Merge branch 'master' of github.com:pfloos/EPAWTFT
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Hugh Burton
commit
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\makeatletter
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\providecommand \@ifxundefined [1]{%
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@ -530,12 +530,6 @@
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{Wigner}},\ }\href {\doibase 10.1103/PhysRev.46.1002} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {46}},\
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\bibinfo {pages} {1002} (\bibinfo {year} {1934})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Goodson}(2011)}]{Goodson_2011}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
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{Goodson}},\ }\href {\doibase 10.1002/wcms.92} {\bibfield {journal}
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{\bibinfo {journal} {{WIREs} Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
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{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2007)\citenamefont {Cejnar},
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\citenamefont {Heinze},\ and\ \citenamefont {Macek}}]{Cejnar_2007}%
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\BibitemOpen
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@ -545,6 +539,12 @@
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{\doibase 10.1103/PhysRevLett.99.100601} {\bibfield {journal} {\bibinfo
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{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo
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{pages} {100601} (\bibinfo {year} {2007})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Goodson}(2011)}]{Goodson_2011}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
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{Goodson}},\ }\href {\doibase 10.1002/wcms.92} {\bibfield {journal}
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{\bibinfo {journal} {{WIREs} Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
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{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Szabo}\ and\ \citenamefont
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{Ostlund}(1989)}]{SzaboBook}%
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\BibitemOpen
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@ -587,6 +587,12 @@
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{Kryachko}}}\ (\bibinfo {publisher} {Kluwer Academic},\ \bibinfo {address}
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{Dordrecht},\ \bibinfo {year} {2003})\ p.~\bibinfo {pages} {67}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Slater}(1951)}]{Slater_1951}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont
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{Slater}},\ }\href {\doibase 10.1103/PhysRev.82.538} {\bibfield {journal}
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{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {82}},\
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\bibinfo {pages} {538} (\bibinfo {year} {1951})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Coulson}\ and\ \citenamefont
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{Fischer}(1949)}]{Coulson_1949}%
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\BibitemOpen
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@ -610,6 +616,22 @@
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
|
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{Fukutome}},\ }\href {\doibase 10.1002/qua.560200502} {\ \textbf {\bibinfo
|
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{volume} {20}},\ \bibinfo {pages} {955}}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Roothaan}(1951)}]{Roothaan_1951}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~C.~J.}\
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\bibnamefont {Roothaan}},\ }\href {\doibase 10.1103/RevModPhys.23.69}
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||||
{\bibfield {journal} {\bibinfo {journal} {Rev. Mod. Phys.}\ }\textbf
|
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{\bibinfo {volume} {23}},\ \bibinfo {pages} {69} (\bibinfo {year}
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||||
{1951})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Hall}\ and\ \citenamefont
|
||||
{Lennard-Jones}(1951)}]{Hall_1951}%
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\BibitemOpen
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||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~G.}\ \bibnamefont
|
||||
{Hall}}\ and\ \bibinfo {author} {\bibfnamefont {J.~E.}\ \bibnamefont
|
||||
{Lennard-Jones}},\ }\href {\doibase 10.1098/rspa.1951.0048} {\bibfield
|
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{journal} {\bibinfo {journal} {Proc. R. Soc. Lond. A}\ }\textbf {\bibinfo
|
||||
{volume} {205}},\ \bibinfo {pages} {541} (\bibinfo {year}
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{1951})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Hiscock}\ and\ \citenamefont
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{Thom}(2014)}]{Hiscock_2014}%
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\BibitemOpen
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@ -1,13 +1,53 @@
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-11-23 11:07:33 +0100
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%% Created for Pierre-Francois Loos at 2020-11-24 09:46:03 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@article{Hall_1951,
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abstract = { An analysis of the `linear combination of atomic orbitals' approximation using the accurate molecular orbital equations shows that it does not lead to equations of the form usually assumed in the semi-empirical molecular orbital method. A new semi-empirical method is proposed, therefore, in terms of equivalent orbitals. The equations obtained, which do have the usual form, are applicable to a large class of molecules and do not involve the approximations that were thought necessary. In this method the ionization potentials are calculated by treating certain integrals as semi-empirical parameters. The value of these parameters is discussed in terms of the localization of equivalent orbitals and some approximate rules are suggested. As an illustration the ionization potentials of the paraffin series are considered and good agreement between the observed and calculated values is found. },
