reviewing Hugh asym Hubbard example
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@ -293,7 +293,7 @@ In the large-$U$ (or strong correlation) regime, the electron repulsion term bec
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and the electrons localise on opposite sites to minimise their Coulomb repulsion.
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This phenomenon is often referred to as Wigner crystallisation. \cite{Wigner_1934}
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To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$ to give the parameterised Hamiltonian $\hH(\lambda)$.
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To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t \to \lambda t$ to give the parameterised Hamiltonian $\hH(\lambda)$.
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When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies
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\begin{subequations}
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\begin{align}
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@ -599,7 +599,7 @@ The inherent non-linearity in the Fock eigenvalue problem arises from self-consi
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in the HF approximation, and is usually solved through an iterative approach.\cite{Roothaan_1951,Hall_1951}
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Alternatively, the non-linear terms arising from the Coulomb and exchange operators can
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be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the
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transformation $U \rightarrow \lambda\, U$, giving the parametrised Fock operator
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transformation $U \to \lambda\, U$, giving the parametrised Fock operator
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\begin{equation}
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\Hat{f}(\vb{x} ; \lambda) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}).
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\end{equation}
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@ -972,7 +972,7 @@ These numerical values reveal that the UMP ground-state series has $\rc > 1$ for
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However, in the strong correlation limit (large $U$), this radius of convergence tends to unity, indicating that
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the corresponding UMP series becomes increasingly slow.
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Furthermore, the doubly-excited state using the ground-state UHF orbitals has $\rc < 1$ for almost any value
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of $U/t$, reaching the limiting value of $1/2$ for $U/t \rightarrow \infty$, and thus the
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of $U/t$, reaching the limiting value of $1/2$ for $U/t \to \infty$, and thus the
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excited-state UMP series will always diverge.
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% DISCUSSION OF UMP RIEMANN SURFACES
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@ -1264,22 +1264,20 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are
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\subcaption{\label{subfig:rmp_cp_surf}}
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\end{subfigure}
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\caption{%
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\hugh{Modelling the RMP critical point using the asymmetric Hubbard dimer.
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(\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed).
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(\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the
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real axis, giving a sharp avoided crossing on the real axis (solid).
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}
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Modelling the RMP critical point using the asymmetric Hubbard dimer.
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(\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed).
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(\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the
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real axis, giving a sharp avoided crossing on the real axis (solid).
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\label{fig:RMP_cp}}
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\end{figure*}
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%------------------------------------------------------------------%
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% INTRODUCING THE MODEL
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\hugh{%
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The simplified site basis of the Hubbard dimer makes explicilty modelling the ionisation continuum impossible.
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Instead, we can use an asymmetric Hubbard dimer to consider one site as a ``ghost atom'' that acts as a
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Instead, we can use an asymmetric Hubbard dimer \cite{Carrascal_2015,Carrascal_2018} to consider one site as a ``ghost atom'' that acts as a
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destination for ionised electrons.
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In this asymmetric model, we introduce a one-electron potential $-\epsilon$ on the left site to
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represent the attraction between the electrons and the model ``atomic'' nucleus, where we define $\epsilon > 0$.
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represent the attraction between the electrons and the model ``atomic'' nucleus [see Eq.~\eqref{eq:H_FCI}], where we define $\epsilon > 0$.
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%The exact Hamiltonian [Eq.~\eqref{eq:H_FCI}] then becomes
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%\begin{equation}
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%\label{eq:H_FCI_Asymm}
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@ -1292,10 +1290,7 @@ represent the attraction between the electrons and the model ``atomic'' nucleus,
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%\end{pmatrix}.
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%\end{equation}
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The reference Slater determinant for a doubly-occupied atom can be represented using the RHF
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orbitals [Eq.~\eqref{eq:RHF_orbs}] with
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\begin{equation}
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\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0.
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\end{equation}
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orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$.
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%and energy
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%\begin{equation}
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% E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
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@ -1313,18 +1308,17 @@ With this representation, the parametrised RMP Hamiltonian becomes
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\end{pmatrix}.
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\end{equation}
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\end{widetext}
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}
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\titou{Shall we change the symbol $\bH_\text{RMP}\qty(\lambda)$ to avoid confusion? Maybe $\Tilde{\bH}_\text{RMP}\qty(\lambda)$?}
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% DERIVING BEHAVIOUR OF THE CRITICAL SITE
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\hugh{%
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For the ghost site to perfectly represent ionised electrons, the hopping term between the two sites must vanish with $t=0$.
