\section{The order of the transition}
\label{sec:landau}

We now switch on a small coupling and ask what becomes of the flat potential. The analysis is that of AMMPV \cite{AMMPV1}. We give more detail on the three saddle points than is usual, because the comparison with gravity rests on them.

\subsection{The general form of the effective action}

At $\lambda\neq0$ one can still integrate out all the non-zero modes of all the fields on $S^3\times S^1$ in the presence of a constant holonomy $U$. They are massive, with masses of order $1/R$, and perturbation theory in $\lambda$ is well defined. The result is an effective action $S_{\rm eff}[U]$ which is no longer quadratic in the moments $u_n$. Two properties restrict its form.

The first is the centre symmetry. The action is invariant under $U\to e^{2\pi ik/N}U$, which at large $N$ becomes $u_n\to e^{in\alpha}u_n$ for any phase $\alpha$. Each term must carry total charge zero. The allowed terms of low order are $|u_1|^2$, $|u_2|^2$, $u_1^2u_{-2}$ with its conjugate, and $|u_1|^4$, where $u_{-n}=u_n^*$.

The second is large-$N$ counting. A vacuum diagram with $\ell$ loops of momentum has at most $\ell+1$ index loops, and each index loop gives one trace of a power of $U$. At order $\lambda^{\ell-1}$ one therefore finds products of up to $\ell+1$ traces. The quadratic terms are corrected already at one loop in $\lambda$, that is by two-loop diagrams. The cubic term $u_1^2u_{-2}$ needs three traces and also first appears at order $\lambda$. The quartic term needs four traces and first appears at order $\lambda^2$, in three-loop diagrams.

Near the Hagedorn temperature we may then write
\begin{equation}
\frac{S_{\rm eff}}{N^2}=m_1^2|u_1|^2+m_2^2|u_2|^2+\kappa\,\big(u_1^2u_{-2}+{\rm c.c.}\big)+\tilde b\,|u_1|^4+\ldots
\label{eq:Sgeneral}
\end{equation}
with $m_1^2=1-z_1+O(\lambda)$, $m_2^2=\half(1-z_2)+O(\lambda)$, $\kappa=O(\lambda)$ and $\tilde b=O(\lambda^2)$. The dots are terms that involve higher moments or higher powers and do not matter below.

Only $u_1$ is light. The mass $m_2^2$ is about $\half$ near $x_H$, so $u_2$ can be integrated out. It is sourced by $u_1^2$ through the cubic term. Its saddle-point value is $u_2=-\kappa u_1^2/m_2^2$, and substituting back gives \eqref{eq:landau-intro} with
\begin{equation}
b=\tilde b-\frac{\kappa^2}{m_2^2}.
\label{eq:b-from-btilde}
\end{equation}
Both terms are of order $\lambda^2$. The second is negative whatever the sign of $\kappa$. Exchange of a massive mode always produces an attraction.

\subsection{The potential beyond one half}

The expansion \eqref{eq:landau-intro} is in powers of $u_1$, but the transition will take $u_1$ to values near $\half$, so we need the action there. The extra ingredient is the bound \eqref{eq:half}. For $|u_1|>\half$ the eigenvalue density has a gap, the higher moments are forced to be non-zero, and the cost in the measure is larger than $|u_1|^2$.

The exact statement is the following. Consider the matrix integral over $U$ with $|u_1|=u$ held fixed and nothing else in the action. At large $N$ it equals $\exp(-N^2V(u))$ with
\begin{equation}
V(u)=
\begin{cases}
u^2, & 0\le u\le\half,\\[3pt]
\tfrac14-\half\ln\big[2(1-u)\big], & \half\le u<1 .
\end{cases}
\label{eq:V}
\end{equation}
This is the solution of the Gross--Witten--Wadia model \cite{GrossWitten,Wadia} read as a Legendre transform. The two branches join at $u=\half$ with continuous first and second derivatives. The third derivative jumps. Just beyond the junction, with $u=\half+\epsilon$,
\begin{equation}
V(u)-u^2=\tfrac43\epsilon^3+O(\epsilon^4).
\label{eq:Vcubic}
\end{equation}
The gapped branch acts as a wall, which rises as $\epsilon^3$ at first and diverges logarithmically at $u=1$.