|
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author = {Hall, G. G. and Lennard-Jones, John Edward},
|
||||
date-added = {2020-11-24 09:45:15 +0100},
|
||||
date-modified = {2020-11-24 09:45:50 +0100},
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||||
doi = {10.1098/rspa.1951.0048},
|
||||
journal = {Proc. R. Soc. Lond. A},
|
||||
pages = {541-552},
|
||||
title = {The molecular orbital theory of chemical valency VIII. A method of calculating ionization potentials},
|
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volume = {205},
|
||||
year = {1951},
|
||||
Bdsk-Url-1 = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1951.0048},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1098/rspa.1951.0048}}
|
||||
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@article{Roothaan_1951,
|
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author = {Roothaan, C. C. J.},
|
||||
date-added = {2020-11-24 09:43:57 +0100},
|
||||
date-modified = {2020-11-24 09:44:09 +0100},
|
||||
doi = {10.1103/RevModPhys.23.69},
|
||||
journal = {Rev. Mod. Phys.},
|
||||
pages = {69--89},
|
||||
title = {New Developments in Molecular Orbital Theory},
|
||||
volume = {23},
|
||||
year = {1951},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/RevModPhys.23.69},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/RevModPhys.23.69}}
|
||||
|
||||
@article{Slater_1951,
|
||||
author = {Slater, J. C.},
|
||||
date-added = {2020-11-24 09:42:40 +0100},
|
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date-modified = {2020-11-24 09:42:58 +0100},
|
||||
doi = {10.1103/PhysRev.82.538},
|
||||
journal = {Phys. Rev.},
|
||||
pages = {538--541},
|
||||
title = {Magnetic Effects and the Hartree-Fock Equation},
|
||||
volume = {82},
|
||||
year = {1951},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRev.82.538},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRev.82.538}}
|
||||
|
||||
@article{Loos_2019d,
|
||||
author = {P. F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},
|
||||
date-added = {2020-11-23 11:07:32 +0100},
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@ -124,7 +124,7 @@
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\newcommand{\UOX}{Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.K.}
|
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\begin{document}
|
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\title{Perturbation theory in the complex plane: Exceptional points and where to find them}
|
||||
\title{Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them}
|
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\author{Antoine \surname{Marie}}
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\affiliation{\LCPQ}
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@ -195,36 +195,36 @@ We refer the interested reader to the excellent book of Moiseyev for a general o
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\label{sec:TDSE}
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%%%%%%%%%%%%%%%%%%%%%%%
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Within the Born-Oppenheimer approximation, the exact molecular Hamiltonian with $\Ne$ electrons and
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$\Nn$ (clamped) nuclei is defined \hugh{for a given nuclear framework} as
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$\Nn$ (clamped) nuclei is defined for a given nuclear framework as
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\begin{equation}\label{eq:ExactHamiltonian}
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\hugh{\hH(\vb{R})} =
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\hH(\vb{R}) =
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- \frac{1}{2} \sum_{i}^{\Ne} \grad_i^2
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- \sum_{i}^{\Ne} \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}}
|
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+ \sum_{i<j}^{\Ne}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}},
|
||||
\end{equation}
|
||||
where $\vb{r}_i$ defines the position of the $i$-th electron, $\vb{R}_{A}$ and $Z_{A}$ are the position
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and charge of the $A$-th nucleus respectively, \hugh{and $\vb{R} = (\vb{R}_{1}, \dots, \vb{R}_{\Nn})$ is a
|
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collective vector for the nuclear positions.}
|
||||
and charge of the $A$-th nucleus respectively, and $\vb{R} = (\vb{R}_{1}, \dots, \vb{R}_{\Nn})$ is a
|
||||
collective vector for the nuclear positions.
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The first term represents the kinetic energy of the electrons, while
|
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the two following terms account for the electron-nucleus attraction and the electron-electron repulsion.
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% EXACT SCHRODINGER EQUATION
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The exact many-electron wave function \hugh{at a given nuclear geometry} $\Psi(\vb{R})$ corresponds
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The exact many-electron wave function at a given nuclear geometry $\Psi(\vb{R})$ corresponds
|
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to the solution of the (time-independent) Schr\"{o}dinger equation
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\begin{equation}
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\hugh{\hH(\vb{R})\, \Psi(\vb{R}) = E(\vb{R})\, \Psi(\vb{R}),}
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\hH(\vb{R})\, \Psi(\vb{R}) = E(\vb{R})\, \Psi(\vb{R}),
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\label{eq:SchrEq}
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\end{equation}
|
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with the eigenvalues $E(\vb{R})$ providing the exact energies.
