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Hugh Burton 2020-11-19 13:01:37 +00:00
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Manuscript/EPAWTFT.tex
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@ -465,12 +465,16 @@ the total spin $\hat{\mathcal{S}}^2$ operator, leading to so-called ``spin-conta
%Because $Y_0(\theta) = 1/\sqrt{4\pi}$, it is clear that the RHF wave function yields a uniform one-electron density. %Because $Y_0(\theta) = 1/\sqrt{4\pi}$, it is clear that the RHF wave function yields a uniform one-electron density.
% %
%%% FIG 2 (?) %%%
% HF energies as a function of U/t
%%%%%%%%%%%%%%%%%
\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}
\hugh{RHF and UHF energies as a function of the correlation strength $U/t$. \hugh{RHF and UHF energies as a function of the correlation strength $U/t$.
The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot).}} The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot).}}
\end{figure} \end{figure}
%%%%%%%%%%%%%%%%%
Returning to the Hubbard dimer, the UHF energy can be parametrised in terms of two rotation angles $\ta$ and $\tb$ as Returning to the Hubbard dimer, the UHF energy can be parametrised in terms of two rotation angles $\ta$ and $\tb$ as
\begin{equation} \begin{equation}
@ -515,7 +519,7 @@ correctly modelling the physics of the system with the two electrons on opposing
\end{subfigure} \end{subfigure}
\caption{% \caption{%
\hugh{(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$. \hugh{(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$.
Symmetry-broken solutions correspond to an individual sheets and become equivalent at the quasi-EP (black dot). Symmetry-broken solutions correspond to individual sheets and become equivalent at the quasi-EP $\lambda_{\text{c}}$ (black dot).
The RHF solution is independent of $\lambda$, giving constant plane at $\pi/2$. The RHF solution is independent of $\lambda$, giving constant plane at $\pi/2$.
(\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic (\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic
point at the quasi-exceptional point.} point at the quasi-exceptional point.}