Since all the interaction terms in the action are small, to the order we need the full potential for $u=|u_1|$ is
\begin{equation}
\frac{S(u)}{N^2}=V(u)-u^2+m_1^2u^2+b\,u^4 .
\label{eq:Sfull}
\end{equation}
For $u\le\half$ this is \eqref{eq:landau-intro}. For $u>\half$ the wall \eqref{eq:Vcubic} is added.

\begin{figure}[t]
\centering
\includegraphics[width=0.98\textwidth]{figs/potential.pdf}
\caption{The potential \eqref{eq:Sfull} for the Polyakov loop $u=|u_1|$, at several temperatures. The quartic coefficient is exaggerated to $|b|=0.1$ so that the features can be seen. At this value the second minimum appears at $m_1^2=0.053$ and becomes degenerate with the first at $m_1^2=0.032$, to be compared with the leading-order values $|b|/2$ and $|b|/4$ derived below. The dashed vertical line is $u=\half$, where the eigenvalue density develops a gap. Left, $b<0$: a second minimum appears just beyond $u=\half$ at $T_{\min}$, becomes degenerate with $u=0$ at $T_c$, and the barrier between them disappears at $T_H$. Right, $b>0$: the minimum moves continuously away from $u=0$ above $T_H$.}
\label{fig:potential}
\end{figure}

\subsection{Negative \texorpdfstring{$b$}{b}: three saddles and three temperatures}

Take $b<0$ and lower $m_1^2$ from a large positive value, which is what raising the temperature does. Figure~\ref{fig:potential} shows the sequence.

For $u<\half$ the potential $m_1^2u^2-|b|u^4$ has a minimum at $u=0$ and a maximum at
\begin{equation}
u_s=\sqrt{\frac{m_1^2}{2|b|}} ,
\label{eq:us}
\end{equation}
provided this is less than $\half$, that is for $m_1^2<|b|/2$. For $u=\half+\epsilon$, the condition $S'(u)=0$ with \eqref{eq:Vcubic} gives
\begin{equation}
\epsilon=\half\sqrt{\tfrac12|b|-m_1^2} ,
\label{eq:eps}
\end{equation}
to leading order in $|b|$. This is a minimum, and it too exists for $m_1^2<|b|/2$. At $m_1^2=|b|/2$ the maximum and the new minimum are born together at $u=\half$.

The value of the action at the new minimum is $\tfrac14m_1^2-\tfrac1{16}|b|$, up to terms of order $|b|^{3/2}$ from the wall. It is positive when the minimum is born and changes sign at $m_1^2=|b|/4$. Finally at $m_1^2=0$ the maximum $u_s$ reaches the origin and the minimum at $u=0$ ceases to be one. Table~\ref{tab:regimes} collects the four regimes.

\begin{table}[t]
\centering
\begin{tabular}{lll}
\toprule
range of $m_1^2$ & temperature & saddle points of \eqref{eq:Sfull} \\
\midrule
$m_1^2>\half|b|$ & $T<T_{\min}$ & only $u=0$ \\
$\tfrac14|b|<m_1^2<\half|b|$ & $T_{\min}<T<T_c$ & $u=0$ dominates; $u_s$ unstable; $u\simeq\half$ metastable \\
$0<m_1^2<\tfrac14|b|$ & $T_c<T<T_H$ & $u\simeq\half$ dominates; $u_s$ unstable; $u=0$ metastable \\
$m_1^2<0$ & $T>T_H$ & $u\simeq\half$ only; $u=0$ unstable \\
\bottomrule
\end{tabular}
\caption{The saddle points of the Polyakov-loop potential for $b<0$. The pattern is that of the Hawking--Page problem, with $u=0$ as thermal AdS, $u_s$ as the small black hole and $u\simeq\half$ as the large black hole.}
\label{tab:regimes}
\end{table}

