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\bibstyle{unsrt} \bibdata{Rapport} \bibcite{Bittner_2012}{1} @@ -134,3 +155,16 @@ \bibcite{Stillinger_2000}{40} \bibcite{Baker_1971}{41} \bibcite{Goodson_2004}{42} +\bibcite{Heiss_1988}{43} +\bibcite{Heiss_2002}{44} +\bibcite{Cejnar_2005}{45} +\bibcite{Cejnar_2007}{46} +\bibcite{Cejnar_2009}{47} +\bibcite{Borisov_2015}{48} +\bibcite{Sindelka_2017}{49} +\bibcite{Sachdev_2011}{50} +\bibcite{Cejnar_2016}{51} +\bibcite{Cejnar_2006}{52} +\bibcite{Caprio_2008}{53} +\bibcite{Macek_2019}{54} +\bibcite{Stransky_2018}{55} diff --git a/RapportStage/Rapport.bbl b/RapportStage/Rapport.bbl index 81218fb..4bdae8e 100644 --- a/RapportStage/Rapport.bbl +++ b/RapportStage/Rapport.bbl @@ -238,4 +238,78 @@ David~Z. Goodson and Alexey~V. Sergeev. \newblock In {\em Advances in Quantum Chemistry}, volume~47, pages 193--208. Academic Press. +\bibitem{Heiss_1988} +W.~D. Heiss. +\newblock Exceptional points of a hamiltonian and phase transitions in finite + systems. +\newblock 329(2):133--138. + +\bibitem{Heiss_2002} +W.~D. Heiss and M.~Müller. +\newblock Universal relationship between a quantum phase transition and + instability points of classical systems. +\newblock 66(1):016217. + +\bibitem{Cejnar_2005} +Pavel Cejnar, Stefan Heinze, and Jan Dobeš. +\newblock Thermodynamic analogy for quantum phase transitions at zero + temperature. +\newblock 71(1):011304. + +\bibitem{Cejnar_2007} +Pavel Cejnar, Stefan Heinze, and Michal Macek. +\newblock Coulomb analogy for non-hermitian degeneracies near quantum phase + transitions. +\newblock 99(10):100601. + +\bibitem{Cejnar_2009} +Pavel Cejnar and Jan Jolie. +\newblock Quantum phase transitions in the interacting boson model. +\newblock 62(1):210--256. + +\bibitem{Borisov_2015} +Denis~I. Borisov, František Ružička, and Miloslav Znojil. +\newblock Multiply degenerate exceptional points and quantum phase transitions. +\newblock 54(12):4293--4305. + +\bibitem{Sindelka_2017} +Milan Šindelka, Lea~F. Santos, and Nimrod Moiseyev. +\newblock Excited-state quantum phase transitions studied from a non-hermitian + perspective. +\newblock 95(1):010103. + +\bibitem{Sachdev_2011} +Subir Sachdev. +\newblock {\em Quantum Phase Transitions}. +\newblock Cambridge University Press, 2 edition. + +\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. +\newblock 323(5):1106--1135. + +\bibitem{Macek_2019} +Michal Macek, Pavel Stránský, Amiram Leviatan, and Pavel Cejnar. +\newblock Excited-state quantum phase transitions in systems with two degrees + of freedom. {III}. interacting boson systems. +\newblock 99(6):064323. + +\bibitem{Stransky_2018} +Pavel Stránský, Martin Dvořák, and Pavel Cejnar. +\newblock Exceptional points near first- and second-order quantum phase + transitions. +\newblock 97(1):012112. + \end{thebibliography} diff --git a/RapportStage/Rapport.bib b/RapportStage/Rapport.bib index dd9acbd..4f13d19 100644 --- a/RapportStage/Rapport.bib +++ b/RapportStage/Rapport.bib @@ -595,5 +595,147 @@ date = {1971-10-01}, } +@article{Cejnar_2005, + title = {Thermodynamic analogy for quantum phase transitions at zero temperature}, + volume = {71}, + doi = {10.1103/PhysRevC.71.011304}, + pages = {011304}, + number = {1}, + shortjournal = {Phys. Rev. C}, + author = {Cejnar, Pavel and Heinze, Stefan and Dobeš, Jan}, + date = {2005-01-26}, +} + +@article{Stransky_2018, + title = {Exceptional points near first- and second-order quantum phase transitions}, + volume = {97}, + doi = {10.1103/PhysRevE.97.012112}, + pages = {012112}, + number = {1}, + shortjournal = {Phys. Rev. E}, + author = {Stránský, Pavel and Dvořák, Martin and Cejnar, Pavel}, + date = {2018-01-11}, +} + +@article{Cejnar_2007, + title = {Coulomb Analogy for Non-Hermitian Degeneracies near Quantum Phase Transitions}, + volume = {99}, + doi = {10.1103/PhysRevLett.99.100601}, + pages = {100601}, + number = {10}, + shortjournal = {Phys. Rev. Lett.