mu$, as required by the FLEE. For $T/\mu \gg 1$ we find $\s/s$ rises monotonically to a global maximum, and then $\s/s\to 1^+$ as $LT \to \infty$, consistent with our results from section~\ref{adssc}. As $T/\mu$ decreases and AdS-RN increasingly deviates from AdS-SCH, the global maximum persists, moving to smaller $LT$ while growing taller and narrower, such that $\s/s\to1^+$ for all $T/\mu$. \begin{figure}[t] \centering \includegraphics[width=0.6\textwidth]{rn_strip_d8} ]]>

0.107$, indicating $\s/s$ is monotonic in $LT$, then develops two zeroes for $0.107 > T/\mu > 0.102$, indicating a local minimum and maximum in $\s/s$, and then develops a single zero for $T/\mu < 0.098$, indicating a global maximum in $\s/s$. We find qualitatively similar behavior for the strip in all $d<\dcrit$: at some $(T/\mu)_1$ a local minimum and maximum appear, but $\s/s$ remains below one for all $LT$, at some $(T/\mu)_2<(T/\mu)_1$ a global maximum emerges, but still $\s/s\to1^-$ for $LT\to\infty$, and finally at some $(T/\mu)_3<(T/\mu)_2$ the transition occurs to $\s/s\to1^+$ as $LT \to \infty$. Our numerical estimates for $(T/\mu)_1$, $(T/\mu)_2$, and $(T/\mu)_3$ for $d=3,4,5,6<\dcrit$ appear in table~\ref{tab:adsrn}. \begin{table}[t] \centering \renewcommand{\arraystretch}{1.25} \begin{tabular}[c]{|c|c|c|c|} \hline $d$ & $(T/\mu)_1$ & $(T/\mu)_2$ & $(T/\mu)_3$ \\ \hline $3$ & $6.343 \times 10^{-4}$ & $4.858 \times 10^{-4}$ & $2.967 \times 10^{-4}$ \\ \hline $4$ & $0.107$ & $0.102$ & $0.098$ \\ \hline $5$ & $0.407$ & $0.403$ & $0.399$ \\ \hline $6$ & $1.219$ & $1.215$ & $1.213$ \\ \hline \end{tabular} ]]>

\dcrit \approx 6.7$, $C(z_H)>0$ for all $T/\mu$, indicating area theorem violation. For $d<\dcrit$, as $T/\mu$ decreases $C(z_H)$ changes sign from negative to positive at the $(T/\mu)_3$ in table~\ref{tab:adsrn}, indicating area theorem violation for $T/\mu<(T/\mu)_3$.]]>

\dcrit \approx 6.7$, at all $T/\mu$ we find $C(z_H)>0$, indicating $\Delta \a <0$ and hence the area theorem is violated. For all $d<\dcrit$, at high $T/\mu$ we find $C(z_H) < 0$, indicating $\Delta \a > 0$ and the area theorem is obeyed, but as $T/\mu$ decreases $C(z_H)$ eventually passes through zero, so that at low $T/\mu$ we find $C(z_H)>0$, indicating $\Delta \a >0$ and the area theorem is violated. In each case, the critical $T/\mu$ where $C(z_H)=0$ is precisely the $(T/\mu)_3$ for the strip in table~\ref{tab:adsrn}, as expected. \begin{figure} \centering \setlength{\tabcolsep}{0pt} \begin{tabular}{c c} \includegraphics[width=0.49\textwidth]{rn_extremal_strip_dimensions.pdf} & \includegraphics[width=0.49\textwidth]{rn_extremal_strip_d3_close.pdf} \\ {\small(a)} & {\small(b)} \\ \includegraphics[width=0.49\textwidth]{rn_extremal_sphere_dimensions.pdf} & \includegraphics[width=0.49\textwidth]{rn_extremal_sphere_d3_close.pdf} \\ {\small(c)} & {\small(d)} \end{tabular} ]]>

0$, so that for both the strip and sphere $\s/s \to 1^+$ as $L\mu \to \infty$ and the area theorem is violated. Figure~\ref{fig:adsrnextremal} shows $\s/s$ versus $L\mu$ in AdS-RN with $d=3,4,5,6,7,8$ and $T/\mu=0$, for the strip ((a) and (b)) and the sphere ((c) and (d)). In all cases $\s/s$ indeed has a global maximum and $\s/s\to1^+$ as $L\mu\to\infty$. In summary, in AdS-RN for either $d>\dcrit$ at any $T/\mu$, or for any $d$ and sufficiently small $T/\mu$, we find a global maximum in $\s/s$, and in particular $\s/s \to 1^+$ as $LT \to \infty$, indicating area theorem violation. In other words, as we dial a parameter towards a limiting value in which an IR fixed point appears with different scaling from the UV CFT ($d \to \infty$ or $T/\mu \to 0$), we find area theorem violation, as we saw in AdS-SCH and as we will see in some, but not all, of the following examples. ]]>

(T/\mu)_3$, indicating the area theorem is obeyed, while $C(z_H)>0$ for $T/\mu<(T/\mu)_3$, indicating area theorem violation. As mentioned above, for a solution such as this, with $\zeta \to \infty$, when $T/\mu=0$ the $z \to \infty$ geometry is conformal to ${\rm AdS}_2 \times \mathbb{R}^{d-2}$, with no horizon. In particular, $s=0$ when $T/\mu=0$, so figure~\ref{fig:sachdev2} (d) shows $\s$ in units of $\mu^2 R^2 /G$ versus $L\mu$ at $T/\mu=0$. As $L\mu$ increases, we find a transition at $L\mu\approx 2.37$, from a connected to disconnected minimal surface, similar to the transition in figure~\ref{fig:rgphasetrans}, and the transitions in various geometries conformal to ${\rm AdS}_2 \times \mathbb{R}^{d-2}$ in refs.~\cite{Kulaxizi-ml-2012gy,Erdmenger-ml-2013rca}. Otherwise, however, the overall behavior is the natural extrapolation from $T/\mu>0$, with $\s \propto L$ at small $L\mu$, then as $L\mu$ increases a global maximum appears, and finally $\s \to 0^+$ as $L\mu \to \infty$, indicating area theorem violation. \begin{table}[t] \centering \renewcommand{\arraystretch}{1.2} \begin{tabular}{|c|c|c|c|c|c|} \hline $d$ & $\zeta$ & $\theta$ & $(T/\mu)_1$ & $(T/\mu)_2$ & $(T/\mu)_3$ \\ \hline $4$ & $\infty$ & $-3\zeta$ & $0.336$ & $0.319$ & $0.289$ \\ \hline $3$ & $3$ & $1$ & $0.0629$ & $0.0516$ & $0.0334$ \\ \hline \end{tabular} ]]>

(T/\mu)_3$, indicating the area theorem is obeyed, while $C(z_H)>0$ for $T/\mu<(T/\mu)_3$, indicating area theorem violation. \begin{figure} \centering \setlength{\tabcolsep}{0pt} \begin{tabular}{c c} \includegraphics[width=0.49\textwidth]{sachdev_d3_z3_th1.pdf} & \includegraphics[width=0.49\textwidth]{sachdev_d3_z3_th1_close.pdf} \\ {\small(a)} & {\small(b)} \\ \includegraphics[width=0.49\textwidth]{sachdev_d3_z3_th1_c.pdf} & \includegraphics[width=0.49\textwidth]{sachdev_d3_z3_th1_t0.pdf} \\ {\small(c)} & {\small(d)} \end{tabular} ]]>

