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@ -192,7 +192,7 @@ describe metastable resonance phenomena.\cite{MoiseyevBook}
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Through the methods of complex-scaling\cite{Moiseyev_1998} and complex absorbing
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potentials,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonances can be stabilised as square-integrable
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wave functions.
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\hugh{In these situations, the energy becomes complex-valued, with the real and imaginary components allowing
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\titou{In these situations, the energy becomes complex-valued, with the real and imaginary components allowing
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the resonance energy and lifetime to be computed respectively.}
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We refer the interested reader to the excellent book by Moiseyev for a general overview. \cite{MoiseyevBook}
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@ -299,7 +299,7 @@ unless otherwise stated, atomic units will be used throughout.
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\end{subfigure}
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\caption{%
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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}).
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Only the \hugh{real component of the} interacting closed-shell singlet \hugh{energies} are shown in the complex plane,
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Only the \titou{real component of the} interacting closed-shell singlet \titou{energies} are shown in the complex plane,
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becoming degenerate at the EP (black dot).
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Following a contour around the EP (black solid) interchanges the states, while a second rotation (black dashed)
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returns the states to their original energies.
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@ -347,7 +347,7 @@ E_{\text{S}} &= U.
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\end{align}
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\end{subequations}
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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}).
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\hugh{In contrast, when $\lambda$ is complex, the energies may become complex-valued, with the real components shown in
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\titou{In contrast, when $\lambda$ is complex, the energies may become complex-valued, with the real components shown in
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Fig.~\ref{subfig:FCI_cplx}.
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Although the imaginary component of the energy is linked to resonance lifetimes elsewhere in non-Hermitian
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quantum mechanics, \cite{MoiseyevBook} its physical interpretation in the current context is unclear.
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@ -364,16 +364,16 @@ with energy
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E_\text{EP} = \frac{U}{2}.
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\end{equation}
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These $\lambda$ values correspond to so-called EPs and connect the ground and excited states in the complex plane.
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\hugh{Crucially, the ground- and excited-state wave functions at an EP become \emph{identical} rather than just degenerate.}
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\titou{Crucially, the ground- and excited-state wave functions at an EP become \emph{identical} rather than just degenerate.}
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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}).
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\hugh{Along these branch cuts, the real components of the energies are equivalent and appear to give a seam
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\titou{Along these branch cuts, the real components of the energies are equivalent and appear to give a seam
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of intersection, but a strict degeneracy is avoided because the imaginary components are different.}
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On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda = \Re(\lambda_{\text{EP}})$.
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The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction.
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In the limit $U/t \to 0$, the two EPs converge at $\lep = 0$ to create a conical intersection with
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a gradient discontinuity on the real axis.
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\hugh{This gradient discontinuity defines a critical point in the ground-state energy,
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\titou{This gradient discontinuity defines a critical point in the ground-state energy,
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where a sudden change occurs in the electronic wave function, and can be considered as a zero-temperature quantum phase transition.}
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\cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook}
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@ -484,7 +484,7 @@ Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lam
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a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states.
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The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
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$\lambda$ plane where two states become degenerate.
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\hugh{Additional singularities can also arise at critical points of the energy.
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\titou{Additional singularities can also arise at critical points of the energy.
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A critical point corresponds to the intersection of two energy surfaces
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where the eigenstates remain distinct but a gradient discontinuity occurs in
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the ground-state energy.
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@ -604,7 +604,7 @@ and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
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\begin{equation}
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E_\text{RHF} \equiv E_\text{HF}(\ta_\text{RHF}, \tb_\text{RHF}) = -2t + \frac{U}{2}.
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\end{equation}
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\hugh{Here, the molecular orbitals respectively transform
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\titou{Here, the molecular orbitals respectively transform
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according to the $\Sigma_\text{g}^{+}$ and $\Sigma_\text{u}^{+}$ irreducible representations of
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the $D_{\infty \text{h}}$ point group that represents the symmetric Hubbard dimer.
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We can therefore consider these as symmetry-pure molecular orbitals.}
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@ -654,7 +654,7 @@ with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
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E_\text{UHF} \equiv E_\text{HF}(\ta_\text{UHF}, \tb_\text{UHF}) = - \frac{2t^2}{U}.
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\end{equation}
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\hugh{The molecular orbitals of the lower-energy UHF solution do not transform as an irreducible
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\titou{The molecular orbitals of the lower-energy UHF solution do not transform as an irreducible
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representation of the $D_{\infty \text{h}}$ point group and therefore break spatial symmetry.