To convert the ranges of $m_1^2$ into temperatures we need the slope of $m_1^2$ at its zero. From $m_1^2=1-z(x)+O(\lambda)$ and $x=e^{-1/T}$,
\begin{equation}
m_1^2\simeq\Big[x\frac{dz}{dx}\Big]_{x_H}\,\beta_0\,\frac{T_H-T}{T_H}=\frac32\,\beta_0\,\frac{T_H-T}{T_H}.
\label{eq:slope}
\end{equation}
The value $xz'(x)=\tfrac32$ at $x_H$ follows from \eqref{eq:z} by direct differentiation, and it will be used again. With it,
\begin{equation}
\frac{T_H-T_c}{T_H}=\frac{|b|}{6\beta_0},
\qquad
\frac{T_H-T_{\min}}{T_H}=\frac{|b|}{3\beta_0}.
\label{eq:Tc}
\end{equation}
Both are of order $\lambda^2$. The transition temperature $T_c$ and the Hagedorn temperature $T_H$ agree to first order in $\lambda$ and differ at second order.

At $T_c$ the Polyakov loop jumps from $0$ to $\half+\sqrt{|b|}/4$. The energy jumps too. The free energy is continuous across the transition by construction, and $E=\partial_\beta S_{\rm eff}$ evaluated at the saddle. The only explicit dependence on $\beta$ at leading order is in $m_1^2$, with $\partial_\beta m_1^2=xz'(x)=\tfrac32$, and $u^2=\tfrac14$. The latent heat is therefore
\begin{equation}
\Delta E=\frac38\,\frac{N^2}{R}\,\big(1+O(\lambda)\big).
\label{eq:latent}
\end{equation}
It does not vanish as $\lambda\to0$. What vanishes is the width of the window of metastability \eqref{eq:Tc} and the height of the barrier, which is of order $N^2|b|$.

Table~\ref{tab:regimes} has the structure of the gravitational problem. Below $T_{\min}$ there are no black holes. At $T_{\min}$ a small and a large black hole appear together. The small one is always unstable. The large one takes over from thermal AdS at $T_{\rm HP}$. The one feature of the table that classical gravity does not show is the last line, the temperature $T_H$ at which thermal AdS itself becomes unstable. In string theory that is the Hagedorn temperature of strings in AdS, where, as in flat space \cite{Sathiapalan,Kogan,AtickWitten}, a winding mode on the thermal circle becomes tachyonic \cite{BarbonRabinovici}. In the phenomenological matrix model of \cite{AGLW} it is also where the saddle of the small black hole, which shrinks as the temperature is raised, merges with thermal AdS. The authors expect the small black hole to reach the string scale, the correspondence point of Horowitz and Polchinski \cite{HorowitzPolchinski}, a little below $T_H$ at large $\lambda$, and argue that beyond this point the saddle is better viewed as highly excited strings. In the weakly coupled theory the merger is the statement $u_s\to0$ as $m_1^2\to0$.

\subsection{Positive \texorpdfstring{$b$}{b}: two continuous transitions}

For $b>0$ the potential $m_1^2u^2+bu^4$ has its only minimum at $u=0$ until $m_1^2$ changes sign. Above $T_H$ the minimum is at $u^2=-m_1^2/2b$ and grows continuously. The free energy is $-N^2m_1^4/4b$, quadratic in $T-T_H$, and its second derivative jumps. That is a second-order transition with mean-field exponents.

The minimum reaches $u=\half$ at $m_1^2=-b/2$. There the density develops its gap and the potential changes from one branch of \eqref{eq:V} to the other, with a jump in its third derivative. This is a third-order transition of the Gross--Witten--Wadia kind \cite{GrossWitten,Wadia}, at a temperature above $T_H$ by a relative amount $b/3\beta_0$.

Neither transition has a counterpart in classical gravity. If $b$ were positive at weak coupling, the two lines would have to meet and turn into a first-order line somewhere at intermediate coupling, as on the right of figure~\ref{fig:phase}.

The task is thus defined. We need $m_1^2$ to the order at which $T_H$ is shifted, the cubic coupling $\kappa$ at order $\lambda$, and $\tilde b$ at order $\lambda^2$. The first two are one-loop quantities and we review them next. The third is the new computation.