Update QPT

Manuscript/EPAWTFT.bbl

@@ -6,7 +6,7 @@
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   {\doibase 10.1016/0029-5582(65)90862-X} {\bibfield  {journal} {\bibinfo
   {journal} {Nucl. Phys.}\ }\textbf {\bibinfo {volume} {62}},\ \bibinfo {pages}
   {188} (\bibinfo {year} {1965})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Cejnar}\ and\ \citenamefont
+  {Jolie}(2000)}]{Cejnar_2000a}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
+  {Cejnar}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Jolie}},\
+  }\href {\doibase 10.1103/PhysRevE.61.6237} {\bibfield  {journal} {\bibinfo
+  {journal} {Phys. Rev. E}\ }\textbf {\bibinfo {volume} {61}},\ \bibinfo
+  {pages} {6237} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2003)\citenamefont {Cejnar},
+  \citenamefont {Heinze},\ and\ \citenamefont {Jolie}}]{Cejnar_2003}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
+  {Cejnar}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Heinze}}, \
+  and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Jolie}},\ }\href
+  {\doibase 10.1103/PhysRevC.68.034326} {\bibfield  {journal} {\bibinfo
+  {journal} {Phys. Rev. C}\ }\textbf {\bibinfo {volume} {68}},\ \bibinfo
+  {pages} {034326} (\bibinfo {year} {2003})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Cejnar}\ and\ \citenamefont
+  {Iachello}(2007)}]{Cejnar_2007a}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
+  {Cejnar}}\ and\ \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
+  {Iachello}},\ }\href {\doibase 10.1088/1751-8113/40/4/001} {\bibfield
+  {journal} {\bibinfo  {journal} {J. Phys. A: Math. Theor.}\ }\textbf {\bibinfo
+  {volume} {40}},\ \bibinfo {pages} {581} (\bibinfo {year} {2007})}\BibitemShut
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Manuscript/EPAWTFT.bib

@@ -1168,3 +1168,37 @@
 	volume = {103},
 	year = {2009},
 	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.123008}}
+
+@article{Cejnar_2003,
+        title = {Ground-State Shape Phase Transitions in Nuclei: {{Thermodynamic}} Analogy and Finite-\${{N}}\$ Effects},
+        author = {Cejnar, Pavel and Heinze, Stefan and Jolie, Jan},
+        year = {2003},
+        volume = {68},
+        pages = {034326},
+        publisher = {{American Physical Society}},
+        doi = {10.1103/PhysRevC.68.034326},
+        journal = {Phys. Rev. C}}
+
+@article{Cejnar_2000,
+        title = {Quantum Phase Transitions Studied within the Interacting Boson Model},
+        author = {Cejnar, Pavel and Jolie, Jan},
+        year = {2000},
+        volume = {61},
+        pages = {6237--6247},
+        publisher = {{American Physical Society}},
+        doi = {10.1103/PhysRevE.61.6237},
+        journal = {Phys. Rev. E}}
+
+@article{Cejnar_2007a,
+        title = {Phase Structure of Interacting Boson Models in Arbitrary Dimension},
+        author = {Cejnar, Pavel and Iachello, Francesco},
+        year = {2007},
+        volume = {40},
+        pages = {581--595},
+        publisher = {{IOP Publishing}},
+        doi = {10.1088/1751-8113/40/4/001},
+        journal = {J. Phys. A: Math. Theor.}}
+
+
+
+

Manuscript/EPAWTFT.blg

@@ -1,20 +1,20 @@
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Manuscript/EPAWTFT.tex

@@ -498,6 +498,7 @@ In the previous section, we saw that a careful analysis of the structure of the
 
 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 QPTs, the link between the type of QPT (ground state or excited state, first or higher order) and EPs still need to be clarified. One of the major obstacles that one faces in order to achieve this resides in the 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. Hence, the design of specific methods are required to get information on the location of EPs. Following this idea, Cejnar \textit{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 coworkers proved that the distribution of EPs is characteristic on the order of the QPT. \cite{Stransky_2018} In the thermodynamic limit, some of the EPs converge towards a critical point $\lambda_\text{c}$ on the real axis. They showed that, within the interacting boson model, \cite{Lipkin_1965} EPs associated to first- and second-order QPT behave differently when the number of particles increases. The position of these singularities converge towards the critical point on the real axis at different rates (exponentially and algebraically for the first and second orders, respectively) with respect to the number of particles.
 
+Moreover, Cejnar \textit{et al.}~studied the so-called shape-phase transitions of the IBM model from the QPT's point of view \cite{Cejnar_2000, Cejnar_2003, Cejnar_2007a, Cejnar_2009}. The phase of the ensemble of $s$ and $d$ bosons is characterized by a dynamical symmetry. When a parameter is continously modified the dynamical symmetry of the system can change at a critical value of this parameter, leading to a deformed phase. They showed that at this critical value of the parameter, the system undergoes a QPT. For example, without interaction the ground state is the spherical phase (a condensate of s bosons) and when the interaction increases it leads to a deformed phase constituted of a mixture of s and d bosons states. In particular, we see that the transition from the spherical phase to the axially symmetric one is analog to the symmetry breaking of the wavefunction of the hydrogen molecule when the bond is stretched \cite{SzaboBook}.
 It seems like our understanding of the physics of spatial and/or spin symmetry breaking in HF theory can be enlightened by QPT theory. Indeed, the second derivative of the HF ground-state energy is discontinuous at the point of spin symmetry-breaking which means that the system undergo a second-order QPT. Moreover, the $\beta$ singularities introduced by Sergeev and coworkers 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. Thus, they cannot 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 (usually) the same spatial and spin symmetry as the ground state. We believe that $\alpha$ singularities are connected to states with non-negligible contribution in the CI expansion thus to the dynamical part of the correlation energy, while $\beta$ singularities are linked to symmetry breaking and phase transitions of the wave function, \ie, to the multi-reference nature of the wave function thus to the static part of the correlation energy.