update wednesday
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\bibstyle{unsrt}
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\bibdata{Rapport}
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\bibcite{Bittner_2012}{1}
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@ -134,3 +155,16 @@
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\bibcite{Stillinger_2000}{40}
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\bibcite{Baker_1971}{41}
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\bibcite{Goodson_2004}{42}
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\bibcite{Heiss_1988}{43}
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\bibcite{Heiss_2002}{44}
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\bibcite{Cejnar_2005}{45}
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\bibcite{Cejnar_2007}{46}
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\bibcite{Cejnar_2009}{47}
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\bibcite{Borisov_2015}{48}
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\bibcite{Sindelka_2017}{49}
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\bibcite{Sachdev_2011}{50}
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\bibcite{Cejnar_2016}{51}
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\bibcite{Cejnar_2006}{52}
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\bibcite{Caprio_2008}{53}
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\bibcite{Macek_2019}{54}
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\bibcite{Stransky_2018}{55}
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@ -238,4 +238,78 @@ David~Z. Goodson and Alexey~V. Sergeev.
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\newblock In {\em Advances in Quantum Chemistry}, volume~47, pages 193--208.
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Academic Press.
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\bibitem{Heiss_1988}
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W.~D. Heiss.
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\newblock Exceptional points of a hamiltonian and phase transitions in finite
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systems.
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\newblock 329(2):133--138.
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\bibitem{Heiss_2002}
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W.~D. Heiss and M.~Müller.
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\newblock Universal relationship between a quantum phase transition and
|
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instability points of classical systems.
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\newblock 66(1):016217.
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||||
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||||
\bibitem{Cejnar_2005}
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||||
Pavel Cejnar, Stefan Heinze, and Jan Dobeš.
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||||
\newblock Thermodynamic analogy for quantum phase transitions at zero
|
||||
temperature.
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||||
\newblock 71(1):011304.
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||||
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||||
\bibitem{Cejnar_2007}
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||||
Pavel Cejnar, Stefan Heinze, and Michal Macek.
|
||||
\newblock Coulomb analogy for non-hermitian degeneracies near quantum phase
|
||||
transitions.
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||||
\newblock 99(10):100601.
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||||
|
||||
\bibitem{Cejnar_2009}
|
||||
Pavel Cejnar and Jan Jolie.
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||||
\newblock Quantum phase transitions in the interacting boson model.
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||||
\newblock 62(1):210--256.
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||||
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||||
\bibitem{Borisov_2015}
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||||
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.
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||||
\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.
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||||
|
||||
\end{thebibliography}
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@ -595,5 +595,147 @@
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|||
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,
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||||
location = {Cambridge},
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||||
edition = {2},
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||||
title = {Quantum Phase Transitions},
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||||
publisher = {Cambridge University Press},
|
||||
author = {Sachdev, Subir},
|
||||
date = {2011},
|
||||
doi = {10.1017/CBO9780511973765},
|
||||
}
|
||||
|
||||
|
||||
@article{Cejnar_2016,
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||||
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},
|
||||
}
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||||
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\BOOKMARK [2][-]{subsection.3.4}{The physics of quantum phase transition}{section.3}% 7
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@ -155,12 +155,19 @@ Exceptional points (EPs) \cite{Heiss_1990, Heiss_1999, Heiss_2012, Heiss_2016} a
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CIs are ubiquitous in non-adiabatic processes and play a key role in photo-chemical mechanisms.
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Although Hermitian and non-Hermitian Hamiltonians are closely related, the behavior of their eigenvalues near degeneracies is starkly different.
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\begin{figure}[h!]
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\centering
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\includegraphics[width=0.7\textwidth]{TopologyEP.pdf}
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\caption{\centering A generic EP with the square root branch point topology. A loop around the EP interconvert the states.}
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\label{fig:my_label}
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\end{figure}
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%============================================================%
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\section{Perturbation theory}
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%============================================================%
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\subsection{The physics of quantum phase transition}
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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).
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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).
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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.
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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.
|
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||||
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.
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Then the mono-electronic wave function are expand in the spatial basis set of the zonal spherical harmonics:
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\section{To do list}
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\begin{itemize}
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\item Rajouter label pour les figures et équations cités
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\item Corriger les erreurs dans la biblio
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\item Finir le paragraphe QPT (singularité $\alpha$ ?)
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\end{itemize}
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\newpage
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\contentsline {subsection}{\numberline {3.1}Behavior of the M{\o }ller-Plesset series}{5}{subsection.3.1}%
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\contentsline {subsection}{\numberline {3.2}Cases of divergence}{6}{subsection.3.2}%
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\contentsline {subsection}{\numberline {3.3}The singularity structure}{7}{subsection.3.3}%
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\contentsline {subsection}{\numberline {3.4}The physics of quantum phase transition}{8}{subsection.3.4}%
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Reference in New Issue
Block a user