little update

RapportStage/Rapport.aux

@@ -97,8 +97,8 @@
 \citation{Sindelka_2017}
 \citation{Cejnar_2009}
 \citation{Sachdev_2011}
+\citation{Cejnar_2015}
 \citation{Cejnar_2016}
-\citation{Cejnar_2006}
 \citation{Caprio_2008}
 \citation{Macek_2019}
 \citation{Cejnar_2009}
@@ -163,8 +163,8 @@
 \bibcite{Borisov_2015}{48}
 \bibcite{Sindelka_2017}{49}
 \bibcite{Sachdev_2011}{50}
-\bibcite{Cejnar_2016}{51}
-\bibcite{Cejnar_2006}{52}
+\bibcite{Cejnar_2015}{51}
+\bibcite{Cejnar_2016}{52}
 \bibcite{Caprio_2008}{53}
 \bibcite{Macek_2019}{54}
 \bibcite{Stransky_2018}{55}

RapportStage/Rapport.bbl

@@ -283,18 +283,17 @@ Subir Sachdev.
 \newblock {\em Quantum Phase Transitions}.
 \newblock Cambridge University Press, 2 edition.
 
+\bibitem{Cejnar_2015}
+Pavel Cejnar, Pavel Stránský, and Michal Kloc.
+\newblock Excited-state quantum phase transitions in finite many-body systems.
+\newblock 90(11):114015.
+
 \bibitem{Cejnar_2016}
 Pavel Cejnar and Pavel Stránský.
 \newblock Quantum phase transitions in the collective degrees of freedom:
   nuclei and other many-body systems.
 \newblock 91(8):083006.
 
-\bibitem{Cejnar_2006}
-Pavel Cejnar, Michal Macek, Stefan Heinze, Jan Jolie, and Jan Dobeš.
-\newblock Monodromy and excited-state quantum phase transitions in integrable
-  systems: collective vibrations of nuclei.
-\newblock 39(31):L515--L521.
-
 \bibitem{Caprio_2008}
 M.~A. Caprio, P.~Cejnar, and F.~Iachello.
 \newblock Excited state quantum phase transitions in many-body systems.

RapportStage/Rapport.bib

@@ -662,15 +662,16 @@
 	date = {2017-01-24},
 }
 
-@article{Cejnar_2006,
-	title = {Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei},
-	volume = {39},
-	doi = {10.1088/0305-4470/39/31/L01},
-	pages = {L515--L521},
-	number = {31},
-	shortjournal = {J. Phys. A: Math. Gen.},
-	author = {Cejnar, Pavel and Macek, Michal and Heinze, Stefan and Jolie, Jan and Dobeš, Jan},
-	date = {2006-07},
+
+@article{Cejnar_2015,
+	title = {Excited-state quantum phase transitions in finite many-body systems},
+	volume = {90},
+	doi = {10.1088/0031-8949/90/11/114015},
+	pages = {114015},
+	number = {11},
+	shortjournal = {Phys. Scr.},
+	author = {Cejnar, Pavel and Stránský, Pavel and Kloc, Michal},
+	date = {2015-10},
 }
 
 @article{Cejnar_2009,

RapportStage/Rapport.blg

@@ -38,10 +38,10 @@ Warning--empty year in Borisov_2015
 Warning--empty journal in Sindelka_2017
 Warning--empty year in Sindelka_2017
 Warning--empty year in Sachdev_2011
+Warning--empty journal in Cejnar_2015
+Warning--empty year in Cejnar_2015
 Warning--empty journal in Cejnar_2016
 Warning--empty year in Cejnar_2016
-Warning--empty journal in Cejnar_2006
-Warning--empty year in Cejnar_2006
 Warning--empty journal in Caprio_2008
 Warning--empty year in Caprio_2008
 Warning--empty journal in Macek_2019
@@ -50,24 +50,24 @@ Warning--empty journal in Stransky_2018
 Warning--empty year in Stransky_2018
 You've used 55 entries,
             1791 wiz_defined-function locations,
-            802 strings with 12579 characters,
-and the built_in function-call counts, 13548 in all, are:
-= -- 1115
-> -- 649
+            802 strings with 12505 characters,
+and the built_in function-call counts, 13482 in all, are:
+= -- 1108
+> -- 643
 < -- 2
-+ -- 239
-- -- 184
-* -- 1226
-:= -- 2156
++ -- 237
+- -- 182
+* -- 1218
+:= -- 2146
 add.period$ -- 166
 call.type$ -- 55
 change.case$ -- 53
 chr.to.int$ -- 0
 cite$ -- 100
 duplicate$ -- 511
-empty$ -- 1300
-format.name$ -- 184
-if$ -- 2864
+empty$ -- 1297
+format.name$ -- 182
+if$ -- 2852
 int.to.chr$ -- 0
 int.to.str$ -- 55
 missing$ -- 58
@@ -79,14 +79,14 @@ purify$ -- 0
 quote$ -- 0
 skip$ -- 183
 stack$ -- 0
-substring$ -- 1134
+substring$ -- 1121
 swap$ -- 49
 text.length$ -- 2
 text.prefix$ -- 0
 top$ -- 0
 type$ -- 0
 warning$ -- 45
-while$ -- 128
+while$ -- 127
 width$ -- 57
 write$ -- 563
 (There were 45 warnings)

