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Manuscript/EPAWTFT.bbl
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@ -710,6 +710,61 @@
10.1140/epjb/e2018-90114-9} {\bibfield {journal} {\bibinfo {journal} {Eur. 10.1140/epjb/e2018-90114-9} {\bibfield {journal} {\bibinfo {journal} {Eur.
Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {142} Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {142}
(\bibinfo {year} {2018})}\BibitemShut {NoStop}% (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Heiss}\ and\ \citenamefont
{M{\"u}ller}(2002)}]{Heiss_2002}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~D.}\ \bibnamefont
{Heiss}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{M{\"u}ller}},\ }\href {\doibase 10.1103/PhysRevE.66.016217} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. E}\ }\textbf {\bibinfo {volume}
{66}},\ \bibinfo {pages} {016217} (\bibinfo {year} {2002})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Borisov}\ \emph {et~al.}(2015)\citenamefont
{Borisov}, \citenamefont {Ru{\v z}i{\v c}ka},\ and\ \citenamefont
{Znojil}}]{Borisov_2015}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~I.}\ \bibnamefont
{Borisov}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Ru{\v z}i{\v
c}ka}}, \ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Znojil}},\
}\href {\doibase 10.1007/s10773-014-2493-y} {\bibfield {journal} {\bibinfo
{journal} {Int. J. Theor. Phys.}\ }\textbf {\bibinfo {volume} {54}},\
\bibinfo {pages} {4293} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {{\v S}indelka}\ \emph {et~al.}(2017)\citenamefont
{{\v S}indelka}, \citenamefont {Santos},\ and\ \citenamefont
{Moiseyev}}]{Sindelka_2017}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {{\v
S}indelka}}, \bibinfo {author} {\bibfnamefont {L.~F.}\ \bibnamefont
{Santos}}, \ and\ \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Moiseyev}},\ }\href {\doibase 10.1103/PhysRevA.95.010103} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
{95}},\ \bibinfo {pages} {010103} (\bibinfo {year} {2017})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Carr}(2010)}]{CarrBook}%
\BibitemOpen
\bibinfo {editor} {\bibfnamefont {L.}~\bibnamefont {Carr}},\ ed.,\ \href
{\doibase 10.1201/b10273} {\emph {\bibinfo {title} {Understanding Quantum
Phase Transitions}}}\ (\bibinfo {publisher} {Boca Raton: CRC Press},\
\bibinfo {year} {2010})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Vojta}(2003)}]{Vojta_2003}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Vojta}},\ }\href {\doibase 10.1088/0034-4885/66/12/r01} {\bibfield
{journal} {\bibinfo {journal} {Rep. Prog. Phys.}\ }\textbf {\bibinfo
{volume} {66}},\ \bibinfo {pages} {2069} (\bibinfo {year}
{2003})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sachdev}(1999)}]{SachdevBook}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Sachdev}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Phase
Transitions}}}\ (\bibinfo {publisher} {Cambridge University Press},\
\bibinfo {year} {1999})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gilmore}(1981)}]{GilmoreBook}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Gilmore}},\ }\href@noop {} {\emph {\bibinfo {title} {Catastrophe Theory for
Scientists and Engineers}}}\ (\bibinfo {publisher} {New York, Wiley},\
\bibinfo {year} {1981})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2007)\citenamefont {Cejnar}, \bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2007)\citenamefont {Cejnar},
\citenamefont {Heinze},\ and\ \citenamefont {Macek}}]{Cejnar_2007}% \citenamefont {Heinze},\ and\ \citenamefont {Macek}}]{Cejnar_2007}%
\BibitemOpen \BibitemOpen
@ -1225,61 +1280,6 @@
{Goodson}},\ }\href {\doibase 10.1063/1.2173989} {\bibfield {journal} {Goodson}},\ }\href {\doibase 10.1063/1.2173989} {\bibfield {journal}
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {124}},\ {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {124}},\
\bibinfo {pages} {094111} (\bibinfo {year} {2006})}\BibitemShut {NoStop}% \bibinfo {pages} {094111} (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Heiss}\ and\ \citenamefont
{M{\"u}ller}(2002)}]{Heiss_2002}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~D.}\ \bibnamefont
{Heiss}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{M{\"u}ller}},\ }\href {\doibase 10.1103/PhysRevE.66.016217} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. E}\ }\textbf {\bibinfo {volume}
{66}},\ \bibinfo {pages} {016217} (\bibinfo {year} {2002})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Borisov}\ \emph {et~al.}(2015)\citenamefont
{Borisov}, \citenamefont {Ru{\v z}i{\v c}ka},\ and\ \citenamefont
{Znojil}}]{Borisov_2015}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~I.}\ \bibnamefont
{Borisov}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Ru{\v z}i{\v
c}ka}}, \ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Znojil}},\
}\href {\doibase 10.1007/s10773-014-2493-y} {\bibfield {journal} {\bibinfo
{journal} {Int. J. Theor. Phys.}\ }\textbf {\bibinfo {volume} {54}},\
\bibinfo {pages} {4293} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {{\v S}indelka}\ \emph {et~al.}(2017)\citenamefont
{{\v S}indelka}, \citenamefont {Santos},\ and\ \citenamefont
{Moiseyev}}]{Sindelka_2017}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {{\v
S}indelka}}, \bibinfo {author} {\bibfnamefont {L.~F.}\ \bibnamefont
{Santos}}, \ and\ \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Moiseyev}},\ }\href {\doibase 10.1103/PhysRevA.95.010103} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
{95}},\ \bibinfo {pages} {010103} (\bibinfo {year} {2017})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Carr}(2010)}]{CarrBook}%
\BibitemOpen
\bibinfo {editor} {\bibfnamefont {L.}~\bibnamefont {Carr}},\ ed.,\ \href
{\doibase 10.1201/b10273} {\emph {\bibinfo {title} {Understanding Quantum
Phase Transitions}}}\ (\bibinfo {publisher} {Boca Raton: CRC Press},\
\bibinfo {year} {2010})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Vojta}(2003)}]{Vojta_2003}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Vojta}},\ }\href {\doibase 10.1088/0034-4885/66/12/r01} {\bibfield
{journal} {\bibinfo {journal} {Rep. Prog. Phys.}\ }\textbf {\bibinfo
{volume} {66}},\ \bibinfo {pages} {2069} (\bibinfo {year}
{2003})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sachdev}(1999)}]{SachdevBook}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Sachdev}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Phase
Transitions}}}\ (\bibinfo {publisher} {Cambridge University Press},\
\bibinfo {year} {1999})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gilmore}(1981)}]{GilmoreBook}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Gilmore}},\ }\href@noop {} {\emph {\bibinfo {title} {Catastrophe Theory for
Scientists and Engineers}}}\ (\bibinfo {publisher} {New York, Wiley},\
\bibinfo {year} {1981})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2005)\citenamefont {Cejnar}, \bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2005)\citenamefont {Cejnar},
\citenamefont {Heinze},\ and\ \citenamefont {Dobe{\v s}}}]{Cejnar_2005}% \citenamefont {Heinze},\ and\ \citenamefont {Dobe{\v s}}}]{Cejnar_2005}%
\BibitemOpen \BibitemOpen

136
Manuscript/EPAWTFT.tex
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@ -191,7 +191,9 @@ Non-Hermitian Hamiltonians already have a long history in quantum chemistry and
describe metastable resonance phenomena.\cite{MoiseyevBook} describe metastable resonance phenomena.\cite{MoiseyevBook}
Through the methods of complex-scaling\cite{Moiseyev_1998} and complex absorbing Through the methods of complex-scaling\cite{Moiseyev_1998} and complex absorbing
potentials,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonances can be stabilised as square-integrable potentials,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonances can be stabilised as square-integrable
wave functions with a complex energy that allows the resonance energy and lifetime to be computed. wave functions.
