From 39bb234a4de9bd9f9eb3015dd56f29a80e9d755a Mon Sep 17 00:00:00 2001 From: Hugh Burton Date: Thu, 19 Nov 2020 13:01:37 +0000 Subject: [PATCH] more tweaks --- Manuscript/EPAWTFT.tex | 14 +++++++++----- 1 file changed, 9 insertions(+), 5 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index b43d39b..12416bd 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -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. % +%%% FIG 2 (?) %%% +% HF energies as a function of U/t +%%%%%%%%%%%%%%%%% \begin{figure} -\includegraphics[width=\linewidth]{HF_real.pdf} -\caption{\label{fig:HF_real} -\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).}} + \includegraphics[width=\linewidth]{HF_real.pdf} + \caption{\label{fig:HF_real} + \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).}} \end{figure} +%%%%%%%%%%%%%%%%% Returning to the Hubbard dimer, the UHF energy can be parametrised in terms of two rotation angles $\ta$ and $\tb$ as \begin{equation} @@ -515,7 +519,7 @@ correctly modelling the physics of the system with the two electrons on opposing \end{subfigure} \caption{% \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 $\pi/2$. (\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic point at the quasi-exceptional point.}