Transferred RMP critical point discussion from notebook as skeleton.
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\definecolor{hughgreen}{RGB}{0, 128, 0}
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@ -1239,10 +1240,95 @@ In contrast, $\alpha$ singularities correspond to large avoided crossings that a
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states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT.
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states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT.
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%=======================================
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%=======================================
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%\subsection{Critical Point in the Hubbard Dimer}
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\subsection{Critical Points in the Hubbard Dimer}
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%\label{sec:critical_point_hubbard}
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\label{sec:critical_point_hubbard}
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%=======================================
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%=======================================
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%------------------------------------------------------------------%
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% Figure on the RMP critical point
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%------------------------------------------------------------------%
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\begin{figure*}[t]
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[height=0.65\textwidth]{rmp_critical_point}
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\subcaption{\label{subfig:rmp_cp}}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[height=0.65\textwidth]{rmp_critical_point_surf}
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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{RMP critical point in the Hubbard dimer.}
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\label{fig:RMP_cp}}
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\end{figure*}
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%------------------------------------------------------------------%
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\hughDraft{%
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The simplified site basis of the Hubbard dimer makes explicitly modelling the ionisation continuum
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impossible.
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To model the auto-ionisation in the Hubbard dimer, we need to turn one of the sites into a ghost atom that will act as
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a destination for ionised electrons. We do this by considering an asymmetric Hubbard dimer where the "atomic" site
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has a negative diagonal term in the one-electron Hamiltonian, representing the nuclear attraction. This term is already
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encoded in the Initialisation section such that the core Hamiltonian is
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To model the doubly-occupied atom, we can define our reference HF state as the configuration with $\theta = 0$ and energy
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\begin{equation}
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\frac{1}{2} (2 U - 4 \epsilon)
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\end{equation}
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The RMP Hamiltonian then becomes
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...
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}
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\hughDraft{%
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Now let's think about the physics of the problem...
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For the ghost site to truly represent ionised electrons, we need the hopping term to vanish (or become very small).
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$U$ controls the strength of the HF repulsive potential. A stronger repulsion will encourage the electrons to be forced
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away from the "atomic" site at a less negative value of $\lambda$.
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$\epsilon$ controls the strength of attraction to the atom. A stronger attraction to the nucleus will mean that the electrons
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are more tightly bound to the atom and a more negative $\lambda$ will be needed for auto-ionisation.
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We therefore expect that the position of the RMP critical point will be controlled by the ratio $\epsilon / U$, with smaller ratios
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making ionisation occur closer to $\lambda$ origin, and making the divergence in these cases more likely.
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}
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\hughDraft{%
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Taking the exact case with $t=0$, the RMP energies becomes
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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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\\
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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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\end{subequations}
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By comparison with Fig.~\ref{fig:RMP_cp}, the critical point can be identified as by solving $E_{-} = E_{+}$,
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giving
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\begin{equation}
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\lc = 1 - \epsilon / U.
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\end{equation}
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Thus, as expected, the critical point lies on the real axis and moves closer to the origin for larger
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$U / \epsilon$.
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}
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\hughDraft{%
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The position of the critical point is controlled by the ratio $\epsilon / U$.
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If the magnitude of the ratio becomes greater than one, then the series diverges.
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We can interpret large $U$ as strong electron repulsion effects in electron dense molecules,
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such as \ce{F-}. A small $\epsilon$ is also likely to correspond to
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strong nuclear screening by the core and valence electrons. Both of these factors are
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common in atoms on the right-hand-side of periodic table,
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\eg\ \ce{F}, \ce{O}, \ce{Ne}, etc, as well as negatively-charged species, and so we recover the
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class $\beta$ system classification.
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}
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\hughDraft{%
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In practical calculations, one considers the perturbation energy in a finite basis and the critical point is modelled
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as a cluster of branch points close to the real axis. While we cannot change the size of our basis, we can adjust
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the extent to which the second site behaves as a ghost by making the hopping term larger.
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When $t$ is (slightly) non-zero, our modelled critical point becomes an EP point close to the real axis with a sharp
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associated avoided crossing for real $\lambda$. If $t$ becomes larger, this avoided crossing becomes less sharp and the EPs
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move away from the real axis. This mirrors the discussion of EPs approaching the real axis in the exact basis.
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This effect is shown by the dashed lines in Fig.~\ref{fig:RMP_cp}.
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}
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%%====================================================
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%%====================================================
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%\subsection{The physics of quantum phase transitions}
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%\subsection{The physics of quantum phase transitions}
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%%====================================================
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%%====================================================
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Manuscript/rmp_critical_point.pdf
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Manuscript/rmp_critical_point.pdf
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Manuscript/rmp_critical_point_surf.pdf
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Manuscript/rmp_critical_point_surf.pdf
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