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Pierre-Francois Loos 2020-12-01 16:10:41 +01:00
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Manuscript/EPAWTFT.tex
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@ -503,7 +503,7 @@ the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination''
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{HF_real.pdf} \includegraphics[width=\linewidth]{HF_real.pdf}
\caption{\label{fig:HF_real} \caption{\label{fig:HF_real}
RHF and UHF energies as a function of the correlation strength $U/t$. RHF and UHF energies \titou{in the Hubbard dimer} as a function of the correlation strength $U/t$.
The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot), often known as the Coulson-Fischer point.} The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot), often known as the Coulson-Fischer point.}
\end{figure} \end{figure}
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
@ -553,7 +553,7 @@ modelling the correct physics with the two electrons on opposite sites.
\subcaption{\label{subfig:UHF_cplx_energy}} \subcaption{\label{subfig:UHF_cplx_energy}}
\end{subfigure} \end{subfigure}
\caption{% \caption{%
(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$. (\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$ \titou{in the Hubbard dimer for $U/t = ??$}.
Symmetry-broken solutions correspond to individual sheets and become equivalent at Symmetry-broken solutions correspond to individual sheets and become equivalent at
the \textit{quasi}-EP $\lambda_{\text{c}}$ (black dot). the \textit{quasi}-EP $\lambda_{\text{c}}$ (black dot).
The RHF solution is independent of $\lambda$, giving the constant plane at $\pi/2$. The RHF solution is independent of $\lambda$, giving the constant plane at $\pi/2$.
@ -629,7 +629,7 @@ In contrast, $U < 2t$ yields $\lambda_{\text{c}} > 1$ and corresponds to
the regime where the HF ground state is correctly represented by symmetry-pure orbitals. the regime where the HF ground state is correctly represented by symmetry-pure orbitals.
% COMPLEX ADIABATIC CONNECTION % COMPLEX ADIABATIC CONNECTION
We have recently shown that the complex scaled Fock operator Eq.~\eqref{eq:scaled_fock} We have recently shown that the complex scaled Fock operator \eqref{eq:scaled_fock}
also allows states of different symmetries to be interconverted by following a well-defined also allows states of different symmetries to be interconverted by following a well-defined
contour in the complex $\lambda$-plane.\cite{Burton_2019} contour in the complex $\lambda$-plane.\cite{Burton_2019}
In particular, by slowly varying $\lambda$ in a similar (yet different) manner In particular, by slowly varying $\lambda$ in a similar (yet different) manner