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Hugh Burton 2020-12-01 16:08:11 +00:00
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Manuscript/EPAWTFT.tex
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@ -1285,7 +1285,8 @@ Instead, we can use an asymmetric version of the Hubbard dimer \cite{Carrascal_2
where we consider one of the sites as a ``ghost atom'' that acts as a where we consider one of the sites as a ``ghost atom'' that acts as a
destination for ionised electrons being originally localised on the other site. destination for ionised electrons being originally localised on the other site.
To mathematically model this scenario in this asymmetric Hubbard dimer, we introduce a one-electron potential $-\epsilon$ on the left site to To mathematically model this scenario in this asymmetric Hubbard dimer, we introduce a one-electron potential $-\epsilon$ on the left site to
represent the attraction between the electrons and the model ``atomic'' nucleus [see Eq.~\eqref{eq:H_FCI}], where we define $\epsilon > 0$. represent the attraction between the electrons and the model ``atomic'' nucleus [see Eq.~\eqref{eq:H_FCI}],
where we define $\epsilon \geq 0$.
The reference Slater determinant for a doubly-occupied atom can be represented using the RHF The reference Slater determinant for a doubly-occupied atom can be represented using the RHF
orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$, orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$,
which corresponds to strictly localising the two electrons on the left site. which corresponds to strictly localising the two electrons on the left site.
@ -1321,7 +1322,7 @@ and the RMP energies become
\end{align} \end{align}
\end{subequations} \end{subequations}
as shown in Fig.~\ref{subfig:rmp_cp} (dashed lines). as shown in Fig.~\ref{subfig:rmp_cp} (dashed lines).
The RMP critical point then corresponds to the intersection $E_{-} = E_{+}$, giving the critical $\lambda$ value The RMP critical point then occurs at the intersection $E_{-} = E_{+}$, giving the critical $\lambda$ value
\begin{equation} \begin{equation}
\lc = 1 - \frac{\epsilon}{U}. \lc = 1 - \frac{\epsilon}{U}.
\end{equation} \end{equation}
@ -1410,7 +1411,7 @@ represents the reference double excitation for $\lambda > 0.5.$
% SHARPNESS AND QPT % SHARPNESS AND QPT
The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$. The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites, For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites,
both in the UHF reference and as $\lambda$ increases. both in the UHF reference and as $\lambda$ increases towards the exact solution.
This delocalisation dampens the electron swapping process and leads to a ``shallow'' avoided crossing This delocalisation dampens the electron swapping process and leads to a ``shallow'' avoided crossing
that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}). that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}).
As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping