\section*{Introduction}

Potential energy surfaces (PES) are a widely used tool to describe physical and chemical systems, among others. For example, they are used in DNA chemistry~\cite{Elsawy:2005}, materials chemistry~\cite{Buehler:2000}, chemical physics~\cite{Roy:2007}, astrophysics~\cite{Schwartz:2010}, etc\ldots
Numerous methods exist to extract from a PES local minima, saddle points, and least energy paths between local minima~\cite{Henkelman:2000,Olsen:2004}. To produce a PES, a method giving the energy of a system as a function of a given number of constrained variables is needed. In this paper, we will separate these methods into two main classes: self-consistent and non-self-consistent methods. Surfaces associated with these methods will be called Self-consistent Potential Energy Surfaces (SPES) and Non Self-consistent Energy Surfaces (NSPES), respectively. In nuclear physics, SPES can be obtained for example by constrained Hartree-Fock methods and extensions~\cite{Warda:2002,Rodriguez:2010,Hinohara:2009,Bonneau:2006}, by constrained Relativistic Mean Field method~\cite{Guo:2010}, etc\ldots. NSPES can be produced by several methods, ranging from the historical liquid drop model~\cite{Meitner:1939,Bohr:1939} to the well-known macroscopic-microscopic model, using parametrization of the nuclear mean-field deformation~\cite{Moller:2001,Pyatkov:1997}.

The main difference between these two classes of methods is the way they deal with the non-constrained degrees of freedom of the system. Non self-consistent methods neglect their influence (all non-constrained degrees of freedom take a fixed value), while self-consistent methods perform an automatic minimization of the energy of the system in the non-constrained subspace. The presence or the absence of this automatic minimization leads to one specific problem for each class of methods:
\begin{itemize}
 \item {\bf NSPES problem}: every point of a given NSPES may not be a minimum of the same NSPES with an additional dimension.
 \item {\bf SPES problem}: a saddle point on a trajectory from a SPES is not automatically visible on the same trajectory from the same SPES restricted to a smaller dimension ({\it missing} saddle point), and a saddle point from a SPES is not automatically a saddle point on the same SPES with an additional dimension ({\it fake} saddle point).
\end{itemize}

These two problems have important consequences. For the NSPES problem, one would expect that enriching a NSPES with an additional constraint while conserving a given set of symmetries should improve the quality of the description, instead of invaliding it, but this is not always the case. For example, quantities calculated on a $N$-dimensional NSPES like minima, saddle points, or least energy paths between local minima {\it may differ dramatically} from quantities calculated on a $(N+1)$-dimensional NSPES, with the first $N$ constraints and conserved symmetries being the same for both NSPES.
Let us imagine an hypothetical non-constrained HFB minimum $|\psi\rangle$ with $\langle \psi|\hat{Q}_{70}|\psi\rangle=$~37~b$^{7/2}$. Finding this minimum with non-self-consistent methods will require to explore the deformation degree of freedom corresponding to $\hat{Q}_{70}$, without knowing beforehand that this degree of freedom has a role to play. If this degree of freedom is not taken into account, the resulting minimum can be anything but the searched minimum, and may differ dramatically from it.

Concerning the SPES problem, the obtained saddle points can be {\it fake} or {\it missing}. This last problem is well known (it corresponds to a {\it tipping point} in {\it Catastrophe Theory}~\cite{Gilmore:1981}), and is used sometimes to contest -with good reason- the quality of the results obtained by self-consistent or minimization methods~\cite{Myers:1996}. For example, in~\cite{Moller:2009}, the authors present a schematic case leading to the calculation of a {\it fake} saddle point by a minimization method, and conclude rather quickly that the doubt concerning the reality of saddle points found with self-consistent methods invalidates any production of a SPES by a minimization method: ``[\ldots] in Hartree-Fock-Bogoliubov (HFB) calculations with multiple constraints, saddle-point shapes and energies frequently cannot be determined accurately and are subject to errors of fairly random magnitude".

To our knowledge, there is no numerical or analytical method to fix the NSPES problem. Any result extracted from a NSPES may be invalidated by a NSPES of higher dimension. By comparison, there exists a simple numerical method to fix the SPES problem, that we will present in this paper.

A saddle point that is no more a saddle point in a bigger constrained subspace and a saddle point which vanishes while reducing the constrained subspace are two of the possible consequences of a unique phenomenon that we call a discontinuity of the SPES. A continuous SPES has no {\it fake} or {\it missing} saddle point. In this paper we first show such discontinuities, explain some of their consequences, and explain their origins. We then propose a method to quantitatively estimate if a SPES is continuous or not, and to identify the discontinuities if present. Finally we discuss what are the best ways to get a continuous SPES from a discontinuous one.

\section{Discontinuities}

\subsection{Model used}

Every calculation presented in this paper has been obtained by a two-center basis Hartree-Fock-Bogoliubov model \cite{Ring:1980} with Gogny D1S nucleon-nucleon effective interaction~\cite{Decharge:1980,Berger:1991}, based on the minimization principle of the energy functional, namely
\begin{equation}
\delta\langle \psi|\hat{H}-\lambda_N\hat{N}-\lambda_Z\hat{Z}-\sum_l\lambda_l\hat{Q}_{l0}|\psi\rangle=0\label{HFB1},
\end{equation}
where $\hat{H}$ is the nuclear microscopic Hamiltonian, $\hat{Q}_{l0}$ is a mass multipole operator, and $\lambda_N$, $\lambda_Z$ and $\lambda_l$ are the Lagrange multipliers corresponding to constraints on the neutron number $N$, proton number $Z$, and mass multipole moment $q_{l0}$, respectively. The corresponding constrained equations are
\begin{eqnarray}
\langle\psi|\hat{Z}|\psi\rangle&=&Z,\label{HFB2}\\
\langle\psi|\hat{N}|\psi\rangle&=&N,\\
\langle\psi|\hat{Q}_{l0}|\psi\rangle&=&q_{l0}\label{HFB4},
\end{eqnarray}
with $\hat{Q}_{l0}$ defined as
\begin{eqnarray}
\hat{Q}_{l0}&\equiv&(1+\delta_{l,2})\sqrt{\frac{4\pi}{2l+1}}\sum_{i=1}^{A}r_i^lY_{l0}(\theta_i,\phi_i).
\end{eqnarray}
The HFB energy of the constrained system $|\psi_{(N,Z,\{q_{l0}\})}\rangle$ is
\begin{eqnarray}
E^{HFB}_{(N,Z,\{q_{l0}\})}&=&\langle\psi_{(N,Z,\{q_{l0}\})}|\hat{H}|\psi_{(N,Z,\{q_{l0}\})}\rangle.
\end{eqnarray}
The HFB equations (Eqs.~(\ref{HFB1}) to (\ref{HFB4})) are solved iteratively by expanding the single-particle wave-functions onto an axial two-center harmonic oscillator basis. The basis parameters have been automatically optimized for each set of constraints. The Lagrange multipliers are adjusted at each iteration in order that the constraint conditions (Eqs.~(\ref{HFB2}) to (\ref{HFB4})) are fulfilled, and the generalized density matrix is corrected accordingly. This technique allows us to solve the constrained HFB equations in the restricted variational space orthogonal to the PES for which the constraint conditions are fulfilled. In this way, the problems of non-uniform mesh and inaccessible regions mentioned in e.g. \cite{Staszczak:2010} are completely avoided. For more information on the adjustment of the Lagrange multipliers in the case of multiple linear constraints, see for example Appendix~A of~\cite{Younes:2009}.

\subsection{Example of a discontinuity}

During the production of a self-consistent potential energy surface, a strange behavior can sometimes be observed. For a given set of constraints, starting the iterative process from two different points can lead to two very different solutions, both converged. This is clearly not what is expected from a self-consistent method, which is supposed to always converge to the only solution minimizing the total binding energy under the action of the constraints, whatever the starting point. Such a strange behavior is shown on Fig. \ref{fig1}, where two one-dimensional SPES have been produced, each point being the starting point for its left (squares) or right (triangles) neighbor. One can see that for a rather large range of constraint values, two different solutions can exist. The ``bow tie'' shape formed by these two curves is characteristic of a discontinuity in the SPES.

\begin{figure}[t]
 \begin{center}
  \includegraphics[width=9cm]{figure1.eps}
 \end{center}
 \caption{\label{fig1} (Color online) One-dimensional SPES for $^{226}$Th nucleus with a $q_{30}=52\textrm{ b}^{3/2}$ constraint. Propagation of the calculation is from the left neighbor (black triangles) or from the right neighbor (red squares). Upper panel: total binding energy; lower panel: mean value of the hexadecapole operator $\hat{Q}_{40}$.}
\end{figure}

\begin{figure}[t]
 \begin{center}
  \includegraphics[angle=-90,width=13cm]{figure2.eps}
 \end{center}
 \caption{\label{fig2} (Color online) Illustration of the solutions found when the propagation is from the left neighbor (panels (a) and (d)), from the right neighbor (panels (b) and (e)), and in the ideal case of a perfect minimization (panels (c) and (f)). The upper panels show the same schematic two-dimensional SPES as contour lines, with two valleys labeled A and B. The lower panels show the total binding energy as a function of the $Q_{20}$ values. In panel (f), the point with the maximum total binding energy can be the point $a$ or $b$ from panel (c).}
\end{figure}

\subsection{Explanation}

For an hypothetical system, let us consider the two-dimensional SPES plotted as contour lines on the upper panels (a), (b) and (c) of Fig. \ref{fig2}, showing two valleys A and B in the ($\hat{Q}_{20}$, $\hat{Q}_{xx}$) subspace. Let us now try to produce the one-dimensional SPES with increasing values of $q_{20}$, i.e. each point being the starting point for the calculation of its right neighbor. The resulting total binding energy is plotted on the panel (d), and the mean value $\langle Q_{xx} \rangle$ is plotted as dots on the panel (a). One can see that the converged solutions follow valley A until the system pass the barrier and ``falls'' into valley B. If we do the same calculations with decreasing values of $q_{20}$, we obtain the curves on panels (b) and (e). This time, the system stays longer in valley B before falling into valley A. If we superpose the two curves on the panels (d) and (e), we see the same ``bow tie'' shape as in Fig. \ref{fig1}.