|
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\hugh{The energy $E(\vb{R})$ can be considered as a ``one-to-many'' function since each input nuclear geometry
|
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yields several eigenvalues corresponding to the ground and excited states of the exact spectrum.}
|
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The energy $E(\vb{R})$ can be considered as a ``one-to-many'' function since each input nuclear geometry
|
||||
yields several eigenvalues corresponding to the ground and excited states of the exact spectrum.
|
||||
However, exact solutions to Eq.~\eqref{eq:SchrEq} are only possible in the simplest of systems, such as
|
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the one-electron hydrogen atom and some specific two-electron systems with well-defined mathematical
|
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properties.\cite{Taut_1993,Loos_2009b,Loos_2010e,Loos_2012}
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\hugh{In practice, approximations to the exact Schr\"{o}dinger equation must be introduced, including
|
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In practice, approximations to the exact Schr\"{o}dinger equation must be introduced, including
|
||||
the perturbation theories and Hartree--Fock approximation considered in this review
|
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In what follows, we will drop the parametric dependence on the nuclear geometry and,
|
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unless otherwise stated, atomic units will be used throughout.}
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unless otherwise stated, atomic units will be used throughout.
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%===================================%
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\subsection{Exceptional Points in the Hubbard Dimer}
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@ -249,10 +249,8 @@ unless otherwise stated, atomic units will be used throughout.}
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\end{figure*}
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To illustrate the concepts discussed throughout this article, we consider the symmetric Hubbard dimer at half filling, \ie\ with two opposite-spin fermions.
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Analytic\trashHB{ally solvable} model systems are essential in theoretical chemistry and physics as their \hugh{mathematical} simplicity \trashHB{of the
|
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mathematics} compared to realistic systems (e.g., atoms and molecules) allows new concepts and methods to be
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easily \trashHB{illustrated and} tested while retaining the key physical phenomena.
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\hugh{(HGAB: This sentence felt too long to me. Feel free to re-instate words if you think they are neccessary)}
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Analytically solvable models are essential in theoretical chemistry and physics as their mathematical simplicity compared to realistic systems (e.g., atoms and molecules) allows new concepts and methods to be
|
||||
easily tested while retaining the key physical phenomena.
|
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Using the (localised) site basis, the Hilbert space of the Hubbard dimer comprises the four configurations
|
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\begin{align}
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@ -274,12 +272,11 @@ where $t$ is the hopping parameter and $U$ is the on-site Coulomb repulsion.
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We refer the interested reader to Refs.~\onlinecite{Carrascal_2015,Carrascal_2018} for more details about this system.
|
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The parameter $U$ controls the strength of the electron correlation.
|
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In the weak correlation regime (small $U$), the kinetic energy dominates and the electrons are delocalised over both sites.
|
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In the large-$U$ (or strong correlation) regime, the electron repulsion term \hugh{becomes dominant} \trashHB{drives the physics}
|
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In the large-$U$ (or strong correlation) regime, the electron repulsion term becomes dominant
|
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and the electrons localise on opposite sites to minimise their Coulomb repulsion.
|
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This phenomenon is often referred to as Wigner crystallisation. \cite{Wigner_1934}
|
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|
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To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$
|
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\hugh{to give the parameterised Hamiltonian $\hH(\lambda)$.}
|
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To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$ to give the parameterised Hamiltonian $\hH(\lambda)$.
|
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When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies
|
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\begin{subequations}
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\begin{align}
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@ -321,7 +318,7 @@ As a result, completely encircling an EP leads to the interconversion of the two
|
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Additionally, the wave functions pick up a geometric phase in the process, and four complete loops are required to recover their starting forms.\cite{MoiseyevBook}
|
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|
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% LOCATING EPS
|
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\hugh{To locate EPs in practice, one must simultaneously solve
|
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To locate EPs in practice, one must simultaneously solve
|
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\begin{subequations}
|
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\begin{align}
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\label{eq:PolChar}
|
||||
@ -336,15 +333,15 @@ Equation \eqref{eq:PolChar} is the well-known secular equation providing the (ei
|
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If the energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy value is at least two-fold degenerate.
|
||||
These degeneracies can be conical intersections between two states with different symmetries
|
||||
for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the
|
||||
same symmetry for complex values of $\lambda$.}
|
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same symmetry for complex values of $\lambda$.
|
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|
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|
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%============================================================%
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\subsection{Rayleigh-Schr\"odinger Perturbation Theory}
|
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%============================================================%
|
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|
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\hugh{One of the most common routes to approximately solving the Schr\"odinger equation
|
||||
is to introduce a perturbative expansion of the exact energy.}
|
||||
One of the most common routes to approximately solving the Schr\"odinger equation
|
||||
is to introduce a perturbative expansion of the exact energy.
