From 3b2891b2c36d79f3882f68c2d470f2f6c37c6645 Mon Sep 17 00:00:00 2001 From: Hugh Burton Date: Tue, 1 Dec 2020 16:08:11 +0000 Subject: [PATCH] local changes --- Manuscript/EPAWTFT.tex | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 738448c..b3ba19a 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -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 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 -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 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. @@ -1321,7 +1322,7 @@ and the RMP energies become \end{align} \end{subequations} 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} \lc = 1 - \frac{\epsilon}{U}. \end{equation} @@ -1410,7 +1411,7 @@ represents the reference double excitation for $\lambda > 0.5.$ % SHARPNESS AND QPT 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, -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 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