diff --git a/RapportStage/Rapport.aux b/RapportStage/Rapport.aux index 15688d2..2bd4589 100644 --- a/RapportStage/Rapport.aux +++ b/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} diff --git a/RapportStage/Rapport.bbl b/RapportStage/Rapport.bbl index 4bdae8e..61d242d 100644 --- a/RapportStage/Rapport.bbl +++ b/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. diff --git a/RapportStage/Rapport.bib b/RapportStage/Rapport.bib index 4f13d19..7bc90aa 100644 --- a/RapportStage/Rapport.bib +++ b/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, diff --git a/RapportStage/Rapport.blg b/RapportStage/Rapport.blg index b357fcf..93cac5b 100644 --- a/RapportStage/Rapport.blg +++ b/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) diff --git a/RapportStage/Rapport.log b/RapportStage/Rapport.log index dca981e..a62d712 100644 --- a/RapportStage/Rapport.log +++ b/RapportStage/Rapport.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Arch Linux) (preloaded format=pdflatex 2020.5.9) 15 JUL 2020 11:04 +This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Arch Linux) (preloaded format=pdflatex 2020.5.9) 15 JUL 2020 11:26 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -883,19 +883,19 @@ LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10.95> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 317. [9] [10] (./Rapport.bbl [11] [12]) -Package atveryend Info: Empty hook `BeforeClearDocument' on input line 346. +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 347. [13] -Package atveryend Info: Empty hook `AfterLastShipout' on input line 346. +Package atveryend Info: Empty hook `AfterLastShipout' on input line 347. (./Rapport.aux) -Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 346. -Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 346. +Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 347. +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 347. Package rerunfilecheck Info: File `Rapport.out' has not changed. (rerunfilecheck) Checksum: E23D25853611534D3863D871FF4D4B69;645. LaTeX Warning: There were multiply-defined labels. -Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 346. +Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 347. ) (\end occurred inside a group at level 1) @@ -930,7 +930,7 @@ cm-super/sfrm1095.pfb> -Output written on Rapport.pdf (13 pages, 1179185 bytes). +Output written on Rapport.pdf (13 pages, 1179150 bytes). PDF statistics: 421 PDF objects out of 1000 (max. 8388607) 360 compressed objects within 4 object streams diff --git a/RapportStage/Rapport.pdf b/RapportStage/Rapport.pdf index 212488d..3868c04 100644 Binary files a/RapportStage/Rapport.pdf and b/RapportStage/Rapport.pdf differ diff --git a/RapportStage/Rapport.synctex.gz b/RapportStage/Rapport.synctex.gz index 4f1afe9..1579c24 100644 Binary files a/RapportStage/Rapport.synctex.gz and b/RapportStage/Rapport.synctex.gz differ diff --git a/RapportStage/Rapport.tex b/RapportStage/Rapport.tex index d50f106..37f91f8 100644 --- a/RapportStage/Rapport.tex +++ b/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}