\hugh{In these situations, the energy becomes complex-valued, with the real and imaginary components allowing
the resonance energy and lifetime to be computed respectively.}
We refer the interested reader to the excellent book by Moiseyev for a general overview. \cite{MoiseyevBook} We refer the interested reader to the excellent book by Moiseyev for a general overview. \cite{MoiseyevBook}
% EXCEPTIONAL POINTS % EXCEPTIONAL POINTS
@ -297,7 +299,8 @@ unless otherwise stated, atomic units will be used throughout.
\end{subfigure} \end{subfigure}
\caption{% \caption{%
Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}). Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}).
Only the interacting closed-shell singlets are shown in the complex plane, becoming degenerate at the EP (black dot). Only the \hugh{real component of the} interacting closed-shell singlet \hugh{energies} are shown in the complex plane,
becoming degenerate at the EP (black dot).
Following a contour around the EP (black solid) interchanges the states, while a second rotation (black dashed) Following a contour around the EP (black solid) interchanges the states, while a second rotation (black dashed)
returns the states to their original energies. returns the states to their original energies.
\label{fig:FCI}} \label{fig:FCI}}
@ -344,6 +347,12 @@ E_{\text{S}} &= U.
\end{align} \end{align}
\end{subequations} \end{subequations}
While the open-shell triplet ($E_{\text{T}}$) and singlet ($E_{\text{S}}$) are independent of $\lambda$, the closed-shell singlet ground state ($E_{-}$) and doubly-excited state ($E_{+}$) couple strongly to form an avoided crossing at $\lambda=0$ (see Fig.~\ref{subfig:FCI_real}). While the open-shell triplet ($E_{\text{T}}$) and singlet ($E_{\text{S}}$) are independent of $\lambda$, the closed-shell singlet ground state ($E_{-}$) and doubly-excited state ($E_{+}$) couple strongly to form an avoided crossing at $\lambda=0$ (see Fig.~\ref{subfig:FCI_real}).
\hugh{In contrast, when $\lambda$ is complex, the energies may become complex-valued, with the real components shown in
Fig.~\ref{subfig:FCI_cplx}.
Although the imaginary component of the energy is linked to resonance lifetimes elsewhere in non-Hermitian
quantum mechanics, \cite{MoiseyevBook} its physical interpretation in the current context is unclear.
Throughout this work, we will generally consider and plot only the real component of any complex-valued energies.
}
At non-zero values of $U$ and $t$, these closed-shell singlets can only become degenerate at a pair of complex conjugate points in the complex $\lambda$ plane At non-zero values of $U$ and $t$, these closed-shell singlets can only become degenerate at a pair of complex conjugate points in the complex $\lambda$ plane
\begin{equation} \begin{equation}
@ -355,11 +364,18 @@ with energy
E_\text{EP} = \frac{U}{2}. E_\text{EP} = \frac{U}{2}.
\end{equation} \end{equation}
These $\lambda$ values correspond to so-called EPs and connect the ground and excited states in the complex plane. These $\lambda$ values correspond to so-called EPs and connect the ground and excited states in the complex plane.
Crucially, the energy surface becomes non-analytic at $\lambda_{\text{EP}}$ and a square-root singularity forms with two branch cuts running along the imaginary axis from $\lambda_{\text{EP}}$ to $\pm \i \infty$ (see Fig.~\ref{subfig:FCI_cplx}). \hugh{Crucially, the ground- and excited-state wave functions at an EP become \emph{identical} rather than just degenerate.}
Furthermore, the energy surface becomes non-analytic at $\lambda_{\text{EP}}$ and a square-root singularity forms with two branch cuts running along the imaginary axis from $\lambda_{\text{EP}}$ to $\pm \i \infty$ (see Fig.~\ref{subfig:FCI_cplx}).
\hugh{Along these branch cuts, the real components of the energies are equivalent and appear to give a seam
of intersection, but a strict degeneracy is avoided because the imaginary components are different.}
On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda = \Re(\lambda_{\text{EP}})$. On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda = \Re(\lambda_{\text{EP}})$.
The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction. The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction.
In the limit $U/t \to 0$, the two EPs converge at $\lep = 0$ to create a conical intersection with In the limit $U/t \to 0$, the two EPs converge at $\lep = 0$ to create a conical intersection with
a gradient discontinuity on the real axis. a gradient discontinuity on the real axis.
\hugh{This gradient discontinuity defines a critical point in the ground-state energy,
where a sudden change occurs in the electronic wave function, and can be considered as a zero-temperature quantum phase transition.}
\cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook}
Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states (see Fig.~\ref{subfig:FCI_cplx}). Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states (see Fig.~\ref{subfig:FCI_cplx}).