3$~\cite{Mozaffara-ml-2016iwm}. \looseness=-1 In the two-parameter solution space, we focus on two one-parameter subspaces: extremal solutions, $T/\mu=0$ with $\mu/\gamma$ fixed, and uncharged solutions, $\mu/T=0$ with $T/\gamma$ fixed. Figure~\ref{fig:transt0} (a) shows $\s/s$ versus $L\g$ for the strip in the extremal case for various $\mu/\g$. For sufficiently large $\mu/\g$, the effects of $\g$ are small, so $\s/s$ resembles that of extremal AdS-RN with $d=3$: as $L\g$ increases, $\s/s$ rises linearly from zero, reaches a global maximum, and then $\s/s\to1^+$ as $L\g \to \infty$, indicating area theorem violation. However, as $\mu/\gamma$ decreases, the effects of $\g$ grow prominent, especially at small $L\g$. Specifically, as $\mu/\g$ decreases the intercept, $\lim_{L\g\to0} \s/s$, increases, and moreover the slope at small $L\g$ changes sign from positive to negative. To see why, we use $\partial \s/\partial L \propto m$ from eq.~\eqref{eq:andrade_small_L}. Solving eq.~\eqref{eq:transT} (with $T=0$) for $z_H$, plugging the result into $g(z_H)=0$ in eq.~\eqref{eq:andrade_metric_functions}, and solving for $m$ gives \beq \frac{\partial\sigma}{\partial L} \propto m = \frac{(\mu^2 - \gamma^2)\sqrt{\mu^2 + 2 \gamma^2}}{6\sqrt{3}}, \eeq \looseness=-1 which clearly changes from positive to negative as $\mu/\g$ decreases. Meanwhile, for all $\mu/\g$ area theorem violation occurs: $\s/s\to1^+$ as $L\g\to\infty$. That is unsurprising since the near-horizon geometry is ${\rm AdS}_2 \times \mathbb{R}^{d-1}$, which we know from extremal AdS-RN exhibits area theorem violation, and changing $\mu/\g$ just changes $R_{{\rm AdS}_2}$ in eq.~\eqref{eq:transads2}. As confirmation, figure~\ref{fig:transt0} (b) shows $C(z_H)$ versus $\mu/\g$ for extremal solutions, where indeed $C(z_H)>0$ for all $\mu/\g$. \begin{figure} \centering \setlength{\tabcolsep}{0pt} \begin{tabular}{c c} \includegraphics[width=0.49\textwidth]{andrade_withers_t0.pdf} & \includegraphics[width=0.49\textwidth]{andrade_withers_t0_c.pdf} \\ {\small(a)} & {\small(b)} \end{tabular} ]]>

0$ for all $\mu/\g$, indicating area theorem violation.]]>

0.034$ and $C(z_H)>0$ for $T/\g<0.034$, indicating area theorem violation. \fussy \begin{figure} \centering \setlength{\tabcolsep}{0pt} \begin{tabular}{c c} \includegraphics[width=0.49\textwidth]{andrade_withers_mu0.pdf} & \includegraphics[width=0.49\textwidth]{andrade_withers_mu0_c.pdf} \\ {\small(a)} & {\small(b)} \end{tabular} ]]>

0.89$ (red dashed). (b) The differences in area, $\Delta A$, normalized by $\textrm{Vol}\left(\mathbb{R}\right)R^3/z_0$, between the two connected extremal surfaces (blue solid and red dashed) and the disconnected extremal surfaces (black dot-dashed), and the minimal surface with the same $L$ in compactified ${\rm AdS}_5$, versus $L/z_0$. A ``first-order phase transition'' occurs at \mbox{$L/z_0\approx 0.615$}~\cite{Bueno-ml-2016rma} from a connected (blue solid) to a disconnected (black dot-dashed) extremal surface.]]>

0$ for all $d$, consistent with the area theorem. In summary, the AdS soliton is our only example with a mass gap, i.e.\ no massless IR degrees of freedom. The AdS soliton is not Lorentz-invariant, and hence the proofs of the area theorem in refs.~\cite{Ryu-ml-2006ef,Myers-ml-2012ed,Casini-ml-2012ei,Casini-ml-2016udt} do not apply. Nevertheless, the AdS soliton has $\Delta \a>0$, consistent with the idea that the area term's coefficient counts degrees of freedom: with zero IR degrees of freedom, $\Delta \a$ should be positive. The question of whether $\Delta \a>0$ in all (holographic) systems with a mass gap we leave for future research. ]]>