The typical hysteresis figure shown on the lower panel of Fig.~\ref{fig1} or in panels (a) and (b) of Fig.~\ref{fig2} is due to the fact that the system can stay in its valley of origin even when the solution is no longer the global minimum. This phenomenon is mainly related to the way the minimization is performed. In the ideal case of a hypothetical perfect minimization, the global minimum is always found, leading to curves in the panels (c) and (f).

To solve the hysteresis problem in the case of a behavior like the one shown in Fig. \ref{fig1}, one has to discard the non-global minima, keeping the states with the lowest binding energies. This solution leads us to the previous ideal case: every constrained HFB solution is an absolute minimum, there is no more hysteresis, thus no more influence of the starting points. In the following, we will assume that the minimization is perfect, or that hysteresis problems have been tracked and solved. However, even with a perfect minimization, the passage from one valley to the other (from point $a$ to point $b$) is not continuous, since the mean value $\langle Q_{xx} \rangle$ is quite different for these two points. This discontinuity is not visible in the total binding energy curve, since point $a$ and point $b$ can be found to have very close (if not the same) total binding energies. A discontinuity in a SPES can have serious consequences.

\subsection{Consequences}

We define a path in a subspace as a 1-dimensional set of contiguous points in this subspace. In the case of a non-regular mesh, the contiguousness condition becomes a closest neighbors condition (cf. \ref{delaunay} for a definition of this condition). If a path in the constrained subspace crosses a discontinuity, there exists a subspace in which its set of points is not a path anymore, since the points before and after the discontinuity belong to different valleys and are no more contiguous. A description of the evolution of a system along a path that crosses a discontinuity misses the description of the passage between these valleys. This missing passage may correspond to a high-energy saddle point.

One can generalize this statement to the case of $N$-dimensional SPES: any description of the evolution of a system on a discontinuous SPES is an incomplete description, since when crossing a discontinuity, the real system does not ``teleport'' itself from one valley to the other.

An other consequence is that the energy barrier between valleys A and B is necessarily wrongly evaluated, since there can be a huge energy barrier or a low-energy path between both valleys. The barrier height that can be seen on the panel (f) of Fig.~\ref{fig2} may be totally different from the real barrier, and there is no way to estimate this difference by only considering the evolution of the total binding energy like on Fig.~\ref{fig1} or on the panel (f) of Fig.~\ref{fig2}. In other words, the visible saddle point $a$ or $b$ on the panel (f) is a {\it fake} saddle point. If the evolution of the energy on the panel (f) were to be monotonic, a saddle point would not even be visible, and point $a$ or $b$ would correspond to a {\it missing} saddle point.

\section{Finding the discontinuities}

In this section we propose a method to estimate if a given SPES is continuous or not, and to localize and identify the discontinuities.

\subsection{Definitions}

What is usually called a self-consistent potential energy surface (SPES) is a set of $N$ self-consistent solutions $\{|\psi_i\rangle\}_{i=1,N}$ under the action of $N_c$ constraint operators $\{\hat{Q}_j\}_{j=1,N_c}$ with constraint values $q_{i,j}$.
This set can be regularly or randomly distributed on the constrained subspace.
We suppose that a boundary of this SPES is given, separating the deformation subspace in an interior $I$ containing the SPES and an exterior. For example, a SPES with constraint values on $\hat{Q}_{20}$ regularly spaced from 100 b to 200 b with a step $q_{20}^{s}=10$ b has an interior defined as $I=[100,200]$.

Let us introduce the idealized self-consistent potential energy surface (ISPES): for a given SPES, the unique corresponding ISPES is the infinite set of self-consistent solutions $\{|\psi'_x\rangle\}$ under the action of the same $N_c$ constraint operators $\{\hat{Q}_j\}_{j=1,N_c}$ with constraint values $q_{x,j}$ taking all possible values in the same interior. The $x$ quantity is a $N_c$-dimensional vector of coordinates. The ISPES corresponding to the SPES in the previous example can be obtained by taking the limit $q_{20}^{s}\rightarrow 0$:

\begin{eqnarray}
\{|\psi'_x\rangle\}_{x\in I}&=&\lim_{q_{20}^{s}\rightarrow 0} \left(\{|\psi_i\rangle\}\right)\label{plop1}.
\end{eqnarray}

We define the continuity of an ISPES: an ISPES is continuous if the mean values of every one-body multipole operator applied to the system are continuous with respect to each constraint coordinate:

\begin{eqnarray}
&&\forall \hat{Q}_{\lambda\mu}, \forall x\in I,\forall j \in\{1,N_c\},\nonumber\\
&&\lim_{d \rightarrow 0} \langle \psi'(q_{x,j}+d)|\hat{Q}_{\lambda\mu}|\psi'(q_{x,j}+d)\rangle\nonumber\\
&&=\langle \psi'(q_{x,j})|\hat{Q}_{\lambda\mu}|\psi'(q_{x,j})\rangle\label{plop2}
\end{eqnarray}
For the ISPES of the previous example, the continuity condition is
\begin{eqnarray}
&&\forall \hat{Q}_{\lambda\mu}, \forall x\in I,\nonumber\\
&&\lim_{d \rightarrow 0} \langle \psi'(q_{20}(x)+d)|\hat{Q}_{\lambda\mu}|\psi'(q_{20}(x)+d)\rangle\nonumber\\
&&=\langle \psi'(q_{20}(x))|\hat{Q}_{\lambda\mu}|\psi'(q_{20}(x))\rangle\label{plop3}
\end{eqnarray}

We then define the continuity of a SPES: a SPES is continuous if its corresponding ISPES is continuous.

If a SPES is not continuous, one can identify a couple of neighboring states that surround a discontinuity of the mean-value of one of the possible operators on the corresponding ISPES, thus localizing this discontinuity on the SPES.

For a given discontinuity, at least one of the mean-values of the usual multipole moments must be discontinuous. We call the lowest-order discontinuous multipole moment operator the {\it signature of the discontinuity}.

\subsection{Density distance}

As a way to numerically estimate the difference between two neighboring states $|\psi\rangle$ and $|\psi'\rangle$, we define the density distance $D_{\rho\rho'}$ as
\begin{equation}
D_{\rho\rho'}\equiv\int d\vec{r}\textrm{ }|\rho(\vec{r})-\rho'(\vec{r})|
\end{equation}
with $\rho(\vec{r})$ and $\rho'(\vec{r})$ being the local spatial density of the $|\psi\rangle$ and $|\psi'\rangle$ states, respectively. This distance is dimensionless, and can be seen as the geometrical difference between two states, measured in nucleons. Any state of $A$ nucleons has a density distance of $A$ with the void state ($\int d\tau^3 \rho(\vec{r})=A$ and $\rho'(\vec{r})=0$).

\subsection{Maximum density distance}

We define the maximum density distance $D_{\textrm{max}}$ of a SPES as
\begin{eqnarray}
&&D_{\textrm{max}}\equiv \textrm{max}(D_{\rho_i\rho_j})\nonumber\\
&&\textrm{with }i\in\{1,N\},j\in \mathcal{N}_i,
\end{eqnarray}
$\mathcal{N}_i$ being the set of closest neighbors of states $|\psi\rangle_i$. In the case of a regular mesh, two points are closest neighbors if only one of their coordinates differ and if this difference is minimal. In the case of a non-regular mesh, two points are considered as closest neighbors if there exists a Delaunay cell containing them~\cite{Barber:1996}\label{delaunay}. Being given a $N_c$-dimensional set of points and an Euclidean distance function, the Delaunay triangulation is unique (if no points are co-circular). The distance function used for the Delaunay cells computation has to be carefully chosen in order to take into account variations in all $N_c$ possible directions. For example, to find the closest neighbors in the case of a non-regular 4-dimensional mesh with constraints on ($\hat{Q}_{20}$,$\hat{Q}_{30}$,$\hat{Q}_{40}$ and $\hat{Q}_{60}$), we have used the following pseudo-Euclidean distance function:
\begin{eqnarray}
d^2&\equiv& \left(\frac{q_{20}-q'_{20}}{1 \textrm{b}}\right)^2+\left(\frac{q_{30}-q'_{30}}{1 \textrm{b}^{3/2}}\right)^2\nonumber\\
 &+&\left(\frac{q_{40}-q'_{40}}{1 \textrm{b}^2}\right)^2+\left(\frac{q_{60}-q'_{60}}{1 \textrm{b}^3}\right)^2
\end{eqnarray}
with the values of $q_{20}$, $q_{30}$, $q_{40}$ and $q_{60}$ being given in b, b$^{3/2}$, b$^2$ and b$^3$, respectively. A 2-dimensional example of a Delaunay triangulation using the usual Euclidean distance is given on Fig.~\ref{figdelaunay}.
\begin{figure}[t]
 \begin{center}
  \includegraphics[width=7cm]{figure3.eps}
 \end{center}
 \caption{\label{figdelaunay} Example of a Delaunay triangulation in a 2-dimensional space, using the usual Euclidean distance. Points $E$, $F$ and $L$ belong to the same Delaunay cell. In a $N$-dimensional space, a set of $(N+1)$ points is a Delaunay cell if its circumscribed circle/sphere/\ldots contains no other point. The circumscribed circle of the Delaunay cell $EFL$ is shown in red.}
\end{figure}

