update monday

RapportStage/Rapport.aux

@@ -81,13 +81,14 @@
 \citation{Sergeev_2005}
 \citation{Sergeev_2006}
 \citation{Stillinger_2000}
+\citation{Baker_1971}
 \@writefile{toc}{\contentsline {subsection}{\numberline {3.3}The singularity structure}{7}{subsection.3.3}\protected@file@percent }
 \citation{Goodson_2004}
 \citation{Olsen_2000}
-\citation{SzaboBook}
 \@writefile{toc}{\contentsline {subsection}{\numberline {3.4}The physics of quantum phase transition}{8}{subsection.3.4}\protected@file@percent }
 \@writefile{toc}{\contentsline {section}{\numberline {4}The spherium model}{8}{section.4}\protected@file@percent }
-\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Weak correlation regime}{8}{subsection.4.1}\protected@file@percent }
+\citation{SzaboBook}
+\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Weak correlation regime}{9}{subsection.4.1}\protected@file@percent }
 \bibstyle{unsrt}
 \bibdata{Rapport}
 \bibcite{Bittner_2012}{1}
@@ -130,4 +131,5 @@
 \bibcite{Sergeev_2005}{38}
 \bibcite{Sergeev_2006}{39}
 \bibcite{Stillinger_2000}{40}
-\bibcite{Goodson_2004}{41}
+\bibcite{Baker_1971}{41}
+\bibcite{Goodson_2004}{42}

RapportStage/Rapport.bbl

@@ -226,6 +226,12 @@ Frank~H. Stillinger.
   chemistry.
 \newblock {\em J. Chem. Phys.}, 112(22):9711--9715.
 
+\bibitem{Baker_1971}
+{GEORGE}~A. {BAKER}.
+\newblock Singularity structure of the perturbation series for the ground-state
+  energy of a many-fermion system.
+\newblock 43(4):479--531.
+
 \bibitem{Goodson_2004}
 David~Z. Goodson and Alexey~V. Sergeev.
 \newblock Singularity structure of møller-plesset perturbation theory.

RapportStage/Rapport.bib

@@ -584,7 +584,16 @@
 	date = {1996-01-01},
 }
 
-
+@article{Baker_1971,
+	title = {Singularity Structure of the Perturbation Series for the Ground-State Energy of a Many-Fermion System},
+	volume = {43},
+	doi = {10.1103/RevModPhys.43.479},
+	pages = {479--531},
+	number = {4},
+	shortjournal = {Rev. Mod. Phys.},
+	author = {{BAKER}, {GEORGE} A.},
+	date = {1971-10-01},
+}
 
 
 

RapportStage/Rapport.blg

@@ -20,46 +20,48 @@ Warning--empty year in Christiansen_1996
 Warning--empty year in Sergeev_2005
 Warning--empty year in Sergeev_2006
 Warning--empty year in Stillinger_2000
+Warning--empty journal in Baker_1971
+Warning--empty year in Baker_1971
 Warning--empty year in Goodson_2004
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+(There were 20 warnings)

RapportStage/Rapport.log

@@ -1,4 +1,4 @@
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RapportStage/Rapport.tex

@@ -287,7 +287,7 @@ H(\lambda)=H_0 + \lambda (H_\text{phys} - H_0)
     H(\lambda)=\sum\limits_{j=1}^{n}\left[-\frac{1}{2}\grad_j^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_j-\vb{R}_k|} + (1-\lambda)V_j^{(scf)}+\lambda\sum\limits_{j<l}^{n}\frac{1}{|\vb{r}_j-\vb{r}_l|} \right]
 \end{equation}
 
-The first two terms, the kinetic energy and the electron-nucleus attraction, form the mono-electronic core Hamiltonian which is independant of $\lambda$. The third term is the mean field repulsion of the Hartree-Fock calculation done to get $H_0$ and the last term is the Coulomb repulsion. If $\lambda$ is negative, the Coulomb interaction becomes attractive but the mean field stays repulsive as it is proportional to $(1-\lambda)$. If $\lambda$ becomes more and more negative the mean field becomes more and more repulsive so the nucleus can't bind the electrons anymore because the electron-nucleus attraction is not scaled with $\lambda$. The repulsive mean field is localized around nucleus whereas the electrons interactions persist away from nucleus. There is a real negative value $\lambda_c$ where the electrons form a bound cluster and goes to infinity. According to Baker this value is a critical point of the system and by analogy with thermodynamics the energy $E(\lambda)$ exhibits a singularity at $\lambda_c$. Beyond $\lambda_c$ there is a continuum of eigenstates with electrons dissociated from the nucleus.
+The first two terms, the kinetic energy and the electron-nucleus attraction, form the mono-electronic core Hamiltonian which is independant of $\lambda$. The third term is the mean field repulsion of the Hartree-Fock calculation done to get $H_0$ and the last term is the Coulomb repulsion. If $\lambda$ is negative, the Coulomb interaction becomes attractive but the mean field stays repulsive as it is proportional to $(1-\lambda)$. If $\lambda$ becomes more and more negative the mean field becomes more and more repulsive so the nucleus can't bind the electrons anymore because the electron-nucleus attraction is not scaled with $\lambda$. The repulsive mean field is localized around nucleus whereas the electrons interactions persist away from nucleus. There is a real negative value $\lambda_c$ where the electrons form a bound cluster and goes to infinity. According to Baker this value is a critical point of the system and by analogy with thermodynamics the energy $E(\lambda)$ exhibits a singularity at $\lambda_c$ \cite{Baker_1971}. At this point the system undergo a phase transition and a symmetry breaking. Beyond $\lambda_c$ there is a continuum of eigenstates with electrons dissociated from the nucleus.
 
 This reasoning is done on the exact Hamiltonian and energy, this is the exact energy which exhibits this singularity on the negative real axis. But in finite basis set, one can prove that for a Hermitian Hamiltonian the singularities of $E(\lambda)$ occurs in complex conjugate pair with non-zero imaginary parts. Sergeev and Goodson proved, as predicted by Stillinger, that in a finite basis set the critical point on the real axis is modeled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane. They explain that Olsen et al., because they used a $2\times2$ model, only observed the first singularity of this cluster of singularities causing the divergence.
 
@@ -295,7 +295,15 @@ Finally, it was shown that $\beta$ singularities are very sensitive to the basis
 
 \subsection{The physics of quantum phase transition}
 
-At a sharp avoided crossings the two eigenstates undergo 
+In the previous section, we seen 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 physicist proved that EPs are connected to quantum phase transitions. 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. 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. 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. More recently Stransky and co-workers proved
+
+Ajouter la biblio
+Singularity $\beta$ and quantum phase transition ?
+Coulson-Fisher point second QPT
+Singularity $\beta$ + général
+
 
 %============================================================%
 \section{The spherium model}

RapportStage/Rapport.toc

@@ -7,4 +7,4 @@
 \contentsline {subsection}{\numberline {3.3}The singularity structure}{7}{subsection.3.3}% 
 \contentsline {subsection}{\numberline {3.4}The physics of quantum phase transition}{8}{subsection.3.4}% 
 \contentsline {section}{\numberline {4}The spherium model}{8}{section.4}% 
-\contentsline {subsection}{\numberline {4.1}Weak correlation regime}{8}{subsection.4.1}% 
+\contentsline {subsection}{\numberline {4.1}Weak correlation regime}{9}{subsection.4.1}%