}, + author = {Cejnar, Pavel and Heinze, Stefan and Macek, Michal}, + date = {2007-09-07}, +} + +@article{Heiss_1988, + title = {Exceptional points of a Hamiltonian and phase transitions in finite systems}, + volume = {329}, + doi = {10.1007/BF01283767}, + pages = {133--138}, + number = {2}, + journaltitle = {Zeitschrift für Physik A Atomic Nuclei}, + shortjournal = {Z. Physik A - Atomic Nuclei}, + author = {Heiss, W. D.}, + date = {1988-06-01}, +} + +@article{Heiss_2002, + title = {Universal relationship between a quantum phase transition and instability points of classical systems}, + volume = {66}, + doi = {10.1103/PhysRevE.66.016217}, + pages = {016217}, + number = {1}, + shortjournal = {Phys. Rev. E}, + author = {Heiss, W. D. and Müller, M.}, + date = {2002-07-26}, +} + +@article{Sindelka_2017, + title = {Excited-state quantum phase transitions studied from a non-Hermitian perspective}, + volume = {95}, + doi = {10.1103/PhysRevA.95.010103}, + pages = {010103}, + number = {1}, + shortjournal = {Phys. Rev. A}, + author = {Šindelka, Milan and Santos, Lea F. and Moiseyev, Nimrod}, + 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_2009, + title = {Quantum phase transitions in the interacting boson model}, + volume = {62}, + doi = {10.1016/j.ppnp.2008.08.001}, + pages = {210--256}, + number = {1}, + shortjournal = {Progress in Particle and Nuclear Physics}, + author = {Cejnar, Pavel and Jolie, Jan}, + date = {2009-01-01}, +} + +@article{Borisov_2015, + title = {Multiply Degenerate Exceptional Points and Quantum Phase Transitions}, + volume = {54}, + doi = {10.1007/s10773-014-2493-y}, + pages = {4293--4305}, + number = {12}, + shortjournal = {Int J Theor Phys}, + author = {Borisov, Denis I. and Ružička, František and Znojil, Miloslav}, + date = {2015-12-01}, +} + +@article{Caprio_2008, + title = {Excited state quantum phase transitions in many-body systems}, + volume = {323}, + doi = {10.1016/j.aop.2007.06.011}, + pages = {1106--1135}, + number = {5}, + shortjournal = {Annals of Physics}, + author = {Caprio, M. A. and Cejnar, P. and Iachello, F.}, + date = {2008-05-01}, +} + +@article{Macek_2019, + title = {Excited-state quantum phase transitions in systems with two degrees of freedom. {III}. Interacting boson systems}, + volume = {99}, + doi = {10.1103/PhysRevC.99.064323}, + pages = {064323}, + number = {6}, + shortjournal = {Phys. Rev. C}, + author = {Macek, Michal and Stránský, Pavel and Leviatan, Amiram and Cejnar, Pavel}, + date = {2019-06-21}, +} +@book{Sachdev_2011, + location = {Cambridge}, + edition = {2}, + title = {Quantum Phase Transitions}, + publisher = {Cambridge University Press}, + author = {Sachdev, Subir}, + date = {2011}, + doi = {10.1017/CBO9780511973765}, +} + + +@article{Cejnar_2016, + title = {Quantum phase transitions in the collective degrees of freedom: nuclei and other many-body systems}, + volume = {91}, + doi = {10.1088/0031-8949/91/8/083006}, + pages = {083006}, + number = {8}, + shortjournal = {Phys. Scr.}, + author = {Cejnar, Pavel and Stránský, Pavel}, + date = {2016-07}, +} \ No newline at end of file diff --git a/RapportStage/Rapport.blg b/RapportStage/Rapport.blg index afc7839..b357fcf 100644 --- a/RapportStage/Rapport.blg +++ b/RapportStage/Rapport.blg @@ -23,45 +23,70 @@ Warning--empty year in Stillinger_2000 Warning--empty journal in Baker_1971 Warning--empty year in Baker_1971 Warning--empty year in Goodson_2004 -You've used 42 entries, +Warning--empty journal in Heiss_1988 +Warning--empty year in Heiss_1988 +Warning--empty journal in Heiss_2002 +Warning--empty year in Heiss_2002 +Warning--empty journal in Cejnar_2005 +Warning--empty year in Cejnar_2005 +Warning--empty journal in Cejnar_2007 +Warning--empty year in Cejnar_2007 +Warning--empty journal in Cejnar_2009 +Warning--empty year in Cejnar_2009 +Warning--empty journal in Borisov_2015 +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_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 +Warning--empty year in Macek_2019 +Warning--empty journal in Stransky_2018 +Warning--empty year in Stransky_2018 +You've