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@ -1268,29 +1268,20 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are
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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}) 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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(\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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real axis, giving a sharp avoided crossing on the real axis (solid).
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\titou{vertical axe label wrong in b.}
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\label{fig:RMP_cp}}
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\label{fig:RMP_cp}}
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\end{figure*}
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\end{figure*}
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%------------------------------------------------------------------%
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%------------------------------------------------------------------%
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% INTRODUCING THE MODEL
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% INTRODUCING THE MODEL
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The simplified site basis of the Hubbard dimer makes explicilty modelling the ionisation continuum impossible.
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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 \cite{Carrascal_2015,Carrascal_2018} to consider one site as a ``ghost atom'' that acts as a
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Instead, we can use an asymmetric version of the Hubbard dimer \cite{Carrascal_2015,Carrascal_2018}
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destination for ionised electrons.
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where we consider one of the sites as a ``ghost atom'' that acts as a
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In this asymmetric model, we introduce a one-electron potential $-\epsilon$ on the left site to
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destination for ionised electrons being originally localised on the other site.
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To mathematically model this scenario, in this asymmetric Hubbard dimer, 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 [see Eq.~\eqref{eq:H_FCI}], 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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%\bH =
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%\begin{pmatrix}
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% U-2\epsilon & -t & -t & 0 \\
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% -t & -\epsilon & 0 & -t \\
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% -t & 0 & -\epsilon & -t \\
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% 0 & -t & -t & U \\
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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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The reference Slater determinant for a doubly-occupied atom can be represented using the RHF
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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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orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$, which corresponds to strictly localise the two electrons on the left site.
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%and energy
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%and energy
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%\begin{equation}
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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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% E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
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@ -1299,7 +1290,7 @@ With this representation, the parametrised RMP Hamiltonian becomes
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\begin{widetext}
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\begin{widetext}
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\begin{equation}
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\begin{equation}
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\label{eq:H_RMP}
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\label{eq:H_RMP}
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\bH_\text{RMP}\qty(\lambda) =
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\Tilde{\bH}_\text{RMP}\qty(\lambda) =
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\begin{pmatrix}
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\begin{pmatrix}
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2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\
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2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\
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-\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\
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-\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\
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@ -1308,17 +1299,16 @@ With this representation, the parametrised RMP Hamiltonian becomes
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\end{pmatrix}.
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\end{pmatrix}.
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\end{equation}
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\end{equation}
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\end{widetext}
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\end{widetext}
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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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% DERIVING BEHAVIOUR OF THE CRITICAL SITE
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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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For the ghost site to perfectly represent ionised electrons, the hopping term between the two sites must vanish (\ie, $t=0$).
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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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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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and the RMP energies become
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\begin{subequations}
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\begin{subequations}
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\begin{align}
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\begin{align}
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E_{-} &= 2U - 2 \epsilon - U \lambda,
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E_{-} &= 2(U - \epsilon) - \lambda U,
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\\
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\\
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E_{\text{S}} &= U - \epsilon - U \lambda,
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E_{\text{S}} &= (U - \epsilon) - \lambda U,
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\\
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\\
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E_{+} &= U \lambda,
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E_{+} &= U \lambda,
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\end{align}
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\end{align}
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@ -1376,19 +1366,21 @@ set representations of the MP critical point.\cite{Sergeev_2006}
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%------------------------------------------------------------------%
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%------------------------------------------------------------------%
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% RELATIONSHIP BETWEEN QPT AND UMP
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% RELATIONSHIP BETWEEN QPT AND UMP
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\titou{Returning to the symmetric Hubbard dimer?}
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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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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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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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These EPs have positive real components and small imaginary components (see Fig.~\ref{fig:UMP}), suggesting a potential
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connection to MP critical points and QPTs (see Sec.~\ref{sec:MP_critical_point}).
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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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For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's
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Hamiltonian displayed in 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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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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\titou{Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent
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points where 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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are successively expelled from the molecule.\cite{Sergeev_2006}}
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\titou{T2: I'd like to discuss that with you.}
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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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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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Consider one of the two reference UHF solutions 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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right sites respectively.
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The spin-up HF potential will then be a repulsive interaction from the spin-down electron
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The spin-up HF potential will then be a repulsive interaction from the spin-down electron
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density that is centred around the right site (and vice-versa).
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density that is centred around the right site (and vice-versa).
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@ -1396,9 +1388,10 @@ As $\lambda$ becomes greater than 1 and the HF potentials become attractive, the
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driving force for the electrons to swap sites.
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driving force for the electrons to swap sites.
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This swapping process can also be represented as a double excitation, and thus an avoided crossing will occur
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This swapping process can also be represented as a double excitation, and thus an avoided crossing will occur
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for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
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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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\titou{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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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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represents the double excitation for $\lambda > 0.5.$}
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\titou{T2: I find it hard to understand. I'd like to discuss this as well.}
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% SHARPNESS AND QPT
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% SHARPNESS AND QPT
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The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
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The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
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@ -1408,12 +1401,15 @@ This delocalisation dampens the electron swapping process and leads to a ``shall
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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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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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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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term to make electron delocalisation less favourable.
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\titou{In other words, the electron localises on each site forming a so-called Wigner crystal.
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T2: is it worth saying again?}
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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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These effects create a stronger driving force for the electrons to swap sites until eventually this swapping
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occurs exactly at $\lambda = 1$.
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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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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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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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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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a new type of MP critical point \titou{and represents a QPT in the perturbation approximation}.
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\titou{T2: what do you mean by this?}
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Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the
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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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radius of convergence (see Fig.~\ref{fig:RadConv}).
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