An example of a discontinuity for the symmetric fission of $^{256}$Fm is shown on Fig.~\ref{fig3}. The evolution of the HFB energy is rather smooth (top panel of Fig.\ref{fig3}). There is no way to suspect a possible discontinuity just by considering the HFB energy curve. However, a spike of the $D_{\rho\rho'}$ value (bottom panel of Fig.~\ref{fig3}) is observed for $q_{20}=260$ b and $q_{20}=261$ b, consequence of a discontinuity. We can see that the mean value of the hexadecapole operator $\hat{Q}_{40}$ changes at this discontinuity (middle panel of Fig.~\ref{fig3}), so we can conclude that this discontinuity has $\hat{Q}_{40}$ for signature. Since the evolution of the energy is monotone, no saddle point is visible, but there has to be a saddle point between both valleys in the $\hat{Q}_{40}$ subspace. This is an example of a {\it missing} saddle point. The local densities before and after the discontinuity are shown on Fig.~\ref{fig4}. The main visible change in the local densities seems to be a stronger neck between the nascent fission fragments.
\begin{figure}[t]
 \begin{center}
  \includegraphics[width=9cm]{figure4.eps}
 \end{center}
 \caption{\label{fig3} Example of a discontinuity for the symmetric scission of $^{256}$Fm. The upper panel shows the total binding energy, the center panel shows the mean value of the hexadecapole operator $\hat{Q}_{40}$, and the lower panel shows the maximum total density distance between each point and its closest neighbors. The cylindrical symmetry is conserved.}
\end{figure}
\begin{figure}[t]
 \begin{center}
  \includegraphics[width=9cm]{figure5a.eps}
  \includegraphics[width=9cm]{figure5b.eps}
 \end{center}
 \caption{\label{fig4}Local densities for $^{256}$Fm in the ($z$, $r$) plane before (top) and after (bottom) the discontinuity shown on Fig.~\ref{fig3}. The isolines are separated by 0.02~fm$^{-3}$.}
\end{figure}

An other example of a discontinuity is shown on Fig.~\ref{fig5}, for the symmetric first barrier of ${}^{240}$Pu.
We would like to stress that this axial $1$-dimensional SPES is shown as an example only, and must not be considered as a prediction of the real first barrier of $^{240}$Pu.
The influence of the triaxial degree of freedom is neglected here, and it has been shown that this degree of freedom significantly lowers the first barrier of several actinides \cite{Abusara:2010,Girod:1983,Pashkevich:1969,Moller:1970,Rutz:1995,Egido:2000,Bonneau:2004,Lu:2011}.
The situation is very similar to the one on Fig.~\ref{fig3}: the evolution of the HFB energy is smooth (top panel of Fig.~\ref{fig5}), but a spike of the $D_{\rho\rho'}$ value (bottom panel of Fig.~\ref{fig5}) indicates that there is a discontinuity between $q_{20}=48$ b and $q_{20}=50$ b.
The signature of this discontinuity is the hexadecapole operator $\hat{Q}_{40}$, since its mean value is discontinuous as shown in the middle panel of Fig.~\ref{fig5}.
The local densities corresponding to the states before and after the discontinuity are shown on Fig.~\ref{fig6}.
The visible difference between these states seems to be the quantity of matter around the $z=0$ plane.
Since the discontinuity occurs at a saddle point, this point is what we have called a {\it fake} saddle point in the Introduction.

Since the step for the $q_{20}$ constraint value is smaller for the SPES of Fig.~\ref{fig3} compared to the SPES of Fig.~\ref{fig5}, it is important to remark that the average value of $D_{\rho\rho'}$ for continuous points is smaller (around 0.3 for $^{256}$Fm versus 1.0 for $^{240}$Pu).
To check the continuity of a SPES, the used steps must be small enough to be able to see spikes corresponding to discontinuities.

\begin{figure}[t]
 \begin{center}
  \includegraphics[width=9cm]{figure6.eps}
 \end{center}
 \caption{\label{fig5} Example of a discontinuity for the symmetric first barrier of $^{240}$Pu. The upper panel shows the total binding energy, the center panel shows the mean value of the hexadecapole operator $\hat{Q}_{40}$, and the lower panel shows the maximum total density distance between each point and its closest neighbors. The cylindrical symmetry is conserved.}
\end{figure}
\begin{figure}[t]
 \begin{center}
  \includegraphics[width=9cm]{figure7a.eps}
  \includegraphics[width=9cm]{figure7b.eps}
 \end{center}
 \caption{\label{fig6}Local densities for $^{240}$Pu in the ($z$, $r$) plane before (top) and after (bottom) the discontinuity shown on Fig.~\ref{fig5}. The isolines are separated by 0.02~fm$^{-3}$.}
\end{figure}

\section{Obtaining a continuous SPES}

As far as we know, all of the discontinuities we have had to deal with led at least to a density distance of several nucleons. It is then safe to consider that a SPES is continuous if we have $D_{\textrm{max}}<2$. In order to reach this level, all of the discontinuities have to be ``cleaned''. There are two ways to do so.

\subsection{Additional dimension}

The first way to remove a discontinuity is simply to globally extend the mesh with the signature of this discontinuity. For example, if a given one-dimensional SPES along $\hat{Q}_{20}$ has a discontinuity with $\hat{Q}_{40}$ for signature, the two-dimensional SPES in the ($\hat{Q}_{20}$, $\hat{Q}_{40}$) subspace will not have any discontinuity with $\hat{Q}_{40}$ for signature. This method is brute-force, and can lead to a high number of points.
As an example, we have extended the SPES shown on Fig.~\ref{fig5} by taking the signature of the discontinuity $\hat{Q}_{40}$ as a constraint. The resulting SPES is shown on Fig.~\ref{fig7}.
On this figure, the topology of the two valleys (corresponding to the first and second well) is very similar to the one of the schematic valleys shown on Fig.~\ref{fig2}.
One can see that the automatic basis optimization can be a source of rather small local maxima of $D_{\rho\rho'}$ values, since these maxima values are lowered when disabling the basis optimization. Anyway, these local maxima are small enough to let us consider that the SPES shown on Fig.~\ref{fig7} are continuous.

\begin{figure}[t]
 \begin{center}
  \includegraphics[width=11.2cm]{figure8.eps}
 \end{center}
 \caption{\label{fig7} (Color online) Self-consistent potential energy surfaces for the symmetric first barrier of $^{240}$Pu in the ($q_{20}$, $q_{40}$) subspace. The top figures show the evolution of the total binding energy, the lower figures show the corresponding maximum total density distance between each point and its closest neighbors. For the left figures, an automatic optimization of the basis parameters has been used for each point, while the basis parameters have been kept constant for the right figures.}
\end{figure}

\subsection{Connecting points}

The second way to remove a discontinuity is to locally extend the mesh with some missing points at a discontinuity. For the schematic discontinuity of Fig.~\ref{fig2}, one could simply add to the SPES the connecting points between valleys A and B of the panel (f). To find these connecting points, one calculates a small two-dimensional SPES large enough to include points $a$ and $b$ like the one plotted in panel (c), one finds the saddle point between the valleys by calculating the discrete local derivatives for each point, or by a more subtle technique if needed, like for example the {\it immersion technique}~\cite{Moller:2009,Luc:1991}, and one calculates the least-energy paths from this saddle point to the bottom of each of the valleys. The {\it immersion technique} is a fast and reliable method to find the saddle point between two given points on a regular mesh. Its principle is to mimic the way a liquid would fill a valley before flowing into an other one through a saddle point.

\begin{figure}[t]
 \begin{center}
  \includegraphics[angle=-90,width=6cm]{figure9.eps}
 \end{center}
 \caption{\label{fig8} Illustration of the ``connecting points'' method. The saddle point $a$ is found, then points $a$ to $b$ and $a$ to $c$ are found as least action paths. Original points before $b$, original points after $c$, and both least action paths ($a$ to $b$ and $a$ to $c$) are then joined to form a continuous path. The constraint value $q_{20}$ from Fig.~\ref{fig2} has been replaced with a curvilinear abscissa $s$ in lower panel.}
\end{figure}

The schematic result is shown on Fig.~\ref{fig8}. It is worth mentioning that $\langle \hat{Q}_{20}\rangle$ is no longer a good variable for a one-dimensional description, for two reasons. Firstly, it is possible that the least-energy path ``backbends'', i.e. there can be multiple points with the same $\langle \hat{Q}_{20}\rangle$ value. Secondly, the connecting points are not anymore minimizing the total binding energy for a given $\langle \hat{Q}_{20}\rangle$ value. This is why a curvilinear abscissa is used instead of $\langle \hat{Q}_{20}\rangle$, and why the original points between $b$ and $c$ are discarded and replaced with points from the least action paths. The resulting set of points can no longer be called a SPES, but can be called an effective PES.

This method is rather convenient in the case of one-dimensional SPES, but can be hard to automatize for SPES of higher dimension. The fact that the constraint variables are no longer good variables can also be problematic when trying to perform dynamical propagations on the resulting effective PES.

\subsection{Discussion}

We would like to stress that for a given system, the dimension of the smallest continuous SPES is the dimension of the smallest corresponding continuous ISPES. If the continuous description of a physical process needs $N_p$ degrees of freedom, the dimension of a corresponding continuous SPES must obey $N\ge N_p$. Trying to obtain a continuous description with an insufficient number of degrees of freedom with a change of variables like in the ``connecting points'' method presented above can be dangerous for the physical interpretation of the resulting quantities like the inertia tensor, mainly because the fact that some points do not minimize the total binding energy anymore for a given $\langle \hat{Q}_{20}\rangle$ value forbids the use of this latter as a good variable.

Using the ``additional dimension'' method presented above does not lead to such problems, but can be numerically difficult. For some physical processes, having $N=N_p$ can mean that the number of points of the continuous SPES is huge if the considered mesh is rectangular and regular. As a way to lower this number of points, one can restrict the SPES to the solutions with a low energy (non-rectangular mesh), and/or use multidimensional adaptive meshing (non-regular mesh).

\section*{Conclusion}

Discontinuities are not easily visible, and have important consequences. Any result extracted from a discontinuous SPES should be considered with extreme caution. The method to obtain $D_{\textrm{max}}$ that we have presented in this paper is simple, and allows to check with good confidence if a given SPES is continuous or not, without any additional self-consistent calculation. The value of $D_{\textrm{max}}$ should be specified for any SPES, like the conserved symmetries and the level of convergence reached.

In the near future, we plan to perform dynamical calculations on SPES for the description of the fission process and the prediction of some fission fragment properties, using a Time-Dependent Generator Coordinate Method~\cite{Goutte:2005,Dubray:2008}, and the continuity of the produced SPES will be carefully checked using the method presented here. One of the main advantages of this method is that it can be fully automatized, allowing an on-the-fly detection of discontinuities during the production of a SPES.