used 55 entries, 1791 wiz_defined-function locations, - 726 strings with 10578 characters, -and the built_in function-call counts, 10488 in all, are: -= -- 868 -> -- 520 + 802 strings with 12579 characters, +and the built_in function-call counts, 13548 in all, are: += -- 1115 +> -- 649 < -- 2 -+ -- 191 -- -- 149 -* -- 947 -:= -- 1691 -add.period$ -- 127 -call.type$ -- 42 -change.case$ -- 40 ++ -- 239 +- -- 184 +* -- 1226 +:= -- 2156 +add.period$ -- 166 +call.type$ -- 55 +change.case$ -- 53 chr.to.int$ -- 0 -cite$ -- 62 -duplicate$ -- 393 -empty$ -- 994 -format.name$ -- 149 -if$ -- 2226 +cite$ -- 100 +duplicate$ -- 511 +empty$ -- 1300 +format.name$ -- 184 +if$ -- 2864 int.to.chr$ -- 0 -int.to.str$ -- 42 -missing$ -- 44 -newline$ -- 213 -num.names$ -- 42 -pop$ -- 81 +int.to.str$ -- 55 +missing$ -- 58 +newline$ -- 278 +num.names$ -- 55 +pop$ -- 136 preamble$ -- 1 purify$ -- 0 quote$ -- 0 -skip$ -- 142 +skip$ -- 183 stack$ -- 0 -substring$ -- 865 -swap$ -- 48 +substring$ -- 1134 +swap$ -- 49 text.length$ -- 2 text.prefix$ -- 0 top$ -- 0 type$ -- 0 -warning$ -- 20 -while$ -- 98 -width$ -- 44 -write$ -- 445 -(There were 20 warnings) +warning$ -- 45 +while$ -- 128 +width$ -- 57 +write$ -- 563 +(There were 45 warnings) diff --git a/RapportStage/Rapport.log b/RapportStage/Rapport.log index 1f252aa..dca981e 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) 14 JUL 2020 11:26 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -579,6 +579,9 @@ s \f@nch@O@orf=\skip80 ) (./Rapport.aux +LaTeX Warning: Label `fig:my_label' multiply defined. + + LaTeX Warning: Label `fig:my_label' multiply defined. ) @@ -754,7 +757,7 @@ Package epstopdf Info: Source file: df logoens.eps> (epstopdf) \includegraphics on input line 72. 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LaTeX Warning: There were multiply-defined labels. -Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 331. +Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 346. ) (\end occurred inside a group at level 1) -### simple group (level 1) entered at line 207 ({) +### simple group (level 1) entered at line 214 ({) ### bottom level Here is how much of TeX's memory you used: - 21746 strings out of 492167 - 398366 string characters out of 6131559 - 493612 words of memory out of 5000000 - 25833 multiletter control sequences out of 15000+600000 + 21783 strings out of 492167 + 398929 string characters out of 6131559 + 494057 words of memory out of 5000000 + 25853 multiletter control sequences out of 15000+600000 557560 words of font info for 90 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 52i,12n,71p,1615b,994s stack positions out of 5000i,500n,10000p,200000b,80000s @@ -906,10 +930,10 @@ cm-super/sfrm1095.pfb> -Output written on Rapport.pdf (12 pages, 1011004 bytes). +Output written on Rapport.pdf (13 pages, 1179185 bytes). PDF statistics: - 368 PDF objects out of 1000 (max. 8388607) - 311 compressed objects within 4 object streams - 81 named destinations out of 1000 (max. 500000) - 118 words of extra memory for PDF output out of 10000 (max. 10000000) + 421 PDF objects out of 1000 (max. 8388607) + 360 compressed objects within 4 object streams + 97 named destinations out of 1000 (max. 500000) + 131 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/RapportStage/Rapport.out b/RapportStage/Rapport.out index e4efcbf..f84c5cf 100644 --- a/RapportStage/Rapport.out +++ b/RapportStage/Rapport.out @@ -7,3 +7,4 @@ \BOOKMARK [2][-]{subsection.3.4}{The physics of quantum phase transition}{section.3}% 7 \BOOKMARK [1][-]{section.4}{The spherium model}{}% 8 \BOOKMARK [2][-]{subsection.4.1}{Weak correlation regime}{section.4}% 9 +\BOOKMARK [1][-]{section.5}{To do list}{}% 10 diff --git a/RapportStage/Rapport.pdf b/RapportStage/Rapport.pdf index 633978b..212488d 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 9400542..4f1afe9 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 2363b21..d50f106 100644 --- a/RapportStage/Rapport.tex +++ b/RapportStage/Rapport.tex @@ -155,12 +155,19 @@ Exceptional points (EPs) \cite{Heiss_1990, Heiss_1999, Heiss_2012, Heiss_2016} a CIs are ubiquitous in non-adiabatic processes and play a key role in photo-chemical mechanisms. In the case of auto-ionizing resonances, EPs have a role in deactivation processes similar to CIs in the decay of bound excited states. Although Hermitian and non-Hermitian Hamiltonians are closely related, the behavior of their eigenvalues near degeneracies is starkly different. -For example, encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. +For example, encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy (see Figure 1 for a graphical example). Additionally, the wave function picks up a geometric phase (also known as Berry phase \cite{Berry_1984}) and four loops are required to recover the initial wave function. In contrast, encircling Hermitian degeneracies at CIs only introduces a geometric phase while leaving the states unchanged. More dramatically, whilst eigenvectors remain orthogonal at CIs, at non-Hermitian EPs the eigenvectors themselves become equivalent, resulting in a \textit{self-orthogonal} state. \cite{MoiseyevBook} More importantly here, although EPs usually lie off the real axis, these singular points are intimately related to the convergence properties of perturbative methods and avoided crossing on the real axis are indicative of singularities in the complex plane. \cite{Olsen_1996, Olsen_2000} +\begin{figure}[h!] + \centering + \includegraphics[width=0.7\textwidth]{TopologyEP.pdf} + \caption{\centering A generic EP with the square root branch point topology. A loop around the EP interconvert the states.} + \label{fig:my_label} +\end{figure} + %============================================================% \section{Perturbation theory} %============================================================% @@ -293,9 +300,9 @@ 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 (citation du stransky 2018 sauf lee). 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 (stransky 2017, cejnar 2006 caprio 2008). 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 (cejnar 2009). 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_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). -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 (citation cejnar 2005 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 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. 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 mean 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. @@ -321,6 +328,14 @@ At a critical value of R, called the Coulson-Fischer point, a second unrestricte Then the mono-electronic wave function are expand in the spatial basis set of the zonal spherical harmonics: +\section{To do list} + +\begin{itemize} +\item Rajouter label pour les figures et équations cités +\item Corriger les erreurs dans la biblio +\item Finir le paragraphe QPT (singularité $\alpha$ ?) +\end{itemize} + \newpage \bibliographystyle{unsrt} diff --git a/RapportStage/Rapport.toc b/RapportStage/Rapport.toc index 5719c82..0910181 100644 --- a/RapportStage/Rapport.toc +++ b/RapportStage/Rapport.toc @@ -5,6 +5,7 @@ \contentsline {subsection}{\numberline {3.1}Behavior of the M{\o }ller-Plesset series}{5}{subsection.3.1}% \contentsline {subsection}{\numberline {3.2}Cases of divergence}{6}{subsection.3.2}% \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 {subsection}{\numberline {3.4}The physics of quantum phase transition}{9}{subsection.3.4}% \contentsline {section}{\numberline {4}The spherium model}{9}{section.4}% -\contentsline {subsection}{\numberline {4.1}Weak correlation regime}{9}{subsection.4.1}% +\contentsline {subsection}{\numberline {4.1}Weak correlation regime}{10}{subsection.4.1}% +\contentsline {section}{\numberline {5}To do list}{10}{section.5}% diff --git a/RapportStage/TopologyEP.pdf b/RapportStage/TopologyEP.pdf new file mode 100644 index 0000000..3375ab9 Binary files /dev/null and b/RapportStage/TopologyEP.pdf differ