This behaviour can be seen by expanding the radicand in Eq.~\eqref{eq:singletE} as a Taylor series around $\lambda_{\text{EP}}$ to give This behaviour can be seen by expanding the radicand in Eq.~\eqref{eq:singletE} as a Taylor series around $\lambda_{\text{EP}}$ to give
@ -392,8 +408,6 @@ These degeneracies can be conical intersections between two states with differen
for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the
same symmetry for complex values of $\lambda$. same symmetry for complex values of $\lambda$.
\titou{Although the complex component of the energy is linked to lifetime in non-Hermitian quantum mechanics, \cite{MoiseyevBook} its meaning is unclear to us in the present context.}
%============================================================% %============================================================%
\subsection{Rayleigh--Schr\"odinger Perturbation Theory} \subsection{Rayleigh--Schr\"odinger Perturbation Theory}
%============================================================% %============================================================%
@ -448,7 +462,7 @@ As a result, the radius of convergence for a function is equal to the distance f
in the complex plane, referred to as the ``dominant'' singularity. in the complex plane, referred to as the ``dominant'' singularity.
This singularity may represent a pole of the function, or a branch point (\eg, square-root or logarithmic) This singularity may represent a pole of the function, or a branch point (\eg, square-root or logarithmic)
in a multi-valued function. in a multi-valued function.
\titou{T2: define here critical point.}
For example, the simple function For example, the simple function
\begin{equation} \label{eq:DivExample} \begin{equation} \label{eq:DivExample}
@ -470,6 +484,12 @@ Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lam
a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states. 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 The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
$\lambda$ plane where two states become degenerate. $\lambda$ plane where two states become degenerate.
\hugh{Additional singularities can also arise at critical points of the energy.
A critical point corresponds to the intersection of two energy surfaces
where the eigenstates remain distinct but a gradient discontinuity occurs in
the ground-state energy.
In contrast, at a square-root branch point, both the energies and the associated wave functions
of the intersecting surfaces become identical.}
Later we will demonstrate how the choice of reference Hamiltonian controls the position of these EPs, and Later we will demonstrate how the choice of reference Hamiltonian controls the position of these EPs, and
ultimately determines the convergence properties of the perturbation series. ultimately determines the convergence properties of the perturbation series.
@ -574,7 +594,7 @@ In the weak correlation regime $0 \le U \le 2t$, the angles which minimise the H
\begin{equation} \begin{equation}
\ta_\text{RHF} = \tb_\text{RHF} = \pi/2, \ta_\text{RHF} = \tb_\text{RHF} = \pi/2,
\end{equation} \end{equation}
giving the symmetry-pure molecular orbitals giving the molecular orbitals
\begin{align} \begin{align}
\titou{\psi_{1,\text{RHF}}^{\sigma}} & = \frac{\Lsi + \Rsi}{\sqrt{2}}, \titou{\psi_{1,\text{RHF}}^{\sigma}} & = \frac{\Lsi + \Rsi}{\sqrt{2}},
& &
@ -584,6 +604,10 @@ and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
\begin{equation} \begin{equation}
E_\text{RHF} \equiv E_\text{HF}(\ta_\text{RHF}, \tb_\text{RHF}) = -2t + \frac{U}{2}. E_\text{RHF} \equiv E_\text{HF}(\ta_\text{RHF}, \tb_\text{RHF}) = -2t + \frac{U}{2}.
\end{equation} \end{equation}
\hugh{Here, the molecular orbitals respectively transform
according to the $\Sigma_\text{g}^{+}$ and $\Sigma_\text{u}^{+}$ irreducible representations of
the $D_{\infty \text{h}}$ point group that represents the symmetric Hubbard dimer.
We can therefore consider these as symmetry-pure molecular orbitals.}
However, in the strongly correlated regime $U>2t$, the closed-shell orbital restriction prevents RHF from However, in the strongly correlated regime $U>2t$, the closed-shell orbital restriction prevents RHF from
modelling the correct physics with the two electrons on opposite sites. modelling the correct physics with the two electrons on opposite sites.
@ -610,7 +634,7 @@ modelling the correct physics with the two electrons on opposite sites.
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
As the on-site repulsion is increased from 0, the HF approximation reaches a critical value at $U=2t$ where a symmetry-broken As the on-site repulsion is increased from 0, the HF approximation reaches a critical value at $U=2t$ where an alternative
UHF solution appears with a lower energy than the RHF one. UHF solution appears with a lower energy than the RHF one.
Note that the RHF wave function remains a genuine solution of the HF equations for $U \ge 2t$, but corresponds to a saddle point Note that the RHF wave function remains a genuine solution of the HF equations for $U \ge 2t$, but corresponds to a saddle point
of the HF energy rather than a minimum. of the HF energy rather than a minimum.
@ -629,11 +653,17 @@ with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
\begin{equation} \begin{equation}
E_\text{UHF} \equiv E_\text{HF}(\ta_\text{UHF}, \tb_\text{UHF}) = - \frac{2t^2}{U}. E_\text{UHF} \equiv E_\text{HF}(\ta_\text{UHF}, \tb_\text{UHF}) = - \frac{2t^2}{U}.
\end{equation} \end{equation}
Time-reversal symmetry dictates that this UHF wave function must be degenerate with its spin-flipped counterpart, obtained
\hugh{The molecular orbitals of the lower-energy UHF solution do not transform as an irreducible
representation of the $D_{\infty \text{h}}$ point group and therefore break spatial symmetry.
Allowing different orbitals for the different spins also means that the
overall wave function is no longer an eigenfunction of the $\cS^2$ operator and can be considered to break spin symmetry.
This combined spatial and spin symmetry-breaking occurs for all $U \ge 2t$.}
Furthermore, time-reversal symmetry dictates that this UHF wave function must be degenerate with its spin-flipped counterpart, obtained
by swapping $\ta_{\text{UHF}}$ and $\tb_{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf} and \eqref{eq:tb_uhf}. 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 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} ``oscillates'' between the two symmetry-broken configurations. \cite{GiulianiBook}
Symmetry breaking can also occur in RHF theory when a charge-density wave is formed from an oscillation Spatial 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} between the two closed-shell configurations with both electrons localised on one site or the other.\cite{StuberPaldus,Fukutome_1981}
%===============================================% %===============================================%
@ -896,12 +926,15 @@ The RMP series is convergent \titou{at $\lambda = 1$} for $U = 3.5\,t$ with $\rc
perturbation order in Fig.~\ref{subfig:RMP_cvg}. perturbation order in Fig.~\ref{subfig:RMP_cvg}.