In this study, the definition of $D_{\rho\rho'}$ has been chosen for practical reasons (mainly its numerical simplicity), but other expressions can be used to estimate the ``distance" between two neighboring states, involving for example the non-local density
\begin{equation}
D'_{\rho\rho'}\equiv\int\int d\vec{r}\textrm{ }d\vec{r}'\textrm{ }|\rho(\vec{r},\vec{r}')-\rho'(\vec{r},\vec{r}')|
\end{equation}
or the overlap between neighboring states
\begin{equation}
D''_{\rho\rho'}\equiv\frac{1}{|\langle \psi|\psi'\rangle|}.
\end{equation}
Such expressions can reveal a discontinuity in non-local fields, but their computational cost may be important. One should also note that the proposed expression does not take into account time-odd quantities. When calculating a SPES for an odd nucleus, another expression should be used if one is looking for discontinuities in time-odd quantities.
In Ref.~\cite{Moller:2009}, the authors state that ``[\ldots] minimization does not necessarily lead to the wrong solution; in many simple problems it will give the correct saddle point. Unfortunately, one cannot know {\it a priori} whether a true or incorrect solution will be found''. Thanks to the simple method presented in this paper to calculate $D_{\textrm{max}}$, it is now easy to estimate if and where a discontinuity has occurred while calculating a SPES. One can consider that a SPES with a low enough $D_{\textrm{max}}$ value does not contain such ``incorrect solutions'' ({\it fake} or {\it missing} saddle points), and that the ``Problem SPES" from Introduction is fixed. By comparison, there is to our knowledge no numerical method to fix ``Problem NSPES" from the introduction, in other words, to estimate if partial local extrema obtained by non-self-consistent methods in a given subspace remain partial local extrema in bigger subspaces.

\section*{Acknowledgments}

We gratefully thank D.~Gogny for his useful advices concerning this study.

\bibliographystyle{elsarticle-num}


\section{Introduction}
\noindent
For many good reasons $G_2$ gluodynamics has attracted much attention
recently. For example, the $14$-dimensional exceptional group $G_2$ has a trivial centre,
in contrast to the usually studied $SU(N)$ gauge groups. Indeed
it is the smallest simple and simply connected compact Lie group with this property.
Thus $G_2$ gluodynamics is useful to better understand the relevance of the
centre symmetry for confinement \cite{Pepe:2006er}.
Actually a non-trivial centre is needed in several proposed scenarios for
confinement and hence $G_2$ gluodynamics can be used to test these proposals.
It has been convincingly demonstrated that the theory
shows a first order finite temperature transition without
order parameter from a confining to a deconfining phase
which can be explained by  centre vortices \cite{Greensite:2006sm}.
In this context confinement refers to confinement at intermediate scales, where
a Casimir scaling of string tensions has been reported \cite{Liptak:2008gx}.
But on large scales, deep in the infrared, strings break due to gluon production and the
static inter-quark potential becomes flat \cite{Greensite:2003bk}.
Recently it has been demonstrated that chiral symmetry is broken at low
temperatures and is restored at high temperatures at the thermodynamic
phase transition \cite{Danzer:2008bk}.

$G_2$ gluodynamics has an intriguing connection to $SU(3)$ gauge theory.
When one couples a scalar field in the $7$-dimensional fundamental
representation to the gauge field one can break the $G_2$ gauge symmetry to the $SU(3)$ gauge symmetry
of strong interaction. With increasing hopping parameter $\kappa$ the resulting
Yang-Mills-Higgs theory interpolates smoothly between $G_2$ gluodynamics without
centre symmetry and $SU(3)$ gluodynamics with  $\mathbb{Z}_3$ centre symmetry.
\begin{figure}
\includegraphics{ymTransition}
\caption{Expectation value of the traced Polyakov loop in the
fundamental $7$-dimensional representation in $G_2$ gluodynamics
on a $16^3\!\times\!6$ lattice as obtained via hybrid Monte-Carlo sampling. The
phase transition is located for the Wilson action at a critical
$\beta_\text{c}\approx 9.765$.}
\label{fig:ymTransition}
\end{figure}
For intermediate values of the hopping parameter the theory mimics $SU(3)$ gauge
theory with dynamical quarks and the masses of these `quarks' increase
with increasing hopping parameter. In $G_2$ gluodynamics the Polyakov loop is no longer
an order parameter in the strict sense.
Despite this fact it still serves as an approximate order
parameter separating the confined from the deconfined phase (see Fig.~\ref{fig:ymTransition})
with a rapid change at the phase transition point.

According to the conjecture by Svetitsky and Yaffe
\cite{Svetitsky:1982gs,Yaffe:1982qf} the dynamics at the finite temperature
confinement-decon\-finement transition of a $d\!+\!1$-dimensional pure gauge
theory can be described by an effective spin model in $d$ dimensions. Based on
our earlier results on finite-temperature $SU(2)$ and $SU(3)$ gluodynamics
\cite{Dittmann:2003qt,Heinzl:2005xv,Wozar:2007tz,Velytsky:2008bh,Wozar:2008nv}
there are strong indications that the effective models derived and analysed in
the present work are sufficient to accurately describe the dynamics of Polyakov
loops. The direct connection between the effective spin models and  $G_2$
gluodynamics is postponed to a forthcoming publication.

In Sec.~\ref{sect:g2} we review kinematic aspects of
$G_2$ and the main implications for $G_2$ gluodynamics. Afterwards in
Sec.~\ref{sect:strongCoupling} the strong coupling expansion for
the effective Polyakov loop action is explained and in particular the effective
theory in leading order is introduced. In Sec.~\ref{sect:fundamentalModel} we investigate
the properties of the effective model first by a classical analysis, then by a modified
mean field approximation and finally by extensive Monte-Carlo simulations.
Reducing the continuous spin degrees further to the discrete spins situated at the $3$
edges of the fundamental domain of $G_2$ we end up with a deformed Potts-type
spin model whose phase diagram is explored in Sec.~\ref{sect:pottsModel}.

\section{The group $\bm{G_2}$}
\label{sect:g2}
\noindent
$G_2$ is the smallest of the five exceptional simple Lie groups and can be
viewed as a subgroup of $SO(7)$ subject to seven independent cubic constraints for the
$7$-dimensional matrices $g$ representing $SO(7)$ \cite{Holland:2003jy}:
\begin{equation}
\label{eq:g2constraint}
T_{abc} = T_{def}\,g_{da}\,g_{eb}\,g_{fc}.
\end{equation}
Here $T$ is a total antisymmetric tensor given by
\begin{equation}
T_{127} = T_{154} = T_{163} = T_{235} = T_{264} = T_{374} = T_{576} = 1.
\end{equation}
The constraints \eqref{eq:g2constraint} for the group elements reduce the $21$ generators of $SO(7)$ to
$14$ generators of the group $G_2$ with rank $2$. Its fundamental representations are the defining
$7$-dimensional and the adjoint $14$-dimensional representation with Dynkin
labels
\begin{equation}
(7) = [1,0],\quad (14) = [0,1].
\end{equation}
$G_2$ has a trivial centre and its Weyl group is the dihedral group $D_6$ of order $12$.
Additionally $G_2$ is connected to $SU(3)$
through the embedding of $SU(3)$ as a subgroup of $G_2$ according to \cite{Macfarlane:2002hr}
\begin{equation}
G_2/SU(3) \sim SO(7)/SO(6) \sim S_6.
\end{equation}
So when the $S_6$ part of $G_2$ is frozen out\footnote{This is possible when a
fundamental Higgs field is coupled to the gauge field \cite{Pepe:2006er}.} we
end up at $SU(3)$ gauge theory.

In effective theories for the gauge invariant (traced) Pol\-ya\-kov loops
in the fundamental representations we are aiming at, only the reduced
Haar measure is needed. Based on \cite{Uhlmann:2006ze} this measure can be
given for a parametrisation of the conjugacy classes either by angular variables
or alternatively by the fundamental characters,
\begin{equation}
d\mu \propto J^2d\,\varphi_1\,d\varphi_2 = J\,d\chi_7\,d\chi_{14}.
\end{equation}
The density $J^2$ can be expressed in terms of the fundamental characters,
\begin{equation}
\begin{aligned}
J^2 &= \left(4\chi_7^3-\chi_7^2-2\chi_7-10\chi_7\chi_{14}
+7-10\chi_{14}-\chi_{14}^2\right) \\ &\quad\times
\left(7-\chi_7^2-2\chi_7+4\chi_{14}\right),
\end{aligned}
\end{equation}
where the characters are given in terms of (particularly chosen) angular variables $\varphi_{1,2}$ as
\begin{equation}
\begin{aligned}
\chi_7 &= 1+2\cos(\varphi_1)+2\cos(\varphi_2)+2\cos(\varphi_1+\varphi_2),\\
\chi_{14} &=
2\bigl(1+\cos(\varphi_1) +\cos(\varphi_1-\varphi_2)
+\cos(\varphi_2) \bigr. \\ \bigl. &\;\;+
\cos(\varphi_1+\varphi_2)+\cos(2\varphi_1+\varphi_2) +\cos(\varphi_1+2\varphi_2) \bigr).
\end{aligned}
\end{equation}
The boundary of the fundamental domain is determined by $J=0$ and thus is parametrised by
the three curves (see Fig.~\ref{fig:fundamentalDomain})
\begin{equation}
\begin{aligned}
\chi_{14} &= \frac{1}{4}(\chi_7+1)^2-2,\\
\chi_{14} &= -5(\chi_7+1)\pm 2(\chi_7+2)^{3/2}.
\end{aligned}
\end{equation}
Note that the reduced $G_2$-Haar measure is maximal not at the origin
but for $(\chi_7,\chi_{14})=(-1/5,-2/5)$. The fundamental domain has no
symmetries at all and this expresses the fact that the centre of
$G_2$ is  trivial.

\begin{figure}[h]
\includegraphics{fundamentalDomain}
\caption{Fundamental domain of $G_2$. Darker regions indicate a bigger Haar
measure.}
\label{fig:fundamentalDomain}
\end{figure}

\subsection*{Representation theory and implications for confinement}
\noindent
In the pioneering work \cite{Holland:2003jy} the confining properties  of $G_2$
have been discussed and compared to those of $SU(3)$. Quarks and anti-quarks
in $SU(3)$ transform under the fundamental  representations $3$ and $\bar 3$
such that their charges can only be screened by particles with non-vanishing $3$-ality,
especially \emph{not by gluons}. This explains why in the confining phase of
$SU(3)$ gluodynamics the static inter-quark potential is linearly rising up to arbitrary
long distances. As a consequence the free energy of a single quark gets infinite and
the Polyakov loop expectation value vanishes. Hence the Polyakov loop discriminates
the confining from the deconfining phase and at the same time serves as order
parameter for the $\mathbb{Z}_3$ centre symmetry.