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This limit corresponds to the dissociative regime in the asymmetric Hubbard dimer (see Ref.~\ref{Carrascal_2018}),
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This limit corresponds to the dissociative regime in the asymmetric Hubbard dimer as discussed in Ref.~\onlinecite{Carrascal_2018},
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and the RMP energies become
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\begin{subequations}
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\begin{align}
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E_{-} &= 2U - 2 \epsilon - U \lambda
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E_{-} &= 2U - 2 \epsilon - U \lambda,
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\\
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E_{\text{S}} &= U - \epsilon - U \lambda
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E_{\text{S}} &= U - \epsilon - U \lambda,
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\\
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E_{+} &= U \lambda,
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\end{align}
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@ -1344,22 +1338,19 @@ representing strong screening of the nuclear attraction by core and valence elec
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and again a less negative $\lambda$ is required for ionisation to occur.
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Both of these factors are common in atoms on the right-hand side of the periodic table, \eg\ \ce{F},
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\ce{O}, \ce{Ne}, and thus molecules containing these atoms are often class $\beta$ systems with
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a divergent RMP series due to the MP critical point.
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}
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a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006}
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% EXACT VERSUS APPROXIMATE
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\hugh{%
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The critical point in the exact case $t=0$ lies on the real $\lambda$ axis, mirroring the behaviour of a quantum
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phase transition.\cite{Kais_2006}
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The critical point in the exact case $t=0$ lies on the \titou{negative} real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: dashed lines),
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mirroring the behaviour of a quantum phase transition.\cite{Kais_2006}
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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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The use of a finite basis can be modelled in the asymmetric dimer by making the second site a less
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idealised destination for the ionised electrons with a non-zero hopping term $t$.
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Taking the small value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: solid lines), the critical point becomes a
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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}: solid lines), the critical point becomes a
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sharp avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}).
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In the limit $t \rightarrow 0$, these EPs approach the real axis, mirroring Sergeev's discussion on finite basis
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In the limit $t \to 0$, these EPs approach the real axis, 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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}
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%------------------------------------------------------------------%
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% Figure on the UMP critical point
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@ -1374,33 +1365,28 @@ set representations of the MP critical point.\cite{Sergeev_2006}
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\subcaption{\label{subfig:ump_ep_to_cp}}
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\end{subfigure}
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\caption{%
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\hugh{%
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The UMP ground-state EP becomes a critical point in the strong correlation limit (large $U/t$).
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(\subref{subfig:ump_cp}) As the $U/t$ increases, the avoided crossing on the real $\lambda$ axis
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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_ep_to_cp}) The convergence of the EPs at $\lep$ onto the real axis for $U/t \rightarrow \infty$
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mirrors the formation of the RMP critical point and other QPTs in the infinite basis set limit.
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}
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(\subref{subfig:ump_ep_to_cp}) The 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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\end{figure*}
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%------------------------------------------------------------------%
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% RELATIONSHIP BETWEEN QPT AND UMP
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\hugh{%
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In Section~\ref{sec:spin_cont} we showed that the slow convergence of the UMP series for the strongly correlated
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Hubbard dimer was due to a complex-conjguate pair of EPs just outside the radius of convergence.
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In Sec.~\ref{sec:spin_cont} we showed that the slow convergence of the UMP series for the strongly correlated
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Hubbard dimer was due to a complex-conjugate pair of EPs just outside the radius of convergence.
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These EPs have positive real components and small imaginary components (Fig.~\ref{fig:UMP}), suggesting a potential
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connection to MP critical points and QPTs.
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connection to MP critical points and QPTs (see Sec.~\ref{sec:MP_critical_point}).
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For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's
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Hamiltonian [Eq.~\eqref{eq:HamiltonianStillinger}], while the explicit electron-electron interaction
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Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction
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becomes increasingly repulsive.
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Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent
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points were the two-electron repulsion overcomes the attractive HF potential and an electron
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points where the two-electron repulsion overcomes the attractive HF potential and an electron
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are successively expelled from the molecule.\cite{Sergeev_2006}
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}
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\hugh{%
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Symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
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Consider the reference UHF solution where the spin-up and spin-down electrons are localised on the left and
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right sites respectively.
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@ -1413,14 +1399,12 @@ for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
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Note that, although this appears to be an avoided crossing between the ground and first-excited state,
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the earlier excited-state avoided crossing means that the first-excited state qualitatively
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represents the double excitation for $\lambda > 0.5.$
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}
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% SHARPNESS AND QPT
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\hugh{%
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The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
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For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites,
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both in the UHF reference and as $\lambda$ increases.
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This delocalisataion dampens the electron swapping process and leads to a ``shallow'' avoided crossing
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This delocalisation dampens the electron swapping process and leads to a ``shallow'' avoided crossing
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that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}).
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As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping
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term to make electron delocalisation less favourable.
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@ -1428,11 +1412,10 @@ These effects create a stronger driving force for the electrons to swap sites un
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occurs exactly at $\lambda = 1$.
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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 becomes
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We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes
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a new type of MP critical and represents a QPT in the perturbation approximation.
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Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the
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radius of convergence (see Fig.~\ref{fig:RadConv}).
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}
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%%====================================================
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%\subsection{The physics of quantum phase transitions}
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