In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes divergent \titou{at $\lambda = 1$}. In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes divergent \titou{at $\lambda = 1$}.
The corresponding Riemann surfaces for $U = 3.5\,t$ and $4.5\,t$ are shown in Figs.~\ref{subfig:RMP_3.5} and The corresponding Riemann surfaces for $U = 3.5\,t$ and $4.5\,t$ are shown in Figs.~\ref{subfig:RMP_3.5} and
\ref{subfig:RMP_4.5}, respectively, with the single EP at $\lep$ (black dot) and \titou{the radius of convergence indicated \ref{subfig:RMP_4.5}, respectively, with the single EP at $\lep$ (black dot).
by the vertical cylinder of unit radius (TODO)}. \hugh{We illustrate the surface $\abs{\lambda} = 1$ using a vertical cylinder of unit radius to provide
For the divergent case, $\lep$ lies inside this cylinder of convergence, while in the convergent case $\lep$ lies a visual aid for determining if the series will converge at the physical case $\lambda =1$.}
For the divergent case, $\lep$ lies inside this \hugh{unit} cylinder, while in the convergent case $\lep$ lies
outside this cylinder. outside this cylinder.
In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour
for the two states using the ground-state RHF orbitals is identical. for the two states using the ground-state RHF orbitals is identical.
\hugh{Note that, when $\lep$ lies \emph{on} the unit cylinder, we cannot \textit{a priori} determine
whether the perturbation series will converge or not.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% RADIUS OF CONVERGENCE PLOTS % RADIUS OF CONVERGENCE PLOTS
@ -972,7 +1005,7 @@ The ground-state UMP expansion is convergent in both cases, although the rate of
for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}). for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}).
% EFFECT OF SYMMETRY BREAKING % EFFECT OF SYMMETRY BREAKING
As the UHF orbitals break the spin symmetry, new coupling terms emerge between the electronic states that As the UHF orbitals break the \hugh{spatial and} spin symmetry, new coupling terms emerge between the electronic states that
cause fundamental changes to the structure of EPs in the complex $\lambda$-plane. cause fundamental changes to the structure of EPs in the complex $\lambda$-plane.
For example, while the RMP energy shows only one EP between the ground and For example, while the RMP energy shows only one EP between the ground and
doubly-excited states (Fig.~\ref{fig:RMP}), the UMP energy has two pairs of complex-conjugate EPs: one connecting the ground state with the doubly-excited states (Fig.~\ref{fig:RMP}), the UMP energy has two pairs of complex-conjugate EPs: one connecting the ground state with the
@ -1291,16 +1324,36 @@ Both of these factors are common in atoms on the right-hand side of the periodic
Molecules containing these atoms are therefore often class $\beta$ systems with Molecules containing these atoms are therefore often class $\beta$ systems with
a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006} a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% RMP critical point density
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[b]
\includegraphics[width=\linewidth]{rmp_crit_density}
\caption{
\hugh{Electron density $\rho_\text{atom}$ on the ``atomic'' site of the asymmetric Hubbard dimer with
$\epsilon = 2.5 U$.
The autoionsation process associated with the critical point is represented by the sudden drop on the negative $\lambda$ axis.
In the idealised limit $t=0$, this process becomes increasingly sharp and represents a zero-temperature QPT.}
\label{fig:rmp_dens}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% EXACT VERSUS APPROXIMATE % EXACT VERSUS APPROXIMATE
The critical point in the exact case $t=0$ lies on the negative real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: dashed lines), The critical point in the exact case $t=0$ \hugh{is represented by the gradient discontinuity in the
ground-state energy} on the negative real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: solid lines),
mirroring the behaviour of a quantum phase transition.\cite{Kais_2006} mirroring the behaviour of a quantum phase transition.\cite{Kais_2006}
\hugh{The autoionisation process is manifested by a sudden drop in the ``atomic site''
electron density $\rho_\text{atom}$ (Fig.~\ref{fig:rmp_dens}).}
However, in practical calculations performed with a finite basis set, the critical point is modelled as a cluster However, in practical calculations performed with a finite basis set, the critical point is modelled as a cluster
of branch points close to the real axis. of branch points close to the real axis.
The use of a finite basis can be modelled in the asymmetric dimer by making the second site a less The use of a finite basis can be modelled in the asymmetric dimer by making the second site a less
idealised destination for the ionised electrons with a non-zero (yet small) hopping term $t$. idealised destination for the ionised electrons with a non-zero (yet small) hopping term $t$.
Taking the value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: solid lines), the critical point becomes a Taking the value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: dashed lines), the critical point becomes
sharp avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}). an avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}).
In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}), \hugh{In contrast to the exact critical point with $t=0$, the ground-state energy remains
smooth through this avoided crossing.}
In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}) \hugh{and the
avoided crossing becomes a gradient discontinuity},
mirroring Sergeev's discussion on finite basis mirroring Sergeev's discussion on finite basis
set representations of the MP critical point.\cite{Sergeev_2006} set representations of the MP critical point.\cite{Sergeev_2006}
@ -1327,7 +1380,7 @@ set representations of the MP critical point.\cite{Sergeev_2006}
The UMP ground-state EP in the symmetric Hubbard dimer becomes a critical point in the strong correlation limit (\ie, large $U/t$). The UMP ground-state EP in the symmetric Hubbard dimer becomes a critical point in the strong correlation limit (\ie, large $U/t$).
(\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis (\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis
becomes increasingly sharp. becomes increasingly sharp.
(\subref{subfig:ump_cp_surf}) Complex energy surfaces for $U = 5t$. (\subref{subfig:ump_cp_surf}) \hugh{The avoided crossing at $U=5t$ corresponds to EPs with a non-zero imaginary component.}
(\subref{subfig:ump_ep_to_cp}) Convergence of the EPs at $\lep$ onto the real axis for $U/t \to \infty$. (\subref{subfig:ump_ep_to_cp}) Convergence of the EPs at $\lep$ onto the real axis for $U/t \to \infty$.