To better understand $G_2$ gluodynamics we recall the decomposition of
tensor products into irreducible representations,
\begin{equation}
\label{eq:representationsG2}
\begin{aligned}
(7) \otimes (7)&=(1) \oplus (7) \oplus (14) \oplus (27)\\
(7) \otimes (7) \otimes (7)&=(1) \oplus 4 \cdot (7) \oplus 2 \cdot (14) \oplus 3 \cdot (27) \\
&\quad \oplus 2 \cdot (64) \oplus (77)\\
(14) \otimes (14)&=(1) \oplus (14) \oplus (27) \oplus (77) \oplus (77') \\
(14) \otimes (14)\otimes (14)&=(1)\oplus (7) \oplus 5\cdot (14)\oplus
3\cdot(27)\oplus\dotsb
\end{aligned}
\end{equation}
with Dynkin labels
\begin{equation}
\begin{aligned}
(1) &= [0,0],& (27) &= [2,0],& (64) &= [1,1],\\ (77)&=
[3,0],& (77') &= [0,2].
\end{aligned}
\end{equation}
The quarks in $G_2$ transform under the $7$-dimensional fundamental representation, gluons
under the $14$-dimensional fundamental (and at the same time adjoint) representation. From
\eqref{eq:representationsG2} we see that similarly as in $SU(3)$ two or three quarks
can build a colour singlet (meson or baryon, respectively). In $G_2$ gluodynamics three
centre-blind dynamical gluons can screen the colour charge of a single quark,
\begin{equation}
(7) \otimes (14) \otimes (14) \otimes (14)=(1) \oplus \dotsb.
\end{equation}
Thus the flux tube between two static quarks can break due to gluon production
and the Polyakov loop does not vanish even in the
confining phase \cite{Pepe:2006er}. This shows that the Polyakov loop can at
best be an approximate order parameter (see Fig.~\ref{fig:ymTransition}) which
changes rapidly at the phase transition and is small (but non-zero) in the confining phase.
To characterise confinement we can no longer refer to a non-vanishing
asymptotic string tension and vanishing Polyakov loop. Instead we define
confinement as the absence of free colour charges in the physical spectrum.
In the confining phase the inter-quark potential rises linearly
at intermediate scales \cite{Greensite:2006sm,Liptak:2008gx}.

\section{Effective theories and the strong coupling expansion}
\label{sect:strongCoupling}
\noindent
Based on a conjecture  relating
finite temperature $SU(N)$ gluodynamics in $d\!+\!1$ dimensions at the critical
point with a $\mathbb{Z}_N$ spin model in $d$ dimensions \cite{Svetitsky:1982gs,Yaffe:1982qf},
there have been  extended studies to compare correlation functions of both systems for $SU(2)$
\cite{Dittmann:2003qt,Heinzl:2005xv,Velytsky:2008bh} and $SU(3)$ gluodynamics
\cite{Wozar:2007tz,Wozar:2008nv}, either by using Schwinger-Dyson equations
or demon methods \cite{Creutz:1983ra,Hasenbusch:1994ne}.
The strong coupling expansion for the distribution of the
inhomogeneous Polyakov loops was taken as ansatz for the (exponentiated)
effective Polyakov-loop action. This way effective models for
$SU(3)$ gluodynamics have been derived in \cite{Wozar:2007tz}. Here
we sketch how one arrives at the analogous results for $G_2$ and obtain
the effective continuous spin model in leading order.

Starting with the lattice Wilson action
\begin{equation}
 S_{\text{W}} = \beta \sum_{\square} \left( 1-  \frac{1}{N_\text{C}} \Re \tr
 U_{\square} \right),\quad \beta = \frac{6}{a^4g^2}
\end{equation}
a \emph{strong coupling expansion} (for small
$\beta$) is performed to arrive at an effective theory for the local
Polyakov loops. To do that one inserts a group valued delta
function into the path integral,
\begin{equation}
\begin{aligned}
Z &= \int \mathcal{D} U \exp(-S_{\text{W}}[U])\\
&= \int \mathcal{D} P \int \mathcal{D} U\, \delta\left(P_\vc{x},\prod_{\tau=1}^{N_\text{t}}
U_{\tau,\vc{x}}\right) \exp(-S_{\text{W}}[U])
\\ &\equiv \int \mathcal{D} \mathcal{P}\, \exp(-S_{\text{eff}}[\mathcal{P}]).
\end{aligned}
\end{equation}
Here $\mathcal{D} P$ denotes the product of reduced Haar measures on the sites
of the \emph{spatial lattice}. We do not need the full Haar measure of $G_2$
since the effective action $S_{\text{eff}}$ only depends
on the gauge invariant content of the local Polyakov loop.

In compact form the strong coupling
expansion is then given by
\begin{equation}
\begin{aligned}
  S_{\text{eff}}[\mathcal{P}] &= \sum_r \sum_{\mathcal{R}_1 \dotsc  \mathcal{R}_r} \sum_{\ell_1 \dotsc \ell_r}
  c_{\mathcal{R}_1 \dotsc \mathcal{R}_r}^{\ell_1 \dotsc  \ell_r}(\beta)  \prod_{i=1}^r
  S_{\mathcal{R}_i, \ell_i} \\&= \sum_i \lambda_i S_i
\end{aligned}
\end{equation}
with the basic building blocks
\begin{equation}
  S_{\mathcal{R}, \ell} \equiv \sum \limits_{\vc{x},\vc{y}}\chi_\mathcal{R} (\mathcal{P}_\vc{x}) \chi_\mathcal{R}^* (\mathcal{P}_{\vc{y}}) + \mathrm{c.c.}
  ,  \quad \abs{\vc{x}-\vc{y}}=\ell.
\end{equation}
Here $r$ counts the number of link operators contributing at
each order. The coefficients $c_{\mathcal{R}_1 \dotsc \mathcal{R}_r}^{\ell_1
\dotsc \ell_r}$ couple the operators
$S_{\mathcal{R}_i,\ell_i}$ sitting at sites $\vc{x}_i$ and $\vc{y}_i$ separated
by distance $\ell_i$
in representation $\mathcal{R}_i$. The effective action hence describes a \emph{network of link operators} that are collected into (possibly disconnected)
`polymers' contributing with `weight' $c_{\mathcal{R}_1 \dotsc
\mathcal{R}_r}^{\ell_1 \dotsc \ell_r}$. The resulting `operators'
(Polyakov loop monomials) are \emph{dimensionless} and there is
\emph{no natural} ordering scheme at hand. Our chosen truncation is based
on the strong coupling expansion in powers of $\beta$ which is
closely related to the dimension of the corresponding group
representations and the distance across which the Polyakov loops
are coupled. In the strong coupling expansion truncated
at $\mathcal{O}(\beta^{kN_\text{t}})$ one has $r \le k$ and the additional
restriction $|\mathcal{R}_1| + \dotsb + |\mathcal{R}_r| < k$ with $|\mathcal{R}| \equiv p_1+p_2 $
for a given representation $\mathcal{R}$ of $G_2$ with Dynkin labels
$[p_1,p_2]$. The leading order terms only contain
interactions between nearest neighbours $\nN{\vc{x}\vc{y}}$ and
the two fundamental representations.

For $SU(3)$ the characters of the two fundamental representations are
complex conjugate of each other such that the effective
Polyakov loop action contains just one term in leading order.
In $G_2$ this situation  changes and we find two
independent contributions in leading order. We refer to the
corresponding model containing the two fundamental representations
as \emph{fundamental model}. Its action is explicitly given by
\begin{equation}
\label{eq:fundamentalEffectiveAction}
S_\text{{eff}}=\lambda_7 \underbrace{\sum \limits_{\nN{\vc{x}\vc{y}}}
\chi_7(\mathcal{P}_\vc{x}) \chi_7(\mathcal{P}_\vc{y})}_{S_{7}} +
\lambda_{14} \underbrace{\sum \limits_{\nN{\vc{x}\vc{y}}} \chi_{14}(\mathcal{P}_\vc{x})
\chi_{14}(\mathcal{P}_\vc{y})}_{S_{14}},
\end{equation}
where the couplings $\lambda_7$ and $\lambda_{14}$ are indexed by the dimension
of the involved representation. In next-to leading order
there exist six additional terms with nearest neighbour interactions.
Their explicit forms are dictated by representation theory \eqref{eq:representationsG2}
\begin{equation}
\begin{aligned}
S_{27}&=\sum \limits_{\nN{\vc{x}\vc{y}}} \chi_{27}(\mathcal{P}_\vc{x})
\chi_{27}(\mathcal{P}_\vc{y}), \\
S_{77'}&=\sum \limits_{\nN{\vc{x}\vc{y}}}
\chi_{77'}(\mathcal{P}_\vc{x}) \chi_{77'}(\mathcal{P}_\vc{y}), \\
S_{64}&=\sum
\limits_{\nN{\vc{x}\vc{y}}} \chi_{64}(\mathcal{P}_\vc{x}) \chi_{64}(\mathcal{P}_\vc{y}), \\
S_{7,7}&=\sum \limits_{\nN{\vc{x}\vc{y}}} \left(\chi_7(\mathcal{P}_\vc{x})
\chi_7(\mathcal{P}_\vc{y}) \right)^2, \\
S_{14,14}&=\sum \limits_{\nN{\vc{x}\vc{y}}}
\left(\chi_{14}(\mathcal{P}_\vc{x}) \chi_{14}(\mathcal{P}_\vc{y}) \right)^2, \\
S_{7,14}&=\sum \limits_{\nN{\vc{x}\vc{y}}} \chi_7(\mathcal{P}_\vc{x})\chi_7(\mathcal{P}_\vc{y})
\chi_{14}(\mathcal{P}_\vc{x})  \chi_{14}(\mathcal{P}_\vc{y}).
\end{aligned}
\end{equation}
It the remainder of this work we shall neglect the next-lo leading order
terms and concentrate on the fundamental model
\eqref{eq:fundamentalEffectiveAction}.

