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Manuscript/EPAWTFT.tex
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@ -192,7 +192,7 @@ describe metastable resonance phenomena.\cite{MoiseyevBook}
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
wave functions.
\hugh{In these situations, the energy becomes complex-valued, with the real and imaginary components allowing
\titou{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}
@ -299,7 +299,7 @@ unless otherwise stated, atomic units will be used throughout.
\end{subfigure}
\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}).
Only the \hugh{real component of the} interacting closed-shell singlet \hugh{energies} are shown in the complex plane,
Only the \titou{real component of the} interacting closed-shell singlet \titou{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)
returns the states to their original energies.
@ -347,7 +347,7 @@ E_{\text{S}} &= U.
\end{align}
\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}).
\hugh{In contrast, when $\lambda$ is complex, the energies may become complex-valued, with the real components shown in
\titou{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.
@ -364,16 +364,16 @@ with energy
E_\text{EP} = \frac{U}{2}.
\end{equation}
These $\lambda$ values correspond to so-called EPs and connect the ground and excited states in the complex plane.
\hugh{Crucially, the ground- and excited-state wave functions at an EP become \emph{identical} rather than just degenerate.}
\titou{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
\titou{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 $\lambda = \Re(\lambda_{\text{EP}})$.
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
a gradient discontinuity on the real axis.
\hugh{This gradient discontinuity defines a critical point in the ground-state energy,
\titou{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}
@ -484,7 +484,7 @@ 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.
The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
$\lambda$ plane where two states become degenerate.
\hugh{Additional singularities can also arise at critical points of the energy.
\titou{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.
@ -604,7 +604,7 @@ and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
\begin{equation}
E_\text{RHF} \equiv E_\text{HF}(\ta_\text{RHF}, \tb_\text{RHF}) = -2t + \frac{U}{2}.
\end{equation}
\hugh{Here, the molecular orbitals respectively transform
\titou{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.}
@ -654,7 +654,7 @@ with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
E_\text{UHF} \equiv E_\text{HF}(\ta_\text{UHF}, \tb_\text{UHF}) = - \frac{2t^2}{U}.
\end{equation}
\hugh{The molecular orbitals of the lower-energy UHF solution do not transform as an irreducible
\titou{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.
@ -927,13 +927,13 @@ 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 $\lambda = 1$}.
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).
\hugh{We illustrate the surface $\abs{\lambda} = 1$ using a vertical cylinder of unit radius to provide
\titou{We illustrate the surface $\abs{\lambda} = 1$ using a vertical cylinder of unit radius to provide
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
For the divergent case, $\lep$ lies inside this \titou{unit} cylinder, while in the convergent case $\lep$ lies
outside this cylinder.
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.
\hugh{Note that, when $\lep$ lies \emph{on} the unit cylinder, we cannot \textit{a priori} determine
\titou{Note that, when $\lep$ lies \emph{on} the unit cylinder, we cannot \textit{a priori} determine
whether the perturbation series will converge or not.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -1005,7 +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}).
% EFFECT OF SYMMETRY BREAKING
As the UHF orbitals break the \hugh{spatial and} spin symmetry, new coupling terms emerge between the electronic states that
As the UHF orbitals break the \titou{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.
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
@ -1330,19 +1330,19 @@ a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_
\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
\titou{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.
The autoionisation 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
The critical point in the exact case $t=0$ \hugh{is represented by the gradient discontinuity in the
The critical point in the exact case $t=0$ \titou{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}
\hugh{The autoionisation process is manifested by a sudden drop in the ``atomic site''
\titou{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
of branch points close to the real axis.
@ -1350,9 +1350,9 @@ The use of a finite basis can be modelled in the asymmetric dimer by making the
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}: dashed lines), the critical point becomes
an avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}).
\hugh{In contrast to the exact critical point with $t=0$, the ground-state energy remains
\titou{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
In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}) \titou{and the
avoided crossing becomes a gradient discontinuity},
mirroring Sergeev's discussion on finite basis
set representations of the MP critical point.\cite{Sergeev_2006}
@ -1380,7 +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 $U/t$).
(\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis
becomes increasingly sharp.
(\subref{subfig:ump_cp_surf}) \hugh{The avoided crossing at $U=5t$ corresponds to EPs with a non-zero imaginary component.}
(\subref{subfig:ump_cp_surf}) \titou{The avoided crossing at $U=5t$ corresponds to EPs with non-zero imaginary components.}
(\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.
\label{fig:UMP_cp}}
@ -1408,8 +1408,8 @@ and a single electron dissociates from the molecule (see Ref.~\onlinecite{Sergee
\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}).
\titou{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
@ -1429,7 +1429,7 @@ for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
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
represents the reference double excitation for $\lambda > 1/2$.
\hugh{We can visualise this swapping process by considering the difference in the
\titou{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},
@ -1451,7 +1451,7 @@ As $U/t$ becomes larger, the HF potentials become stronger and the on-site repul
term to make electron delocalisation less favourable.
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
occurs suddenly at $\lambda = 1$, \hugh{as shown for $U= 50 t$ in Fig.~\ref{fig:ump_dens} (dashed lines).}
occurs suddenly at $\lambda = 1$, \titou{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
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
@ -1511,7 +1511,7 @@ More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as
= \frac{\sum_{k=0}^{d_A} a_k\, \lambda^k}{1 + \sum_{k=1}^{d_B} b_k\, \lambda^k},
\end{equation}
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}.
\titou{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
chemistry\cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles,
which appear at the roots of $B(\lambda)$.

5
Response_Letter/Response_Letter.tex
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@ -45,7 +45,7 @@
\justifying
Please find attached a revised version of the manuscript entitled
\begin{quote}
\textit{``Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them''}.
\textit{Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them.}
\end{quote}
We thank the reviewers for their constructive comments.
Our detailed responses to their comments can be found below.
@ -203,7 +203,8 @@ Finally, we have endeavoured to illustrate the UMP critical point by considering
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}
\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.
A minor modification has been performed to clarify this.}
\begin{formal}
Throughout the manuscript, the figures are excellent and really help the understanding.