%mirrors the formation of the RMP critical point and other QPTs in the complete basis set limit. %mirrors the formation of the RMP critical point and other QPTs in the complete basis set limit.
\label{fig:UMP_cp}} \label{fig:UMP_cp}}
@ -1335,6 +1388,7 @@ set representations of the MP critical point.\cite{Sergeev_2006}
\end{figure*} \end{figure*}
%------------------------------------------------------------------% %------------------------------------------------------------------%
% RELATIONSHIP BETWEEN QPT AND UMP % RELATIONSHIP BETWEEN QPT AND UMP
Returning to the symmetric Hubbard dimer, we showed in Sec.~\ref{sec:spin_cont} that the slow Returning to the symmetric Hubbard dimer, we showed in Sec.~\ref{sec:spin_cont} that the slow
convergence of the strongly correlated UMP series convergence of the strongly correlated UMP series
@ -1344,11 +1398,27 @@ connection to MP critical points and QPTs (see Sec.~\ref{sec:MP_critical_point})
For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's
Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction
becomes increasingly repulsive. becomes increasingly repulsive.
Closed-shell critical points along the positive real $\lambda$ axis then represent Closed-shell critical points along the positive real $\lambda$ axis may then represent
points where the two-electron repulsion overcomes the attractive HF potential points where the two-electron repulsion overcomes the attractive HF potential
and a single electron dissociates from the molecule (see Ref.~\onlinecite{Sergeev_2006}). and a single electron dissociates from the molecule (see Ref.~\onlinecite{Sergeev_2006}).
In contrast, symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% UMP critical point density
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[b]
\includegraphics[width=\linewidth]{ump_crit_density}
\caption{
\hugh{ Difference in the electron densities on the left and right sites for the UMP ground-state in the symmetric Hubbard dimer
(see Eq.~\eqref{eq:ump_dens}).
At $\lambda = 1$, the spin-up electron transfers from the right site to the left site, while the spin-down
electron transfers in the opposite direction.
In the strong correlation limit (large $U/t$), this process becomes increasingly sharp and represents a
zero-temperature QPT.}
\label{fig:ump_dens}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In contrast, spin symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
Consider one of the two reference UHF solutions where the spin-up and spin-down electrons are localised on the left and right sites respectively. Consider one of the two reference UHF solutions where the spin-up and spin-down electrons are localised on the left and right sites respectively.
The spin-up HF potential will then be a repulsive interaction from the spin-down electron The spin-up HF potential will then be a repulsive interaction from the spin-down electron
density that is centred around the right site (and vice-versa). density that is centred around the right site (and vice-versa).
@ -1359,23 +1429,34 @@ for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
While this appears to be an avoided crossing between the ground and first-excited state, While this appears to be an avoided crossing between the ground and first-excited state,
the presence of an earlier excited-state avoided crossing means that the first-excited state qualitatively the presence of an earlier excited-state avoided crossing means that the first-excited state qualitatively
represents the reference double excitation for $\lambda > 1/2$. represents the reference double excitation for $\lambda > 1/2$.
\hugh{We can visualise this swapping process by considering the difference in the
electron density on the left and right sites, defined for each spin as
\begin{equation}
\Delta \rho^{\sigma} = \rho_\mathcal{R}^{\sigma} - \rho_\mathcal{L}^{\sigma},
\label{eq:ump_dens}
\end{equation}
where $\rho_{\mathcal{L}}^{\sigma}$ ($\rho_{\mathcal{R}}^{\sigma}$) is the spin-$\sigma$ electron density
on the left (right) site.
This density difference is shown for the UMP ground-state at $U = 5 t$ in Fig.~\ref{fig:ump_dens} (solid lines).
Here, the transfer of the high-spin electron from the right site to the left site can be seen as $\lambda$ passes through 1
(and similarly for the low-spin electron).}
% SHARPNESS AND QPT % SHARPNESS AND QPT
The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$. The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites, For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites,
both in the UHF reference and the exact wave function. both in the UHF reference and the exact wave function.
This delocalisation dampens the electron swapping process and leads to a ``shallow'' avoided crossing This delocalisation dampens the electron swapping process and leads to a ``shallow'' avoided crossing (solid lines in Fig.~\ref{subfig:ump_cp})
that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}). that corresponds to EPs with non-zero imaginary components (Fig.~\ref{subfig:ump_cp_surf}).
As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping
term to make electron delocalisation less favourable. term to make electron delocalisation less favourable.
In other words, the electrons localise on individual sites to form a Wigner crystal. In other words, the electrons localise on individual sites to form a Wigner crystal.
These effects create a stronger driving force for the electrons to swap sites until eventually this swapping These effects create a stronger driving force for the electrons to swap sites until, eventually, this swapping
occurs exactly at $\lambda = 1$. occurs suddenly at $\lambda = 1$, \hugh{as shown for $U= 50 t$ in Fig.~\ref{fig:ump_dens} (dashed lines).}
In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided
crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}). crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}).
We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes
a new type of MP critical point and represents a QPT as the perturbation parameter $\lambda$ is varied. a new type of MP critical point and represents a QPT as the perturbation parameter $\lambda$ is varied.
Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the Remarkably, this argument explains why the dominant UMP singularity lies so close, but always outside, the
radius of convergence (see Fig.~\ref{fig:RadConv}). radius of convergence (see Fig.~\ref{fig:RadConv}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -1429,7 +1510,8 @@ More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as
E_{[d_A/d_B]}(\lambda) = \frac{A(\lambda)}{B(\lambda)} E_{[d_A/d_B]}(\lambda) = \frac{A(\lambda)}{B(\lambda)}
= \frac{\sum_{k=0}^{d_A} a_k\, \lambda^k}{1 + \sum_{k=1}^{d_B} b_k\, \lambda^k}, = \frac{\sum_{k=0}^{d_A} a_k\, \lambda^k}{1 + \sum_{k=1}^{d_B} b_k\, \lambda^k},
\end{equation} \end{equation}
where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms for each power of $\lambda$. where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting
\hugh{and comparing terms for each power of $\lambda$ with the low-order terms in the Taylor series expansion}.
Pad\'e approximants are extremely useful in many areas of physics and Pad\'e approximants are extremely useful in many areas of physics and
chemistry\cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles, chemistry\cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles,
which appear at the roots of $B(\lambda)$. which appear at the roots of $B(\lambda)$.