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Allowing different orbitals for the different spins also means that the
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overall wave function is no longer an eigenfunction of the $\cS^2$ operator and can be considered to break spin symmetry.
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@ -927,13 +927,13 @@ perturbation order in Fig.~\ref{subfig:RMP_cvg}.
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In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes divergent \titou{at $\lambda = 1$}.
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The corresponding Riemann surfaces for $U = 3.5\,t$ and $4.5\,t$ are shown in Figs.~\ref{subfig:RMP_3.5} and
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\ref{subfig:RMP_4.5}, respectively, with the single EP at $\lep$ (black dot).
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\hugh{We illustrate the surface $\abs{\lambda} = 1$ using a vertical cylinder of unit radius to provide
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\titou{We illustrate the surface $\abs{\lambda} = 1$ using a vertical cylinder of unit radius to provide
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a visual aid for determining if the series will converge at the physical case $\lambda =1$.}
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For the divergent case, $\lep$ lies inside this \hugh{unit} cylinder, while in the convergent case $\lep$ lies
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For the divergent case, $\lep$ lies inside this \titou{unit} cylinder, while in the convergent case $\lep$ lies
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outside this cylinder.
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In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour
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for the two states using the ground-state RHF orbitals is identical.
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\hugh{Note that, when $\lep$ lies \emph{on} the unit cylinder, we cannot \textit{a priori} determine
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\titou{Note that, when $\lep$ lies \emph{on} the unit cylinder, we cannot \textit{a priori} determine
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whether the perturbation series will converge or not.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -1005,7 +1005,7 @@ The ground-state UMP expansion is convergent in both cases, although the rate of
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for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}).
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% EFFECT OF SYMMETRY BREAKING
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As the UHF orbitals break the \hugh{spatial and} spin symmetry, new coupling terms emerge between the electronic states that
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As the UHF orbitals break the \titou{spatial and} spin symmetry, new coupling terms emerge between the electronic states that
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cause fundamental changes to the structure of EPs in the complex $\lambda$-plane.
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For example, while the RMP energy shows only one EP between the ground and
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doubly-excited states (Fig.~\ref{fig:RMP}), the UMP energy has two pairs of complex-conjugate EPs: one connecting the ground state with the
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@ -1330,19 +1330,19 @@ a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_
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\begin{figure}[b]
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\includegraphics[width=\linewidth]{rmp_crit_density}
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\caption{
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\hugh{Electron density $\rho_\text{atom}$ on the ``atomic'' site of the asymmetric Hubbard dimer with
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\titou{Electron density $\rho_\text{atom}$ on the ``atomic'' site of the asymmetric Hubbard dimer with
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$\epsilon = 2.5 U$.
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The autoionsation process associated with the critical point is represented by the sudden drop on the negative $\lambda$ axis.
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The autoionisation process associated with the critical point is represented by the sudden drop on the negative $\lambda$ axis.
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In the idealised limit $t=0$, this process becomes increasingly sharp and represents a zero-temperature QPT.}
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\label{fig:rmp_dens}}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% EXACT VERSUS APPROXIMATE
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The critical point in the exact case $t=0$ \hugh{is represented by the gradient discontinuity in the
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The critical point in the exact case $t=0$ \titou{is represented by the gradient discontinuity in the
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ground-state energy} on the negative real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: solid lines),
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mirroring the behaviour of a quantum phase transition.\cite{Kais_2006}
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\hugh{The autoionisation process is manifested by a sudden drop in the ``atomic site''
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\titou{The autoionisation process is manifested by a sudden drop in the ``atomic site''
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electron density $\rho_\text{atom}$ (Fig.~\ref{fig:rmp_dens}).}
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However, in practical calculations performed with a finite basis set, the critical point is modelled as a cluster
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of branch points close to the real axis.
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@ -1350,9 +1350,9 @@ The use of a finite basis can be modelled in the asymmetric dimer by making the
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idealised destination for the ionised electrons with a non-zero (yet small) hopping term $t$.
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Taking the value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: dashed lines), the critical point becomes
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an avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}).