\section{The fundamental model}
\label{sect:fundamentalModel}
\noindent
For the fundamental effective model \eqref{eq:fundamentalEffectiveAction} we
shall localise the symmetric, ferromagnetic and anti-ferromagnetic phases
with coexistence lines in order to find
the region in the space of couplings $\lambda_7,\lambda_{14}$
where a connection to $G_2$ gluodynamics can be established.

\subsection{Classical analysis}
\noindent
For strong couplings the fluctuations of the Polyakov loops are suppressed
and the spin system behaves almost classically. Thus for large  $\vert\lambda_7\vert$ and
$\vert\lambda_{14}\vert$ we may compute the phase diagram by minimising the
classical action. Anticipating that there are anti-ferromagnetic phases we
introduce the odd and even sublattices
\begin{equation}
\begin{aligned}
\Lambda_\text{o}&=\left \lbrace \vc{x} \,\vert\,x_1+x_2+x_3\text{ odd}
\right \rbrace\quad\text{and}\\
\Lambda_\text{e}&=\left \lbrace \vc{x}\,\vert\,
x_1+x_2+x_3\text{ even}\right \rbrace.
\end{aligned}
\end{equation}
On each sublattice the Polyakov loop is assumed to
have a \emph{constant} value and the two values are denoted by
$\mathcal{P}_\text{o}$ and $\mathcal{P}_\text{e}$, respectively. We denote the corresponding characters
in the fundamental domain of $G_2$ by
\begin{equation}
\begin{aligned}
\vc{\chi}_\text{e}&=\begin{pmatrix}\chi_{7,\text{e}}\\ \chi_{14,\text{e}}\end{pmatrix}=
\begin{pmatrix}\chi_{7}\\ \chi_{14}\end{pmatrix}(\mathcal{P}_\text{e}) \quad\text{and} \\
\vc{\chi}_\text{o}&=\begin{pmatrix}\chi_{7,\text{o}}\\ \chi_{14,\text{o}}\end{pmatrix}
=\begin{pmatrix}\chi_{7}\\ \chi_{14}\end{pmatrix}(\mathcal{P}_\text{o}).
\end{aligned}
\end{equation}
With this assumption and notation the action of the fundamental model
\eqref{eq:fundamentalEffectiveAction}
reads
\begin{equation}
S_{\text{eff}}=V
\vc{\chi}^\mathsf{T}_\text{e}K\vc{\chi}_\text{o},\quad\text{with}\quad
K=3\begin{pmatrix}\lambda_7&0\\0&\lambda_{14}\end{pmatrix}.\label{actioncl}
\end{equation}
To localise the different phases we may assume that the Polyakov loop on
one sublattice, say $\Lambda_\text{o}$, is equal to the group-identity with maximal
characters, $\chi_{7,\text{o}}=7$ and $\chi_{14,\text{o}}=14$.
For given couplings $\lambda_7$ and $\lambda_{14}$ the corresponding
thermodynamic phase is then determined by that Polyakov loop on $\Lambda_\text{e}$
for which the linear function $7\lambda_7\chi_{7,\text{e}}+14\lambda_{14}\chi_{14,\text{e}}$
is minimal. If the minimising characters are the same on
both sublattices then the phase is ferromagnetic,
else it is anti-ferromagnetic. The minimum of the linear function is
attained for $\vc{\chi}_\text{e}$ on one of the three corners of the fundamental
domain in Fig.~\ref{fig:fundamentalDomain} or on the curve connecting the
corners $(-1,-2)^\mathsf{T}$  and $(7,14)^\mathsf{T}$. Depending on the sign of
$\lambda_7$ and the slope $\xi\equiv\lambda_{14}/\lambda_7$ one finds the following phases:
\begin{itemize}
\item  For $\lambda_7>0$ and $\xi<-1/2$ or for $\lambda_{7}<0$ and
$\xi>-1/8$ we find the \emph{ferromagnetic phase} F with
$\vc{\chi}^\mathsf{T}_\text{e}=(7,14)$.
\item For $\lambda_{7}>0$ and $\xi>1/14$
we find a \emph{anti-ferromagnetic phase} AF1 with
$\vc{\chi}^\mathsf{T}_\text{e}=(-1,-2)$.
\item For $\lambda_7>0$ and $-1/2<\xi<1/14$ we find a second
\emph{anti-ferromagnetic phase} AF2 with $\vc{\chi}^\mathsf{T}_\text{e}=(-2,5)$.
\item For $\lambda_{7}<0$ and $\xi<-1/8$ the characters
$\vc{\chi}^\mathsf{T}_\text{e}=(-1-1/\xi, 1/(2\xi)^2-2)$ change continuously from $(7,14)$ to $(-1,-2)$
along the connecting boundary curve of the fundamental domain. This \emph{transition
phase} is denoted by F~$\to$~AF1.
\end{itemize}
The phase portrait is depicted in Fig.~\ref{fig:effectiveClassical}
where we also included the expected symmetric phase for weak
couplings. Since in a symmetric phase entropy wins over energy
it cannot be seen in any classical analysis.

\begin{figure}[h]
\includegraphics{effectiveClassical}
\caption{The classical phase diagram of the fundamental effective
$G_2$ model. In addition to the calculated ferromagnetic
and anti-ferromagnetic phases we expect a symmetric phase
for weak couplings.}
\label{fig:effectiveClassical}
\end{figure}

\subsection{Mean field analysis}
\noindent
The classical analysis is refined by a modified mean field approximation
in which one allows for inhomogeneous mean fields.  First we
recall the main aspects of the method. Here we are interested
in expectation values of the form
\begin{equation}
\begin{aligned}
\vev{A} &= \frac{1}{Z}\int \mathcal{D} \mathcal{P} \exp(-S_{\text{eff}}[\mathcal{P}]) A[\mathcal{P}],\\
\mathcal{D} \mathcal{P} &\equiv \prod_\vc{x} d\mu(\mathcal{P}_\vc{x}),
\end{aligned}
\end{equation}
where the gauge invariant function $A(\mathcal{P})$ depends on the Polyakov
loop via the the fundamental characters $\chi_7(\mathcal{P})$
and $\chi_{14}(\mathcal{P})$ and $d\mu$
is the reduced Haar measure of $G_2$. The equilibrium probability measure $\mathcal{D} \mathcal{P}
\exp(-S_{\text{eff}})/Z$ is the unique minimum to the variational problem
\begin{equation}
\label{eq:variationalProblem}
\inf_p \vev{S_{\text{eff}}+\ln p}_p,
\end{equation}
where the $p$-indexed expectation value is calculated with the
integration measure $p[\mathcal{P}]\,\mathcal{D}\mathcal{P}$, whose probability
density $p$ is to be varied. Expectation values of observables can
then be computed as
\begin{equation}
\vev{A}_p =\int \mathcal{D} \mathcal{P} \,p[\mathcal{P}] A[\mathcal{P}].
\end{equation}
In this scope a Monte-Carlo simulation is just the approximation of
the probability density $p[\mathcal{P}]\propto\exp(-S_{\text{eff}}[\mathcal{P}])$ with a finite
set of configurations which give
$p[\mathcal{P}]\approx N_\text{MC}^{-1}\times\sum_{t=1}^{N_\text{MC}}
\delta(\mathcal{P}-\mathcal{P}_t)$, where $\mathcal{P}_t$ is the configuration in the $t$'th MC step and $N_\text{MC}$
is the number of MC steps.

In a variational approach the mean field approximation amounts to the restriction
of the admissible densities $p$ to product form
\begin{equation}
p[\mathcal{P}]\rightarrow \prod_\vc{x} p_\vc{x}(\mathcal{P}_\vc{x}).
\end{equation}
Then expectation values factorise and the computation can be done site by site.
Due to the translational invariance of the action one may believe, that the
minimising density is translational invariant, $p_\vc{x}(\mathcal{P})=p(\mathcal{P})$. However,
this assumption is only justified for the symmetric and ferromagnetic phases with
constant mean fields.