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@ -4,6 +4,37 @@
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\usepackage[strict]{changepage}
% for formal definitions
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{\ignorespaces''\end{quotation}}
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{%
\vspace{2pt}\end{adjustwidth}\endMakeFramed%
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\begin{document} \begin{document}
\begin{letter}% \begin{letter}%
@ -25,152 +56,162 @@ We look forward to hearing from you.
\closing{Sincerely, the authors.} \closing{Sincerely, the authors.}
%%% REVIEWER 1 %%% %%% REVIEWER 1 %%%
\newpage
\noindent \textbf{\large Authors' answer to Reviewer \#1} \noindent \textbf{\large Authors' answer to Reviewer \#1}
\begin{itemize} \begin{formal}
In this manuscript, the authors provide a comprehensive and thorough discussion of the role of exceptional point singularities in determining the performance of perturbation methods of electronic structure theory.
The main subject considered is the well-known Moller-Plesset (MP) theory to calculate the electronic correlation energy, and various resummation techniques aimed to improve the convergence of the MP series.
The system used to illustrate the application of MP theory is the Hubbard dimer.
The authors suggest how the combination of resummation methods and acceleration techniques, such as the Shanks transformation, can significantly improve the accuracy of perturbative approaches.
The manuscript is well written and has the format of a very useful and informative review of the progress done in the field.
For that reason, I was worried that it could not fit the audience of the Journal of Physics Condensed Matter journal, and it might be better placed in a different journal, focusing on reviews.
Besides that point, I found some merit in the application of the Shanks transformation to different approximants for both the restricted and unrestricted versions of the theory, which focus on the corresponding set of orbitals.
Thus, I am glad to recommend publication.
\end{formal}
\item \noindent{We thank Reviewer \#1 for their support of our manuscript and for recommending publication.}
{In this manuscript, the authors provide a comprehensive and thorough discussion of the role of exceptional point singularities in determining the performance of perturbation methods of electronic structure theory.
The main subject considered is the well-known Moller-Plesset (MP) theory to calculate the electronic correlation energy, and various resummation techniques aimed to improve the convergence of the MP series.
The system used to illustrate the application of MP theory is the Hubbard dimer.
The authors suggest how the combination of resummation methods and acceleration techniques, such as the Shanks transformation, can significantly improve the accuracy of perturbative approaches.
The manuscript is well written and has the format of a very useful and informative review of the progress done in the field.
For that reason, I was worried that it could not fit the audience of the Journal of Physics Condensed Matter journal, and it might be better placed in a different journal, focusing on reviews.
Besides that point, I found some merit in the application of the Shanks transformation to different approximants for both the restricted and unrestricted versions of the theory, which focus on the corresponding set of orbitals.
Thus, I am glad to recommend publication.}
\\
\alert{We thank Reviewer \#1 for recommending publication of the present review.}
\end{itemize}
%%% REVIEWER 2 %%% %%% REVIEWER 2 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#2} \noindent \textbf{\large Authors' answer to Reviewer \#2}
\begin{itemize} \begin{formal}
The topical review "Perturbation theory in the complex plane: Exceptional points and where to find them" reviews the origin and meaning of exceptional points in quantum chemistry.
The Hubbard dimer is used throughout as an illustrative example for the restricted and unrestricted cases, exact solution (FCI), Hartree-Fock, and Rayleigh-Schrodinger perturbation theory.
The authors review the ways that MP series in common quantum chemical applications can behave, using the classification schemes developed in the literature.
Finally, they discuss alternative methods to approximate the Taylor series of RS theory by other functions which may have better, faster, or otherwise more robust convergence properties, including their own calculations based on the Shanks transformation.
\item \noindent This topical review is very well done and I recommend it for publication as is.
{The topical review "Perturbation theory in the complex plane: Exceptional points and where to find them" reviews the origin and meaning of exceptional points in quantum chemistry. I list my comments below for the authors to address if they choose.
The Hubbard dimer is used throughout as an illustrative example for the restricted and unrestricted cases, exact solution (FCI), Hartree-Fock, and Rayleigh-Schrodinger perturbation theory. Because this manuscript is meant as a review and to have instructional value, my comments are a bit more pedagogical or curiosity driven than may be expected for a normal review.
The authors review the ways that MP series in common quantum chemical applications can behave, using the classification schemes developed in the literature. I am not an expert in this exact area, so my questions are meant to represent those a typical reader might have.
Finally, they discuss alternative methods to approximate the Taylor series of RS theory by other functions which may have better, faster, or otherwise more robust convergence properties, including their own calculations based on the Shanks transformation. My questions do not need to all be addressed before the manuscript is published.
\end{formal}
This topical review is very well done and I recommend it for publication as is. \noindent {Again, we thank Reviewer \#2 for recommending publication and for their constructive comments.
I list my comments below for the authors to address if they choose. Below, we address these comments:}
Because this manuscript is meant as a review and to have instructional value, my comments are a bit more pedagogical or curiosity driven than may be expected for a normal review.
I am not an expert in this exact area, so my questions are meant to represent those a typical reader might have.
My questions do not need to all be addressed before the manuscript is published.}
\\
\alert{Again, we thank Reviewer \#2 for recommending publication and for his/her constructive comments.
Below, we address these comments.}
\item \begin{formal}
{I.\\ How am I to actually interpret the electronic energy $E$ of such a non-Hermitian Hamiltonian?
How am I to actually interpret the electronic energy $E$ of such a non-Hermitian Hamiltonian?
Is the energy associated with just the real part of the Riemann surface $E(\lambda)$ at every point, is it computable from some combination of Re and Im parts, or is the ordinary meaning of the energy lost completely? Is the energy associated with just the real part of the Riemann surface $E(\lambda)$ at every point, is it computable from some combination of Re and Im parts, or is the ordinary meaning of the energy lost completely?
Is there a lifetime interpretation for the imaginary part of this non-Hermitian Hamiltonian in the wave function problem as in Green's function theory? Is there a lifetime interpretation for the imaginary part of this non-Hermitian Hamiltonian in the wave function problem as in Green's function theory?
Maybe something can be said about this because I suspect the imaginary part is somehow related to losses. Maybe something can be said about this because I suspect the imaginary part is somehow related to losses.
