Update on Overleaf.

2 files changed, 8 insertions, 2 deletions

diff --git a/bezout.tex b/bezout.tex
index 3032076..62a773a 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -214,11 +214,17 @@ for $\delta=\kappa a^{-(p-2)}$.
 Another instrument we have to study this problem is to compute the following partition function:
 
 \begin{eqnarray}
-    Z= \int \Pi_i dx_i dy_i \; e^{-\beta_{R} \Re H_0 -\beta_I \Im H_0}
-    \delta(\sum_i z_i^2-N) \delta\left(\sum_i y_i^2 -N \frac{a-1}{2}\right) 
+    Z(a,\beta)&=& \int \Pi_i dx_i dy_i \; e^{-\beta_{R} \Re H_0 -\beta_I \Im H_0}\nonumber\\
+& &    \delta(\sum_i z_i^2-N) \delta\left(\sum_i y_i^2 -N \frac{a-1}{2}\right) 
 \end{eqnarray}
 The energy $\Re H_0, \Im H_0$ are in a one-to one relation with the temperatures $\beta_R,\beta_I$. The entropy $S(a,H_0) = \ln Z+ +\beta_{R} \langle \Re H_0 \rangle +\beta_I \langle \Im H_0\rangle$
 is the logarithm of the number of configurations of a given $(a,H_0)$.
+This problem may be solved exactly with replicas, {\em but it may also be simulated}
+Consider for example the ground-state energy for given $a$, that is, the energy in the limit $\beta_R \rightarrow \infty$ taken after  $\beta_I \rightarrow \infty$. For $a=1$ this coincides with the ground-state of the real problem.
+
+\begin{center}
+    \includegraphics[width=4cm]{phase.pdf}
+\end{center}
 
 }
 \bibliographystyle{apsrev4-2}
diff --git a/phase.pdf b/phase.pdf
new file mode 100644
index 0000000..2673d2f
--- /dev/null
+++ b/phase.pdf