RapportStage/Rapport.log

@@ -1,4 +1,4 @@
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-Output written on Rapport.pdf (13 pages, 1179185 bytes).
+Output written on Rapport.pdf (13 pages, 1179150 bytes).
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RapportStage/Rapport.tex

@@ -300,7 +300,7 @@ Finally, it was shown that $\beta$ singularities are very sensitive to the basis
 
 \subsection{The physics of quantum phase transition}
 
-In the previous section, we saw that a reasoning on the Hamiltonian allows us to predict the existence of a critical point. In a finite basis set this critical point is model by a cluster of singularity $\beta$. It is now well-known that this phenomenon is a specific case of a more general phenomenon. Indeed, theoretical physicists proved that EPs are connected to quantum phase transitions \cite{Heiss_1988, Heiss_2002, Cejnar_2005, Cejnar_2007, Cejnar_2009, Borisov_2015, Sindelka_2017}. In quantum mechanics, the Hamiltonian is almost always dependent of a parameter, in some cases the variation of a parameter can lead to abrupt changes at a critical point. Those quantum phase transitions exist both for ground and excited states \cite{Cejnar_2009, Sachdev_2011, Cejnar_2016, Cejnar_2006, Caprio_2008, Macek_2019}. A ground-state quantum phase transition is characterized by the successive derivative of the ground-state energy with respect to a non-thermal control parameter \cite{Cejnar_2009, Sachdev_2011}. The transition is called discontinuous and of first order if the first derivative is discontinuous at the critical parameter value. Otherwise, it is called continuous and of n-th order if the n-th derivative is discontinuous. A quantum phase transition can also be identify by the discontinuity of an appropriate order parameter (or one of its derivative). 
+In the previous section, we saw that a reasoning on the Hamiltonian allows us to predict the existence of a critical point. In a finite basis set this critical point is model by a cluster of singularity $\beta$. It is now well-known that this phenomenon is a specific case of a more general phenomenon. Indeed, theoretical physicists proved that EPs are connected to quantum phase transitions \cite{Heiss_1988, Heiss_2002, Cejnar_2005, Cejnar_2007, Cejnar_2009, Borisov_2015, Sindelka_2017}. In quantum mechanics, the Hamiltonian is almost always dependent of a parameter, in some cases the variation of a parameter can lead to abrupt changes at a critical point. Those quantum phase transitions exist both for ground and excited states \cite{Cejnar_2009, Sachdev_2011, Cejnar_2015, Cejnar_2016, Caprio_2008, Macek_2019}. A ground-state quantum phase transition is characterized by the successive derivative of the ground-state energy with respect to a non-thermal control parameter \cite{Cejnar_2009, Sachdev_2011}. The transition is called discontinuous and of first order if the first derivative is discontinuous at the critical parameter value. Otherwise, it is called continuous and of n-th order if the n-th derivative is discontinuous. A quantum phase transition can also be identify by the discontinuity of an appropriate order parameter (or one of its derivative). 
 
 The presence of an EP close to the real axis is characteristic of a sharp avoided crossings. Yet at such an avoided crossings eigenstates change abruptly. Although it is now well understood that EPs are closely related to quantum phase transitions, the link between the type of QPT (ground state or excited state, first or superior order) and EPs still need to be clarify. One of the major challenge in order to do this reside in our ability to compute the distribution of EPs. The numerical assignment of an EP to two energies on the real axis is very difficult in large dimensions. Cejnar et al. developped a method based on a Coulomb analogy giving access to the density of EP close to the real axis \cite{Cejnar_2005, Cejnar_2007}. More recently Stransky and co-workers proved that the distribution of EPs is not the same around a QPT of first or second order \cite{Stransky_2018}. Moreover, that when the dimension of the system increases they tends towards the real axis in a different manner, meaning respectively exponentially and algebraically.
 
@@ -333,6 +333,7 @@ Then the mono-electronic wave function are expand in the spatial basis set of th
 \begin{itemize}
 \item Rajouter label pour les figures et équations cités
 \item Corriger les erreurs dans la biblio
+\item Changer de bibliographystyle
 \item Finir le paragraphe QPT (singularité $\alpha$ ?)
 \end{itemize}