This is hinted at in the introduction but was the first question that entered my mind.} This is hinted at in the introduction but was the first question that entered my mind.
\\ \end{formal}
\alert{In most figures, we indeed plot the real component of the energy.
By going complex, the energies lost their meaning as there is no relation of order for complex numbers.
In non-Hermitian processes like resonances or scattering, the imaginary part of the energy is indeed linked to lifetimes [see the book of Moiseyev].
This is mention in the introduction section.
However, the meaning of the imaginary part of the energy in the present context is unclear and we prefer not to speculate on this.
We have added a comment on this point in the revised version of the manuscript.}
\item \noindent{In most figures, we indeed plot the real component of the energy, as shown on the relevant
{IIe.\\ figure axes.
Could the authors make it very clear what is meant here by ``symmetry broken"? By going complex, the energies lose their meaning as there is no ordering relationship for complex numbers.
This term is seen and used very frequently but can be kind of ambiguous and rarely explained completely. In non-Hermitian processes that consider resonances or scattering, the imaginary part of the energy is indeed linked to lifetimes [see the book of Moiseyev].
What is the symmetry that is broken? We mention this in the Introduction section.
Also, what exactly is meant by ``symmetry-pure" when referring to the molecular orbitals? This refers to equal composition of left and right sites, but maybe this can be said explicitly. However, the meaning of the imaginary part of the energy in the present context is unclear and we prefer not to speculate on this.
A useful counterexample would be a mixing angle of zero, in which case the orbitals are either definitely left or definitely right. We have added a comment on this point in Section II.B of the revised manuscript.}
I personally can get confused because both spatial and spin symmetries are in play (or so I think).
At high $U/t$, left and right sites are both open shell with one electron each.
Am I to think of this as a broken spatial symmetry, broken spin symmetry, or both?}
\\
\alert{bla bla bla}
\item \begin{formal}
{It could be emphasized that Eqs. 20, 21a, and 21b are valid for both the RHF and UHF cases, which is why these equations lack an RHF or UHF subscript. Could the authors make it very clear what is meant here by ``symmetry broken"?
Also, in the general case (either RHF or UHF), the bonding orbitals for the two sigma values are occupied and the two antibonding orbitals are unoccupied. This term is seen and used very frequently but can be kind of ambiguous and rarely explained completely.
Bonding and antibonding notation is not so common in the condensed matter literature and it may be missed that these correspond to occupied and empty single-particle states.} What is the symmetry that is broken?
\\ Also, what exactly is meant by ``symmetry-pure" when referring to the molecular orbitals? This refers to equal composition of left and right sites, but maybe this can be said explicitly.
\alert{We now mention that Eqs. 20, 21a, and 21b are valid for both RHF and UHF. A useful counterexample would be a mixing angle of zero, in which case the orbitals are either definitely left or definitely right.
We have also changed the notations for the occupied and empty single-particle states.} I personally can get confused because both spatial and spin symmetries are in play (or so I think).
At high $U/t$, left and right sites are both open shell with one electron each.
Am I to think of this as a broken spatial symmetry, broken spin symmetry, or both?
\end{formal}
\item \noindent{The Reviewer is correct that both spatial and spin symmetries are being broken.
{IIf.\\ We consider spatial symmetry breaking to occur when the orbitals do not transform as an
What am I to think of values of $\lambda$ where $|\lambda|>1$? irreducible transformation of the relevant point group ($D_{\infty\text{h}}$ in the case of
I guess this is mathematically well defined and useful for understanding $r_c$, but the physical system always corresponds to $|\lambda|=1$, even in the complex case. the symmetric Hubbard dimer).
I suppose one needs to understand the singularities at other values of $\lambda$ in order to know the convergence of the series at the value we want, $\lambda = 1$. Similarly, we consider spin symmetry breaking to occur when the wave function is not an
Some discussion could already be made that these values are not necessarily physical but determine the behavior of the physical series. eigenfunction of the $\mathcal{S}^2$ operator.
Certain values away from $\lambda=1$ can also have physical meaning, like the autoionization process discussed later.} We have clarified these definitions in the discussion of the symmetry-broken UHF wave functions.}
\\
\alert{Only the $\lambda = 0$ (non-interacting system in some cases) and $\lambda = 1$ (fully interacting system) are physical systems as mentioned in the manuscript.
However, it can be useful to study the systems at different $\lambda$ values as we do in Sec.~II.F.}
\item \begin{formal}
{IIIc.\\ It could be emphasized that Eqs. 20, 21a, and 21b are valid for both the RHF and UHF cases, which is why these equations lack an RHF or UHF subscript.
To be extra careful, should it be said that the RMP series is either convergent or divergent at the physical coupling value $|\lambda|=1$? Also, in the general case (either RHF or UHF), the bonding orbitals for the two sigma values are occupied and the two antibonding orbitals are unoccupied.
I guess for either value of $U$, the series is convergent if you are close enough to the origin. Bonding and antibonding notation is not so common in the condensed matter literature and it may be missed that these correspond to occupied and empty single-particle states.
The language describing the cylinder with unit radius could be improved from: "the radius of convergence indicated by the vertical cylinder of unit radius". \end{formal}
I first thought the cylinder was marking the radius of convergence. It is meant to show the position of the EPs and indicate if the radius of convergence is $>1$ or $<1$.
The language could be improved.}
\\
\alert{The language has been improved and we have mentioned in several places that we are talking about a convergent/divergent MP series at $\lambda = 1$.}
\item \noindent{We now mention that Eqs. 20, 21a, and 21b are valid for both RHF and UHF. We have also changed the notations for the occupied and empty single-particle states.}
{IIId.\\
The classification of front-door and back-door intruder states is not totally clear.
I raise this point because the journal is JPCM, and these terms are not common in condensed matter literature.
Are intruder states equivalent to states creating exceptional points that ruin the series convergence?}
\\
\alert{We believe that the classification of front-door and back-door intruder states has been clearly defined in the original manuscript:
"Following this theory, a singularity in the unit circle is designated as an intruder state, with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative)."}
\item \begin{formal}
{IIIe.\\ What am I to think of values of $\lambda$ where $|\lambda|>1$?