Anticipating the existence of additional anti-ferromagnetic phases we
partition the lattice into its even and odd sub-lattices, as we did in the classical
analysis, and allow for different densities on the sublattices,
\begin{equation}
p_\vc{x}(\mathcal{P}_\vc{x}) = \begin{cases}
           p_\text{e}(\mathcal{P}_\vc{x})&: \vc{x}\in\Lambda_\text{e}\\
           p_\text{o}(\mathcal{P}_\vc{x})&: \vc{x}\in\Lambda_\text{o}.
           \end{cases}
\end{equation}
The classical analysis is then recovered by allowing only $\delta$-type
point-measures for $p_{\text{e},\text{o}}$. In the modified mean-field analysis
we allow for all $p_{\text{e},\text{o}}$ in the variational principle with prescribed
mean fields $\bar{\vc\chi}_\text{e}$ and $\bar{\vc\chi}_\text{o}$ on the even and odd sublattices. The effective potential $u$ is then obtained by computing
\begin{equation}
u(\bar{\vc{\chi}}_\text{e},\bar{\vc{\chi}}_\text{o}) =\frac{1}{V}\inf_p \vev{S_{\text{eff}}+\ln p}_p,
\end{equation}
subject to the following four constraints for the admitted densities $p_\text{e}$ and $p_\text{o}$:
\begin{equation}
\vev{\vc\chi}_{\text{e},\text{o}}=\bar{\vc\chi}_{\text{e},\text{o}}.
\label{constraints}
\end{equation}
The one-site expectation values $\vev{\dots}_{\text{e},\text{o}}$ are calculated with
$p_{\text{e},\text{o}}\,d\mu$. To actually compute the minimising densities one needs the
expectation value of the action and  entropy, given by
\begin{equation}
\begin{aligned}
\vev{S_{\text{eff}}}_p &=
V\bar{\vc\chi}^\mathsf{T}_\text{e} K\bar{\vc\chi}_\text{o} \\
\vev{\ln p}_p &= \frac{V}{2} \vev{\ln p_\text{e}}_\text{e}+\frac{V}{2}\vev{\ln p_\text{o}}_\text{o},
\end{aligned}
\end{equation}
where $K$ is the matrix given in \eqref{actioncl}.
On each sublattice the variational problem is solved by a
density $p\propto \exp\bigl(\vc{j}\cdot\vc{\chi}(\mathcal{P})\bigr)$
with two Lagrangian multipliers $\vc{j} =(j_7,j_{14})$.
The four multipliers are determined by the four constraints in
\eqref{constraints}.
Using this solution for the densities in the variational principle determines the
effective potential as function of the prescribed mean fields. The expectation
values of the characters on the two sublattices minimise the effective
potential. These minimas solve the following system of coupled gap equations
\begin{equation}
\begin{aligned}
K\bar{\vc\chi}_\text{e}&=-\frac{\partial w(\bar{\vc \chi}_\text{o})}{\partial \bar{\vc\chi}_\text{o}},\quad
K\bar{\vc\chi}_\text{o}=-\frac{\partial w(\bar{\vc \chi}_\text{e})}{\partial \bar{\vc\chi}_\text{e}},\\
w(\bar{\vc\chi})&=\ln\int d\mu(\mathcal{P})\, e^{-\bar{\vc\chi}^\mathsf{T}
K\vc\chi(\mathcal{P})}.
\end{aligned}
\end{equation}
We have calculated the expectation values of $\chi_7$ and $\chi_{14}$
on both sublattices as functions of
the couplings on a $120\!\times\!100$ grid in the rectangle
\begin{equation}
-0.3\leq \lambda_7\leq 0.3\quad\text{and}\quad
-0.25\leq\lambda_{14}\leq 0.25.
\label{grid}
\end{equation}
The contour plot of the expectation value
\begin{equation}
\langle \chi_7\rangle=\frac{1}{2}\left(\langle\chi_{7,\text{e}}\rangle
+\langle\chi_{7,\text{o}}\rangle\right),\label{magnetization}
\end{equation}
called \emph{magnetisation},
is depicted in Fig.~\ref{fig:effectiveMeanfield}. As expected, for weak couplings we find
a symmetric phase with vanishing magnetisation in the centre of the phase diagram.
\begin{figure}[t]
\includegraphics{effectiveMeanfield}
\caption{Polyakov loop $\langle\chi_7\rangle$ of the
\emph{fundamental} $G_2$ spin model in
mean field approximation.}
\label{fig:effectiveMeanfield}
\end{figure}
On the lower left, for negative couplings, we find the ferromagnetic phase
with $\langle\chi_{7,\text{e}}\rangle=\langle\chi_{7,\text{o}}\rangle\approx 7$
or equivalently with a typical $\mathcal{P}_\vc{x}$ near the identity. For an unambiguous
identification of the phases one needs the expectation values
of both $\chi_7$ and $\chi_{14}$ on both sublattices. We have calculated
these four expectation values for the fundamental model on
a grid in the space of coupling constants with extensive Monte Carlo simulations.
Since the numerical simulations and mean field approximation yield almost
identical results we defer the detailed discussion of
phase portrait, and in particular the localisation of the various
anti-ferromagnetic phases for positive couplings,  to the following section.

\subsection{Monte-Carlo results}
\noindent
We performed our Monte-Carlo simulations with about $10\,000$
samples for every point on a $60\!\times\!50$ grid inside
the rectangle \eqref{grid} in the space of coupling constants.
Two neighbouring points on this grid are separated by $0.01$.
First we calculated the magnetisation $\langle\chi_7\rangle$ and
the resulting phase portrait is depicted in Fig.~\ref{fig:effectivePolyakov}.
It looks very similar to the portrait calculated in the mean field approximation,
see Fig.~\ref{fig:effectiveMeanfield}.
\begin{figure}[h]
\includegraphics{effectivePolyakov}
\caption{Polyakov loop $\langle\chi_7\rangle$ of the
fundamental model obtained via MC simulation on an $8^3$ lattice.}
\label{fig:effectivePolyakov}
\end{figure}
For weak couplings entropy wins over energy and the product of
Haar measures of the Polyakov loops become relevant.
In order to unambiguously identify the anti-ferromagnetic
phases for positive couplings we again subdivided the lattice in
the even and odd sublattice, $\Lambda=\Lambda_\text{e}\cup\Lambda_\text{o}$,
and measured the \emph{staggered magnetisation}
\begin{equation}
S=\frac{1}{2}\langle\abs{\chi_{7,\text{e}}-\chi_{7,\text{o}}}\rangle.\label{staggered}
\end{equation}
The corresponding contour plot is exhibited in Fig.~\ref{fig:effectiveStaggered}.
On top and on the right of the plot the staggered magnetisation gets large
and we identify this region as belonging to anti-ferromagnetic phases.
For large absolute values of $\lambda_7,\,\lambda_{14}$ action (energy)
dominates over entropy and this explains why the simulation results agree well
with the classical analysis for strong couplings: all phases but the \emph{transition phase}
F~$\to$~AF1 are already  visible in the classical phase diagram
in Fig.~\ref{fig:effectiveClassical}.
\begin{figure}[t]
\includegraphics{effectiveStaggered}
\caption{Staggered magnetisation $S$ of the fundamental
model obtained via MC simulation on an $8^3$ lattice.}
\label{fig:effectiveStaggered}
\end{figure}
In the Monte-Carlo simulation an additional  ``symmetric phase'' with vanishing
Polyakov loop and vanishing staggered magnetisation appears for
weak couplings, in complete agreement with our mean field
analysis. The resulting phase diagram with one symmetric,
one ferromagnetic and two anti-ferromagnetic phases is depicted
in Fig.~\ref{fig:effectivePhases}.
\begin{figure}[h]
\includegraphics{effectivePhases}
\caption{Phase diagram of the fundamental effective model. The orders of phase
transitions are indicated with full lines (first order) and dotted lines
(second order/crossover).}
\label{fig:effectivePhases}
\end{figure}

Eventually the finite temperature phase transition in $G_2$ gluodynamics
will correspond to a transition between the symmetric and the
ferromagnetic phase in the effective spin model. The dependence of
the effective couplings $\lambda_7,\lambda_{14}$ on the Yang-Mills coupling
$g$ can be calculated with the help of powerful inverse Monte-Carlo
techniques \cite{Wozar:2007tz,Wozar:2008nv}. This will be done in a
forthcoming publication.
However, we anticipate that the confinement-deconfinement phase
transition in $G_2$ gluodynamics will happen near the critical point
$\lambda_{14,\text{c}}=0,\;\lambda_{7,\text{c}}\approx -0.0975(75)$ of the
fundamental model. Thus we have plotted the magnetisation in the vicinity of this
first order transition from the ferromagnetic to the symmetric phase in
Fig.~\ref{fig:symmetricNonVanishing}.
\begin{figure}[t]
\includegraphics{symmetricNonVanishing}
\caption{Polyakov loop $\vev{\chi_7}$ of the fundamental effective model with
coupling $\lambda_{14}=0$ obtained via MC simulation on an $8^3$ lattice.}
\label{fig:symmetricNonVanishing}
\end{figure}
Note that even in the ``symmetric'' phase we find a non-zero magnetisation
$\langle \chi_7\rangle$ which jumps at the critical coupling
$\lambda_{7,\text{c}}$. This parallels the jump of the Polyakov loop in
$G_2$ gluodynamics, see Fig.~\ref{fig:ymTransition}.

The phase diagram in Fig.~\ref{fig:effectivePhases} contains lines of second and
first order transitions and $3$ triple points. The full lines belong to first order
and the dotted lines to second order transitions. Note that we may pass from
the symmetric to the ferromagnetic phase via a first or via a second order
transition. The transitions from the ferromagnetic to the anti-ferromagnetic
phases AF1 and AF2 and between the anti-ferromagnetic phases are always
of second order. In order to determine the orders of the transitions we calculated more than $30$
histograms for the Polyakov loop distribution near the various phase transition curves
and the changes of various `order parameters' when one crosses the transition
lines. A typical scatter plot is depicted in Fig.~\ref{fig:Histo_1}. It shows
the distribution of $\chi_7$ at a transition from the symmetric to the anti-ferromagnetic
phase AF1 with critical couplings $\lambda_{7,\text{c}}=0$ and
$\lambda_{14,\text{c}}=0.1446$. Without further analysis it is already clear
that we are dealing with a first order phase transition.
\begin{figure}[t]
\includegraphics{HistoFO_1}
\caption{Distribution of $\chi_7$ and $\chi_{14}$
in the fundamental domain of $G_2$ at $\lambda_7=0$ and $\lambda_{14}=0.1446$.}
\label{fig:Histo_1}
\end{figure}

The following Fig.~\ref{fig:PolyakovS0_1} shows the behaviour of the magnetisations
$\langle\chi_7\rangle$ and $\langle\chi_{14}\rangle$ near the transition from the
symmetric to the ferromagnetic phase, which happens for $\lambda_{14}=0.13$ and
$\lambda_{7}$ between $-0.18$ and $-0.12$. Both expectation values vary continuously
during the transition and this already suggests that the transition is of second
order. This conclusion is further substantiated by the corresponding histograms for the distributions of
$\chi_7$ and $\chi_{14}$.

\begin{figure}[t]
\includegraphics{PolyakovSO_1}
\caption{Magnetisations $\langle\chi_7\rangle$ (upper curve) and
$\langle\chi_{14}\rangle$ (lower curve) for various $\lambda_7$ at
$\lambda_{14}=0.13$.}
\label{fig:PolyakovS0_1}
\end{figure}

\section{The $\bm{G_2}$ Potts model}
\label{sect:pottsModel}
\noindent
After having collected sufficient information to reconstruct the full phase
diagram of the fundamental continuous spin model with two
effective couplings we now truncate the degrees of freedom further
to arrive at a discrete spin model. In the case of the
well-studied $SU(3)$ Polyakov loop models one projects a Polyakov loop
$\mathcal{P}_\vc{x}$ onto the closest centre elements of $SU(3)$ and arrives
at a $\mathbb{Z}_3$ Potts model with action (energy) given by
\begin{equation}
S_N = -\beta \sum_{\langle \vc{x}\vc{y}\rangle} \delta(\sigma_\vc{x},\sigma_\vc{y}),\quad \sigma_\vc{x}\in \mathbb{Z}_N.
\end{equation}
The continuous and discrete models have coinciding critical exponents at
the second order anti-ferromagnetic phase transition and similar phase structures
\cite{Wozar:2006fi}. Motivated by these earlier successes we perform a similar
reduction of the fundamental $G_2$ spin model and arrive at a discrete
Potts-like $G_2$ spin model.

