update part 3.4 and 4.1

RapportStage/Rapport.bib

@@ -143,7 +143,7 @@
 
 }
 
-@article{goodson_resummation_2012,
+@article{Goodson_2012,
 	title = {Resummation methods},
 	volume = {2},
 	doi = {10.1002/wcms.92},

RapportStage/Rapport.tex

@@ -312,13 +312,11 @@ 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_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). 
+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 close to the real axis 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 at least one 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 crossing. Yet at such an avoided crossing eigen states 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. developed 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.
+The presence of an EP close to the real axis is characteristic of a sharp avoided crossing. Yet at such an avoided crossing 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 resides 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 so methods are required to get information on the location of EPs. Cejnar et al. developed 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.
 
-It seems like our understanding of the physics of spatial and/or spin symmetry breaking in the Hartree-Fock theory can be enlightened by quantum phase transition theory. Indeed, the second derivative of the energy is discontinuous at the Coulson-Fischer point which means that the system undergo a second order quantum phase transition. The $\beta$ singularities introduced by Sergeev to describe the EPs modeling the formation of a bound cluster of electrons are actually a more general class of singularities.
-
-Singularity $\alpha$ and quantum phase transition ?
+It seems like our understanding of the physics of spatial and/or spin symmetry breaking in the Hartree-Fock theory can be enlightened by quantum phase transition theory. Indeed, the second derivative of the energy is discontinuous at the Coulson-Fischer point which means that the system undergo a second order quantum phase transition. Moreover, the $\beta$ singularities introduced by Sergeev to describe the EPs modeling the formation of a bound cluster of electrons are actually a more general class of singularities. The EPs close to the real axis (the so-called $\beta$ singularities) are connected to QPT because they result from a sharp avoided crossings at which the eigenstates change quickly. However the $\alpha$ singularities arise from large avoided crossings therefore they can not be connected to QPT. The avoided crossings generating $\alpha$ singularities generally involve the ground state and low-lying doubly-excited states. Those excited states have a non-negligible contribution to the exact FCI solution because they have the same spatial and spin symmetry as the ground state. We think that $\alpha$ singularities are connected to the multi-reference behavior of the wave function in the same way as $\beta$ singularities are linked to symmetry breaking and phase transition of the wave function. 
 
 %============================================================%
 \section{The spherium model}
@@ -397,14 +395,20 @@ Vérifier minimum, maximum, point selle
 
 \subsubsection{Symmetry-broken solutions}
 
-In addition, there is also the well-known symmetry-broken UHF solution. For $R>3/2$ an other stationary UHF solution appear, this solution is a minimum. This solution corresponds to the configuration with the electron $\alpha$ on one side of the sphere and the electron $\beta$ on the opposite side and the other way round. The electrons can be on opposite sides of the sphere because the choice of $Y_{10}$ induced a privileged axis on the sphere for the electrons. The exact solution for the ground state is a singlet so this wave function does not have the true symmetry. However this solution gives more accurate results for the energy at large R as shown in Table x. In fact at the Coulson-Fischer point, it becomes more efficient to minimize the Coulomb repulsion than the kinetic energy leading to this symmetry breaking. There is a competition between those two effects: keeping the symmetry of the exact wave function and minimize the energy. This type of symmetry breaking is called a spin density wave because the system oscillate between the two symmetry-broken configurations.
+In addition, there is also the well-known symmetry-broken UHF solution. For $R>3/2$ an other stationary UHF solution appear, this solution is a minimum. This solution corresponds to the configuration with the electron $\alpha$ on one side of the sphere and the electron $\beta$ on the opposite side and the other way round. The electrons can be on opposite sides of the sphere because the choice of $Y_{10}$ induced a privileged axis on the sphere for the electrons. The exact solution for the ground state is a singlet so this wave function does not have the true symmetry. However this solution gives more accurate results for the energy at large R as shown in \autoref{tab:ERHFvsEUHF}. In fact at the Coulson-Fischer point, it becomes more efficient to minimize the Coulomb repulsion than the kinetic energy leading to this symmetry breaking. There is a competition between those two effects: keeping the symmetry of the exact wave function and minimize the energy. This type of symmetry breaking is called a spin density wave because the system oscillate between the two symmetry-broken configurations.
 
 \begin{table}[h!]
 \centering
-\begin{tabular}{c}
-a
+\begin{tabular}{c c c c c c c c c}
+$R$ & 0.1 & 1 & 2 & 3 & 5 & 10 & 100 & 1000 \\
+\hline
+RHF & 10 & 1 & 0.5 & 0.3333 & 0.2 & 0.1 & 0.01 & 0.001 \\
+\hline
+UHF & 10 & 1 & 0.490699 & 0.308532 & 0.170833 & 0.078497 & 0.007112 & 0.000703 \\
+\hline
+Exact & 9.783874 & 0.852781 & 0.391959 & 0.247898 & 0.139471 & 0.064525 & 0.005487 & 0.000515 \\
 \end{tabular}
-    \caption{\centering .}
+    \caption{\centering RHF and UHF energies in the minimal basis and exact energies for various R.}
 \label{tab:ERHFvsEUHF}
 \end{table}
 
@@ -442,8 +446,6 @@ PT broken symmetry sb UHF
 
 \begin{itemize}
 \item Corriger les erreurs dans la biblio
-\item Changer de bibliographystyle
-\item Finir le paragraphe QPT (singularité $\alpha$ ?)
 \item tableau nrj uhf, citation spin density wave et charge density wave
 \end{itemize}