review Hugh changes
This commit is contained in:
parent
668b2f1bdb
commit
cf76131ece
@ -6,7 +6,7 @@
|
||||
%Control: page (0) single
|
||||
%Control: year (1) truncated
|
||||
%Control: production of eprint (0) enabled
|
||||
\begin{thebibliography}{131}%
|
||||
\begin{thebibliography}{132}%
|
||||
\makeatletter
|
||||
\providecommand \@ifxundefined [1]{%
|
||||
\@ifx{#1\undefined}
|
||||
@ -1188,6 +1188,15 @@
|
||||
{journal} {\bibinfo {journal} {J. Phys. A: Math. Theor.}\ }\textbf {\bibinfo
|
||||
{volume} {40}},\ \bibinfo {pages} {581} (\bibinfo {year} {2007})}\BibitemShut
|
||||
{NoStop}%
|
||||
\bibitem [{\citenamefont {Kais}\ \emph {et~al.}(2006)\citenamefont {Kais},
|
||||
\citenamefont {Wenger},\ and\ \citenamefont {Wei}}]{Kais_2006}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
||||
{Kais}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Wenger}}, \ and\
|
||||
\bibinfo {author} {\bibfnamefont {Q.}~\bibnamefont {Wei}},\ }\href {\doibase
|
||||
https://doi.org/10.1016/j.cplett.2006.03.035} {\bibfield {journal} {\bibinfo
|
||||
{journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {423}},\ \bibinfo
|
||||
{pages} {45 } (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Goodson}(2019)}]{Goodson_2019}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
|
||||
|
@ -1142,12 +1142,11 @@ regardless of whether this point is inside or outside the complex unit circle. \
|
||||
%=======================================
|
||||
|
||||
% STILLINGER INTRODUCES THE CRITICAL POINT
|
||||
\hugh{%
|
||||
In the early 2000's, Stillinger reconsidered the mathematical origin behind the divergent series with odd-even
|
||||
sign alternation.\cite{Stillinger_2000}
|
||||
This type of convergence behaviour corresponds to Cremer and He's class B systems with closely spaced
|
||||
electron pairs and includes \ce{Ne}, \ce{HF}, \ce{F-}, and \ce{H2O}.\cite{Cremer_1996}
|
||||
Stillinger proposed that the divergence of these series occurs arise from a dominant singularity
|
||||
Stillinger proposed that the divergence of these series arise from a dominant singularity
|
||||
on the negative real $\lambda$ axis, corresponding to a multielectron autoionisation threshold.\cite{Stillinger_2000}
|
||||
To understand Stillinger's argument, consider the paramterised MP Hamiltonian in the form
|
||||
\begin{multline}
|
||||
@ -1171,12 +1170,10 @@ This autoionisation effect is closely related to the critial point for electron
|
||||
atoms (see Ref.~\onlinecite{Baker_1971}).
|
||||
Furthermore, a similar set of critical points exists along the positive real axis, corresponding to single-electron ionisation
|
||||
processes.\cite{Sergeev_2005}
|
||||
}
|
||||
|
||||
% CLASSIFICATIONS BY GOODSOON AND SERGEEV
|
||||
\hugh{%
|
||||
To further develop the link between the critical point and types of MP convergence, Sergeev and Goodson investigated
|
||||
the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004}.
|
||||
the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004}
|
||||
They demonstrated that the dominant singularity in class A corresponds to a dominant EP with a positive real component,
|
||||
with the magnitude of the imaginary component controlling the oscillations in the signs of the MP
|
||||
term.\cite{Goodson_2000a,Goodson_2000b}
|
||||
@ -1187,10 +1184,8 @@ HF potential $v^{\text{HF}}$ is relatively localised and the autoionization is f
|
||||
$\lambda$ values closer to the origin.
|
||||
With these insights, they regrouped the systems into new classes: i) $\alpha$ singularities which have ``large'' imaginary parts,
|
||||
and ii) $\beta$ singularities which have very small imaginary parts.\cite{Goodson_2004,Sergeev_2006}
|
||||
}
|
||||
|
||||
% RELATIONSHIP TO BASIS SET SIZE
|
||||
\hugh{%
|
||||
The existence of the MP critical point can also explain why the divergence observed by Olsen \etal\ in the \ce{Ne} atom
|
||||
and \ce{HF} molecule occured when diffuse basis functions were included.\cite{Olsen_1996}
|
||||
Clearly diffuse basis functions are required for the electrons to dissociate from the nuclei, and indeed using
|
||||
@ -1200,10 +1195,9 @@ by a cluster of sharp avoided crossings between the ground state and high-lying
|
||||
Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' atom also
|
||||
allows the formation of the critical point as the electrons form a bound cluster occuping the ghost atom orbitals.\cite{Sergeev_2005}
|
||||
This effect explains the origin of the divergence in the \ce{HF} molecule as the \ce{F} valence electrons jump to \ce{H}
|
||||
for a sufficiently negative $\lambda$.\cite{Sergeev_2005}
|
||||
for a sufficiently negative $\lambda$ value.\cite{Sergeev_2005}
|
||||
Furthermore, the two-state model of Olsen \etal{}\cite{Olsen_2000} was simply too minimal to understand the complexity of
|
||||
divergences caused by the MP critical point.
|
||||
}
|
||||
|
||||
% BASIS SET DEPENDENCE (INCLUDE?)
|
||||
%Finally, it was shown that $\beta$ singularities are very sensitive to changes in the basis set but not to the bond stretching.
|
||||
@ -1223,21 +1217,19 @@ divergences caused by the MP critical point.
|
||||
%This explains that Olsen \textit{et al.}, because they used a simple two-state model, only observed the first singularity of this cluster of singularities causing the divergence. \cite{Olsen_2000}
|
||||
|
||||
% RELATIONSHIP TO QUANTUM PHASE TRANSITION
|
||||
\hugh{%
|
||||
When a Hamiltonian is parametrised by a variable such as $\lambda$, the existence of abrupt changes in the
|
||||
eigenstates as a function of $\lambda$ indicate the presence of a zero-temperature quantum phase transition (QPT).%
|
||||
\cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook}
|
||||
As an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis.
|
||||
As an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis, eventually forming a critical point.
|
||||
The existence of an EP \emph{on} the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a}
|
||||
Since the MP critical point corresponds to a singularity on the real $\lambda$ axis, it can immediately be
|
||||
recognised a QPT within the perturbation theory approximation.
|
||||
However, a conventional QPT can only occur in the thermodynamic limit, which here is analogous to the complete
|
||||
basis set limit.\cite{Kais_2006}
|
||||
basis set limit in our case.\cite{Kais_2006}
|
||||
The MP critical point $\beta$ singularity in a finite basis must therefore be modelled by pairs of EPs
|
||||
that tend towards the real axis, exactly as described by Sergeev \etal.\cite{Sergeev_2005}
|
||||
that tend towards the real axis, exactly as described by Sergeev \etal\cite{Sergeev_2005}
|
||||
In contrast, $\alpha$ singularities correspond to large avoided crossings that are indicative of low-lying excited
|
||||
states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT.
|
||||
}
|
||||
|
||||
%=======================================
|
||||
\subsection{Critical Point in the Hubbard Dimer}
|
||||
|
Loading…
Reference in New Issue
Block a user