|
||||
% SUMMARY OF RS-PT
|
||||
Within Rayleigh-Schr\"odinger perturbation theory, the time-independent Schr\"odinger equation
|
||||
is recast as
|
||||
@ -401,10 +398,12 @@ However, this series diverges for $x \ge 1$.
|
||||
This divergence occurs because $f(x)$ has four singularities in the complex
|
||||
($\e^{\i\pi/4}$, $\e^{-\i\pi/4}$, $\e^{\i3\pi/4}$, and $\e^{-\i3\pi/4}$) with a modulus equal to $1$, demonstrating
|
||||
that complex singularities are essential to fully understand the series convergence on the real axis.\cite{BenderBook}
|
||||
\titou{Include Antoine's example $\sum_{n=1}^\infty \lambda^n/n$ which is divergent at $\lambda = 1$ but convergent at $\lambda = -1$.}
|
||||
|
||||
The radius of convergence for the perturbation series Eq.~\eqref{eq:E_expansion} is therefore dictated by the magnitude \titou{$\abs{\lambda_0}$} of the
|
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The radius of convergence for the perturbation series Eq.~\eqref{eq:E_expansion} is therefore dictated by the magnitude $\abs{\lambda_c}$ of the
|
||||
singularity in $E(\lambda)$ that is closest to the origin.
|
||||
Note that when $\lambda = \lambda_c$, one cannot \textit{a priori} predict if the series is convergent or not.
|
||||
For example, the series $\sum_{k=1}^\infty \lambda^k/k$ diverges at $\lambda = 1$ but converges at $\lambda = -1$.
|
||||
|
||||
Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lambda)$ represents
|
||||
a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states.
|
||||
The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
|
||||
@ -423,7 +422,7 @@ This Slater determinant is defined as an antisymmetric combination of $\Ne$ (rea
|
||||
\begin{equation}\label{eq:FockOp}
|
||||
\Hat{f}(\vb{x}) \phi_p(\vb{x}) = \qty[ \Hat{h}(\vb{x}) + \Hat{v}_\text{HF}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}).
|
||||
\end{equation}
|
||||
Here the \hugh{(one-electron)} core Hamiltonian is
|
||||
Here the (one-electron) core Hamiltonian is
|
||||
\begin{equation}
|
||||
\label{eq:Hcore}
|
||||
\Hat{h}(\vb{x}) = -\frac{\grad^2}{2} + \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}-\vb{R}_A}}
|
||||
@ -566,8 +565,8 @@ Time-reversal symmetry dictates that this UHF wave function must be degenerate w
|
||||
by swapping $\ta^{\text{UHF}}$ and $\tb^{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf} and \eqref{eq:tb_uhf}.
|
||||
This type of symmetry breaking is also called a spin-density wave in the physics community as the system
|
||||
``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook}
|
||||
\hugh{Symmetry breaking can also occur in RHF theory when a charge-density wave is formed from an oscillation
|
||||
between the two closed-shell configurations with both electrons localised on one site or the other.\cite{StuberPaldus,Fukutome_1981}}
|
||||
Symmetry breaking can also occur in RHF theory when a charge-density wave is formed from an oscillation
|
||||
between the two closed-shell configurations with both electrons localised on one site or the other.\cite{StuberPaldus,Fukutome_1981}
|
||||
|
||||
%============================================================%
|
||||
\subsection{Self-Consistency as a Perturbation} %OR {Complex adiabatic connection}
|
||||
@ -575,12 +574,12 @@ between the two closed-shell configurations with both electrons localised on one
|
||||
|
||||
% INTRODUCE PARAMETRISED FOCK HAMILTONIAN
|
||||
The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency
|
||||
in the HF approximation, and is usually solved through an iterative approach.\cite{Roothaan1951,Hall1951}
|
||||
in the HF approximation, and is usually solved through an iterative approach.\cite{Roothaan_1951,Hall_1951}
|
||||
Alternatively, the non-linear terms arising from the Coulomb and exchange operators can
|
||||
be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the
|
||||
transformation $U \rightarrow \lambda\, U$, giving the parametrised Fock operator
|
||||
\begin{equation}
|
||||
\Hat{f}(\vb{x} \hugh{; \lambda}) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}).
|
||||
\Hat{f}(\vb{x} ; \lambda) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}).
|
||||
\end{equation}
|
||||
The orbitals in the reference problem $\lambda=0$ correspond to the symmetry-pure eigenfunctions of the one-electron core
|
||||
Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the true HF solution.