By projecting the values of $\mathcal{P}_\vc{x}$ to the three group elements with
characters $\chi_7,\chi_{14}$ lying at the extremal points of the
fundamental domain in Fig.~\ref{fig:fundamentalDomain},
we arrive at a model for the tree spins
\begin{equation}
\sigma_\vc{x}\in\left\{
\begin{pmatrix}7\\14\end{pmatrix},
\begin{pmatrix}-1\\-2\end{pmatrix},
\begin{pmatrix}-2\\5\end{pmatrix}
\right\}
\end{equation}
with nearest neighbour interaction determined by the Potts-type action
\begin{equation}
S_\text{Potts} = \sum_{\langle \vc{x}, \vc{y}\rangle}\sigma_\vc{x}^\mathsf{T}
\begin{pmatrix}
\lambda_7 & 0\\ 0 &
\lambda_{14}
\end{pmatrix}
\sigma_\vc{y}\,.
\end{equation}
As expected, the classical phase diagram of the Potts-type model
with discrete spins is similar to the diagram of the fundamental model with continuous spins.
Depending on the sign of $\lambda_7$ and the slope $\xi=\lambda_{14}/\lambda_7$
we find the following phases and phase transition lines:
\begin{itemize}
\item For $\lambda_7>0$ and $\xi>-1/2$ and for $\lambda_7<0$ and
$\xi>-47/206$ we find the \emph{ferromagnetic phase}
F with $\vc{\chi}_\text{e}=\vc{\chi}_\text{o}=(7,14)^\mathsf{T}$.
\item For $\lambda_7<0$ and $-1/2<\xi<-47/206$ we find the \emph{anti-ferromagnetic
phase} AF3 with $\vc{\chi}_\text{e}=(-1,-2)^\mathsf{T}$ and $\vc{\chi}_\text{o}=(-2,5)^\mathsf{T}$.
\item  For $\lambda_7>0$ and $-1/2<\xi<1/14$ we find the \emph{anti-ferromagnetic phase} AF2 wit
$\vc{\chi}_\text{e}=(-2,5)^\mathsf{T}$ and $\vc{\chi}_\text{o}=(7,14)^\mathsf{T}$.
\item For $\lambda_7<0$ and $\xi<-1/2$ and for $\lambda_7>0$ and $\xi>1/14$ we find the \emph{anti-ferromagnetic phase} AF1
with $\vc{\chi}_\text{e}=(-1,-2)^\mathsf{T}$ and $\vc{\chi}_\text{o}=(7,14)^\mathsf{T}$.
\end{itemize}
The phase portrait is depicted in Fig.~\ref{fig:SpinClassical}, where we have
inserted by hand the expected symmetric phase for weak couplings.
\begin{figure}[t]
\includegraphics{SpinClassical}
\caption{The classical phase diagram of the discrete $G_2$ Potts model.}
\label{fig:SpinClassical}
\end{figure}
A striking difference between the diagrams in Fig.~\ref{fig:effectiveClassical}
and in Fig.~\ref{fig:SpinClassical} is the absence of the ``transition phase''
F~$\to$~AF1 in the discrete model for which this phase does not exist by
construction. Instead we find a third anti-ferromagnetic phase denoted by AF3 in Fig.~\ref{fig:SpinClassical}.
In addition, in the symmetric phase of the continuous spin model $\vev{\chi_7}\approx 0$
and in the symmetric phase of the discrete spin model $\vev{\chi_7}\approx 4/3$.

Similarly as for the continuous model we calculated the phase diagram of the discrete model
with the help of the modified  mean field approximation. The contour plot for the magnetisation
is depicted in Fig.~\ref{fig:SpinMeanfield}.
\begin{figure}[t]
\includegraphics{SpinMeanfield}
\caption{Magnetisation $\vev{\chi_7}$ of the discrete $G_2$ Potts model in
mean field approximation.}
\label{fig:SpinMeanfield}
\end{figure}
In the lower part of the plot we can see the ferromagnetic phase for which the Polyakov
loops on both sublattices $\Lambda_\text{e}$ and $\Lambda_\text{o}$ are equal to the identity
with very high probability.

The corresponding contour plot as obtained from Monte-Carlo simulations is
shown in  Fig.~\ref{fig:SpinPolyakov}. Again, the mean field approximation and
the Monte-Carlo simulations fully agree over the whole range of coupling constants. Note that the classical behaviour as depicted in
Fig.~\ref{fig:SpinClassical} can be seen already for rather small coupling constants.
\begin{figure}[htb]
\includegraphics{SpinPolyakov}
\caption{Magnetisation $\vev{\chi_7}$ of the discrete $G_2$ Potts model
obtained via MC simulation on an $8^3$ lattice.}
\label{fig:SpinPolyakov}
\end{figure}

\begin{figure}[t]
\includegraphics{SpinStaggered}
\caption{Staggered magnetisation $S$ of the discrete $G_2$ Potts model
obtained via MC simulation on an $8^3$ lattice.}
\label{fig:SpinStaggered}
\end{figure}
In order to localize the anti-ferromagnetic phases we also measured the staggered magnetisation
introduced in \eqref{staggered}. The resulting values on a grid in coupling
constant space are plotted in  Fig.~\ref{fig:SpinStaggered}.
In accordance with the classical analysis we detect different anti-ferromagnetic
phases for positive coupling constants in the upper right part of the phase
portrait.

As concerning the relation between the Potts-type model and $G_2$ gluodynamics
one caveat should be mentioned. In the discrete spin model there exists no real ``symmetric phase''
with a fixed expectation value of $\vc{\chi}$. Even for very weak coupling do the
magnetisations $\vev{\vc{\chi}}$ depend on the couplings. This is a remnant of the
missing centre symmetry of $G_2$. Nevertheless, there exists a first order
phase transition from one (would be symmetric) ferromagnetic phase at
$\lambda_7\approx\lambda_{14}\approx 0$ to a second
ferromagnetic phase with $\mathcal{P}_\vc{x}$ directed to the $\mathbbm{1}$-element in $G_2$.
Even in the simple discrete model we see very pronounced
what happens in real $G_2$ gluodynamics. In the deconfining and confining phase
there is a non-vanishing Polyakov loop, which still serves as \emph{approximate}
order parameter for confinement since it shows a steep jump at the transition point.

\section{Conclusions}
\label{sect:conclusions}
\noindent
Effective models for confinement with the Polyakov loop as macroscopic degree
of freedom arise naturally from the strong coupling expansion
of $G_2$ gluodynamics. Already the leading order continuous and discrete
effective theories show a rich phase structure with two coexisting phases along transition lines
and three coexisting phases at several triple points. The fundamental
model with continuous spins and the Potts-type model
with discrete spins share many properties, although in the absence
of a centre symmetry they need not be in the same universality class.
The continuous model exhibits a transition from the symmetric to
the ferromagnetic phase with the same behaviour of the
Polyakov loop as in $G_2$ gluodynamics, namely a steep jump from
a small (but non-vanishing) Polyakov loop to a loop near the identity
of $G_2$.

The classical, mean field and Monte-Carlo analysis all lead to a coherent and consistent
picture for both $3$-dimensional effective theories. In particular, the prediction
of the mean field approximation for $\vev{\chi_7}$ and
$\vev{\abs{ \chi_{7,\text{e}}-\chi_{7,\text{o}}}}$ is in excellent agreement
with the corresponding results obtained by detailed Monte-Carlo
simulations. This parallels our findings for $SU(3)$ in \cite{Wozar:2006fi}
and probably is due to the existence of  tricritical points which lower the upper
critical dimension in the vicinity of these points.

As concerning the relationship between the continuous effective
models to the underlying $G_2$ gluodynamics we plan to apply
inverse Monte-Carlo techniques, preferably with demon methods,
to determine the dependence of the coupling in the fundamental model
on the gauge coupling. We hope to present the resulting
curve $\lambda_7(\beta),\lambda_{14}(\beta)$
in a forthcoming publication. However we anticipate that for the critical
Yang-Mills coupling $\beta_\text{c}$ this curve will cross the
transition line between the symmetric and ferromagnetic phases
at small $\lambda_{14}$ and negative $\lambda_7\approx -0.1$ in
Fig.~\ref{fig:effectivePolyakov}.

\begin{acknowledgments}
\noindent
Helpful discussions and earlier collaborations with Tom Heinzl, Tobias
K\"astner,  Kurt Langfeld, and Sebastian Uhl\-mann are gratefully acknowledged.
C. Wozar thanks for the support by the Studienstiftung des
deutschen Volkes. This work has been supported by the
DFG under GRK 1521.
\end{acknowledgments}

\renewcommand{\eprint}[1]{ \href{http://arxiv.org/abs/#1}{[arXiv:#1]}}