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\hugh{In contrast to the exact critical point with $t=0$, the ground-state energy remains
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\titou{In contrast to the exact critical point with $t=0$, the ground-state energy remains
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smooth through this avoided crossing.}
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In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}) \hugh{and the
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In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}) \titou{and the
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avoided crossing becomes a gradient discontinuity},
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mirroring Sergeev's discussion on finite basis
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set representations of the MP critical point.\cite{Sergeev_2006}
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@ -1380,7 +1380,7 @@ set representations of the MP critical point.\cite{Sergeev_2006}
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The UMP ground-state EP in the symmetric Hubbard dimer becomes a critical point in the strong correlation limit (\ie, large $U/t$).
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(\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis
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becomes increasingly sharp.
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(\subref{subfig:ump_cp_surf}) \hugh{The avoided crossing at $U=5t$ corresponds to EPs with a non-zero imaginary component.}
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(\subref{subfig:ump_cp_surf}) \titou{The avoided crossing at $U=5t$ corresponds to EPs with non-zero imaginary components.}
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(\subref{subfig:ump_ep_to_cp}) Convergence of the EPs at $\lep$ onto the real axis for $U/t \to \infty$.
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%mirrors the formation of the RMP critical point and other QPTs in the complete basis set limit.
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\label{fig:UMP_cp}}
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@ -1408,8 +1408,8 @@ and a single electron dissociates from the molecule (see Ref.~\onlinecite{Sergee
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\begin{figure}[b]
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\includegraphics[width=\linewidth]{ump_crit_density}
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\caption{
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\hugh{ Difference in the electron densities on the left and right sites for the UMP ground-state in the symmetric Hubbard dimer
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(see Eq.~\eqref{eq:ump_dens}).
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\titou{Difference in the electron densities on the left and right sites for the UMP ground state in the symmetric Hubbard dimer
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[see Eq.~\eqref{eq:ump_dens}].
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At $\lambda = 1$, the spin-up electron transfers from the right site to the left site, while the spin-down
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electron transfers in the opposite direction.
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In the strong correlation limit (large $U/t$), this process becomes increasingly sharp and represents a
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@ -1429,7 +1429,7 @@ for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
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While this appears to be an avoided crossing between the ground and first-excited state,
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the presence of an earlier excited-state avoided crossing means that the first-excited state qualitatively
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represents the reference double excitation for $\lambda > 1/2$.
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\hugh{We can visualise this swapping process by considering the difference in the
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\titou{We can visualise this swapping process by considering the difference in the
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electron density on the left and right sites, defined for each spin as
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\begin{equation}
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\Delta \rho^{\sigma} = \rho_\mathcal{R}^{\sigma} - \rho_\mathcal{L}^{\sigma},
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@ -1451,7 +1451,7 @@ As $U/t$ becomes larger, the HF potentials become stronger and the on-site repul
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term to make electron delocalisation less favourable.
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In other words, the electrons localise on individual sites to form a Wigner crystal.
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These effects create a stronger driving force for the electrons to swap sites until, eventually, this swapping
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occurs suddenly at $\lambda = 1$, \hugh{as shown for $U= 50 t$ in Fig.~\ref{fig:ump_dens} (dashed lines).}
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occurs suddenly at $\lambda = 1$, \titou{as shown for $U= 50 t$ in Fig.~\ref{fig:ump_dens} (dashed lines).}
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In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided
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crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}).
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We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes
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@ -1511,7 +1511,7 @@ More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as
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= \frac{\sum_{k=0}^{d_A} a_k\, \lambda^k}{1 + \sum_{k=1}^{d_B} b_k\, \lambda^k},
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\end{equation}
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where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting
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\hugh{and comparing terms for each power of $\lambda$ with the low-order terms in the Taylor series expansion}.
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\titou{and comparing terms for each power of $\lambda$ with the low-order terms in the Taylor series expansion}.
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Pad\'e approximants are extremely useful in many areas of physics and
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chemistry\cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles,
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which appear at the roots of $B(\lambda)$.
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@ -45,7 +45,7 @@
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\justifying
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Please find attached a revised version of the manuscript entitled
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\begin{quote}
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\textit{``Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them''}.
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\textit{Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them.}
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\end{quote}
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We thank the reviewers for their constructive comments.
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Our detailed responses to their comments can be found below.
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@ -203,7 +203,8 @@ Finally, we have endeavoured to illustrate the UMP critical point by considering
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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.
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\end{formal}
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\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.
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This is the only knowledge required to compute these.}
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This is the only knowledge required to compute these.
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A minor modification has been performed to clarify this.}
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\begin{formal}
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Throughout the manuscript, the figures are excellent and really help the understanding.
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