Has "critical point" been defined yet?} I guess this is mathematically well defined and useful for understanding $r_c$, but the physical system always corresponds to $|\lambda|=1$, even in the complex case.
\\ I suppose one needs to understand the singularities at other values of $\lambda$ in order to know the convergence of the series at the value we want, $\lambda = 1$.
\alert{Thank you for spotting this. Some discussion could already be made that these values are not necessarily physical but determine the behavior of the physical series.
"Critical point" is now defined in due place.} Certain values away from $\lambda=1$ can also have physical meaning, like the autoionization process discussed later.
\end{formal}
\item \noindent{Only the $\lambda = 0$ (non-interacting system in some cases) and $\lambda = 1$ (fully interacting system) are physical systems, as mentioned in the manuscript.
{IIIf However, it can be useful to study the systems at different $\lambda$ values, as we do in Sec.~II.F.}
The explanation of the critical point as the occurrence of the autoionization process is great and makes sense to me.
I must say, though, I have trouble visualizing this in Figs. 7 and 8.
I can trace any one curve continuously along the $\lambda$ axis, and even their ordering does not change.
How does the critical point appear in these figures?
Presumably, it is because the curves exactly intersect at the CP, but this does not on the surface appear so different than the avoided crossings related to the EPs discussed in earlier sections.
So what exactly in these figures indicates the sudden and abrupt change in the eigenstates that makes it a CP?
Is it the derivative discontinuity in the energy on the real $\lambda$ axis, whereas the derivative on the real axis for an avoided crossing due to an exceptional point is continuous?
Can this gradient discontinuity be used to define the CP?
How do I see something like the autoionization in this figure?
If this is correct, it could be emphasized a bit more by a comment referring specifically to that figure.
Is it possible to circle the CP as one does for an EP?
Is there any meaning to this?}
\\
\alert{bla bla bla}
\item \begin{formal}
{IV. To be extra careful, should it be said that the RMP series is either convergent or divergent at the physical coupling value $|\lambda|=1$?
I guess for either value of $U$, the series is convergent if you are close enough to the origin.
The language describing the cylinder with unit radius could be improved from: "the radius of convergence indicated by the vertical cylinder of unit radius".
I first thought the cylinder was marking the radius of convergence. It is meant to show the position of the EPs and indicate if the radius of convergence is $>1$ or $<1$.
The language could be improved.
\end{formal}
\noindent{The language has been improved and we have mentioned in several places that we are talking about a convergent/divergent MP series at $\lambda = 1$.}
\begin{formal}
The classification of front-door and back-door intruder states is not totally clear.
I raise this point because the journal is JPCM, and these terms are not common in condensed matter literature.
Are intruder states equivalent to states creating exceptional points that ruin the series convergence?
\end{formal}
\noindent{We believe that the classification of front-door and back-door intruder states has been clearly defined in the original manuscript:
\textit{``Following this theory, a singularity in the unit circle is designated as an intruder state, with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative).''}}
\
\begin{formal}%
Has "critical point" been defined yet?
\end{formal}
\noindent{Thank you for spotting this. We have now defined ``critical point'' in the relevant locations.}
\begin{formal}
The explanation of the critical point as the occurrence of the autoionization process is great and makes sense to me.
I must say, though, I have trouble visualizing this in Figs. 7 and 8.
I can trace any one curve continuously along the $\lambda$ axis, and even their ordering does not change.
How does the critical point appear in these figures?
Presumably, it is because the curves exactly intersect at the CP, but this does not on the surface appear so different than the avoided crossings related to the EPs discussed in earlier sections.
So what exactly in these figures indicates the sudden and abrupt change in the eigenstates that makes it a CP?
Is it the derivative discontinuity in the energy on the real $\lambda$ axis, whereas the derivative on the real axis for an avoided crossing due to an exceptional point is continuous?
Can this gradient discontinuity be used to define the CP?
How do I see something like the autoionization in this figure?
If this is correct, it could be emphasized a bit more by a comment referring specifically to that figure.
Is it possible to circle the CP as one does for an EP?
Is there any meaning to this?
\end{formal}
\noindent
We thank the Reviewer for this useful piece of feedback, and attempt to answer some of their questions here.
It is correct to say that the exact critical points correspond to points where the ground-state
energy has a gradient discontinuity.
These points correspond to the intersection of two surfaces rather than an avoided crossing,
leading to the sudden change in the eigenstates.
It is possible to encircle the CP in the complex plane, but unlike an EP, this will leave the eigenstates unchanged.
We have included a new plot of the ``atomic'' site electron density as a function of $\lambda$ to illustrate
the sudden autoionisation process at the critical point.
We have also improved our discussion regarding these features using more explicit references to Figs.~7 and 8.
Finally, we have endeavoured to illustrate the UMP critical point by considering the difference in electron densities on the left and right sites, as shown in an additional figure.
\begin{formal}
Can something more be said about how these methods are fitted (briefly, and in a general way)? Can something more be said about how these methods are fitted (briefly, and in a general way)?
For example, how does one fit the Pad\'e coefficients? For example, how does one fit the Pad\'e coefficients?
Presumably, the exact function is known to some order and/or inside of some radius of convergence. Presumably, the exact function is known to some order and/or inside of some radius of convergence.
The approximant is fitted in this region to match the exact function and then only the approximant is used beyond the original radius of convergence or at higher order.} The approximant is fitted in this region to match the exact function and then only the approximant is used beyond the original radius of convergence or at higher order.
\\ \end{formal}
\alert{As mentioned in the manuscript, the Pad\'e coefficients are determined by solving a set of linear equations involving the coefficients of the Taylor series. \noindent {As mentioned in the manuscript, the Pad\'e coefficients are determined by solving a set of linear equations that relate these coefficients with the low-order terms in the Taylor series.
This is the only knowledge required to compute these.} This is the only knowledge required to compute these.}
\begin{formal}
Throughout the manuscript, the figures are excellent and really help the understanding.
The authors have clearly taken time to prepare the manuscript and it can be published immediately.
\end{formal}
\noindent{Thank you for these kind comments.}
\item
{Throughout the manuscript, the figures are excellent and really help the understanding.
The authors have clearly taken time to prepare the manuscript and it can be published immediately.}
\\
\alert{Thank you for these kind comments.}
\end{itemize}
\end{letter} \end{letter}
\end{document} \end{document}