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minor corrections in QPT

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Pierre-Francois Loos 2020-11-24 09:25:43 +01:00
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12
Manuscript/EPAWTFT.bbl
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@ -530,12 +530,6 @@
{Wigner}},\ }\href {\doibase 10.1103/PhysRev.46.1002} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {46}},\
\bibinfo {pages} {1002} (\bibinfo {year} {1934})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Goodson}(2011)}]{Goodson_2011}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
{Goodson}},\ }\href {\doibase 10.1002/wcms.92} {\bibfield {journal}
{\bibinfo {journal} {{WIREs} Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2007)\citenamefont {Cejnar},
\citenamefont {Heinze},\ and\ \citenamefont {Macek}}]{Cejnar_2007}%
\BibitemOpen
@ -545,6 +539,12 @@
{\doibase 10.1103/PhysRevLett.99.100601} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo
{pages} {100601} (\bibinfo {year} {2007})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Goodson}(2011)}]{Goodson_2011}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
{Goodson}},\ }\href {\doibase 10.1002/wcms.92} {\bibfield {journal}
{\bibinfo {journal} {{WIREs} Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Szabo}\ and\ \citenamefont
{Ostlund}(1989)}]{SzaboBook}%
\BibitemOpen

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Manuscript/EPAWTFT.tex
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@ -1015,7 +1015,7 @@ To the best of our knowledge, the effect of bond stretching on singularities, it
%====================================================
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.
In a finite basis set this critical point is model by a cluster of $\beta$ singularities.
In a finite basis set, this critical point is model by a cluster of $\beta$ singularities.
It is now well known that this phenomenon is a special case of a more general phenomenon.
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}
In quantum mechanics, the Hamiltonian is almost always dependent of, at least, one parameter.
@ -1023,7 +1023,7 @@ In some cases the variation of a parameter can lead to abrupt changes at a criti
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}
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}
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 $m$th order if the $m$th derivative is discontinuous.
Otherwise, it is called continuous and of $m$th order (with $m \ge 2$) if the $m$th derivative is discontinuous.
A QPT can also be identify by the discontinuity of an appropriate order parameter (or one of its derivatives).
The presence of an EP close to the real axis is characteristic of a sharp avoided crossing.
@ -1033,7 +1033,7 @@ One of the major obstacles that one faces in order to achieve this resides in th
The numerical assignment of an EP to two energies on the real axis is very difficult in large dimensions.
Hence, the design of specific methods are required to get information on the location of EPs.
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}
More recently Stransky and coworkers proved that the distribution of EPs is characteristic on the order of the QPT. \cite{Stransky_2018}
More recently Stransky and coworkers proved that the distribution of EPs is characteristic of the QPT order. \cite{Stransky_2018}
In the thermodynamic limit, some of the EPs converge towards a critical point $\lambda_\text{c}$ on the real axis.
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.
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.
@ -1046,6 +1046,7 @@ For example, without interaction the ground state is the spherical phase (a cond
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}
It seems like our understanding of the physics of spatial and/or spin symmetry breaking in HF theory can be enlightened by QPT theory.
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.
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.
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.
However, the $\alpha$ singularities arise from large avoided crossings.