|
||||
@ -700,9 +699,9 @@ i) the Epstein-Nesbet (EN) partitioning which consists in taking the diagonal el
|
||||
%Hence, the off-diagonal elements of $\hH$ are the perturbation operator,
|
||||
ii) the weak correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\hH^{(0)}$ and the two-electron part is the perturbation operator $\hV$, and
|
||||
iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018}
|
||||
\hugh{While an in-depth comparison of these different approaches can offer insight into
|
||||
While an in-depth comparison of these different approaches can offer insight into
|
||||
their relative strengths and weaknesses for various situations, we will restrict our current discussion
|
||||
to the convergence properties of the MP expansion.}
|
||||
to the convergence properties of the MP expansion.
|
||||
|
||||
%=====================================================%
|
||||
\subsection{M{\o}ller-Plesset Convergence in Molecular Systems}
|
||||
@ -955,7 +954,6 @@ To do so, they analysed the relation between the dominant singularity (\ie, the
|
||||
\begin{quote}
|
||||
\textit{``In the limit of large order, the series coefficients become equivalent to the Taylor series coefficients of the singularity closest to the origin. ''}
|
||||
\end{quote}
|
||||
\titou{T2: should we move this theorem earlier?}
|
||||
Following the result of this theorem, the convergence patterns of the MP series can be explained by looking at the dominant singularity.
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A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative).
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@ -1042,7 +1040,7 @@ To the best of our knowledge, the effect of bond stretching on singularities, it
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%====================================================
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In the previous section, we saw that a careful analysis of the structure of the Hamiltonian allows us to predict the existence of a critical point.
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In a finite basis set this critical point is model by a cluster of $\beta$ singularities.
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In a finite basis set, this critical point is model by a cluster of $\beta$ singularities.
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It is now well known that this phenomenon is a special case of a more general phenomenon.
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Indeed, theoretical physicists proved that EPs close to the real axis are connected to \textit{quantum phase transitions} (QPTs). \cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook}
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In quantum mechanics, the Hamiltonian is almost always dependent of, at least, one parameter.
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@ -1050,7 +1048,7 @@ In some cases the variation of a parameter can lead to abrupt changes at a criti
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These QPTs exist both for ground and excited states as shown by Cejnar and coworkers. \cite{Cejnar_2005,Cejnar_2007,Caprio_2008,Cejnar_2009,Sachdev_2011,Cejnar_2015,Cejnar_2016, Macek_2019,Cejnar_2020}
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A ground-state QPT is characterised by the derivatives of the ground-state energy with respect to a non-thermal control parameter. \cite{Cejnar_2009, Sachdev_2011}
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The transition is called discontinuous and of first order if the first derivative is discontinuous at the critical parameter value.
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Otherwise, it is called continuous and of $m$th order if the $m$th derivative is discontinuous.
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Otherwise, it is called continuous and of $m$th order (with $m \ge 2$) if the $m$th derivative is discontinuous.
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A QPT can also be identify by the discontinuity of an appropriate order parameter (or one of its derivatives).
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The presence of an EP close to the real axis is characteristic of a sharp avoided crossing.
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@ -1060,7 +1058,7 @@ One of the major obstacles that one faces in order to achieve this resides in th
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The numerical assignment of an EP to two energies on the real axis is very difficult in large dimensions.
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Hence, the design of specific methods are required to get information on the location of EPs.
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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}
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More recently Stransky and coworkers proved that the distribution of EPs is characteristic on the order of the QPT. \cite{Stransky_2018}
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More recently Stransky and coworkers proved that the distribution of EPs is characteristic of the QPT order. \cite{Stransky_2018}
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In the thermodynamic limit, some of the EPs converge towards a critical point $\lambda_\text{c}$ on the real axis.
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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.
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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.
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@ -1073,6 +1071,7 @@ For example, without interaction the ground state is the spherical phase (a cond
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In particular, we see that the transition from the spherical phase to the axially symmetric one is analog to the symmetry breaking of the wave function of the hydrogen molecule when the bond is stretched. \cite{SzaboBook}
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It seems like our understanding of the physics of spatial and/or spin symmetry breaking in HF theory can be enlightened by QPT theory.
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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.
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Moreover, the $\beta$ singularities introduced by Sergeev and coworkers to describe the EPs modelling the formation of a bound cluster of electrons are actually a more general class of singularities.
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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.
|
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However, the $\alpha$ singularities arise from large avoided crossings.
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|
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