diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 6cf56cd..5d2887a 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -773,6 +773,8 @@ and identified that the slow UMP convergence arises from its failure to correctl low-lying double excitation. This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference UHF wave function, creating the first direct link between spin-contamination and slow UMP convergence.\cite{Gill_1988} + +% LEPETIT CHAT Lepetit \etal\ later analysed the difference between perturbation convergence using the unrestricted MP and EN partitionings. \cite{Lepetit_1988} They argued that the slow UMP convergence for stretched molecules arises from @@ -1195,28 +1197,6 @@ $\lambda$ values closer to the origin. With these insights, they regrouped the systems into new classes: i) $\alpha$ singularities which have ``large'' imaginary parts, and ii) $\beta$ singularities which have very small imaginary parts.\cite{Goodson_2004,Sergeev_2006} -%------------------------------------------------------------------% -% Figure on the RMP critical point -%------------------------------------------------------------------% -\begin{figure*}[t] - \begin{subfigure}{0.49\textwidth} - \includegraphics[height=0.65\textwidth]{rmp_critical_point} - \subcaption{\label{subfig:rmp_cp}} - \end{subfigure} - \begin{subfigure}{0.49\textwidth} - \includegraphics[height=0.65\textwidth]{rmp_critical_point_surf} - \subcaption{\label{subfig:rmp_cp_surf}} - \end{subfigure} - \caption{% - \hugh{Modelling the RMP critical point using the asymmetric Hubbard dimer. - (\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed). - (\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the - real axis, giving a sharp avoided crossing on the real axis (solid). - } - \label{fig:RMP_cp}} -\end{figure*} -%------------------------------------------------------------------% - % RELATIONSHIP TO BASIS SET SIZE The existence of the MP critical point can also explain why the divergence observed by Olsen \etal\ in the \ce{Ne} atom and \ce{HF} molecule occurred when diffuse basis functions were included.\cite{Olsen_1996} @@ -1268,6 +1248,28 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are \label{sec:critical_point_hubbard} %======================================= +%------------------------------------------------------------------% +% Figure on the RMP critical point +%------------------------------------------------------------------% +\begin{figure*}[t] + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth]{rmp_critical_point} + \subcaption{\label{subfig:rmp_cp}} + \end{subfigure} + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth]{rmp_critical_point_surf} + \subcaption{\label{subfig:rmp_cp_surf}} + \end{subfigure} + \caption{% + \hugh{Modelling the RMP critical point using the asymmetric Hubbard dimer. + (\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed). + (\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the + real axis, giving a sharp avoided crossing on the real axis (solid). + } + \label{fig:RMP_cp}} +\end{figure*} +%------------------------------------------------------------------% + % INTRODUCING THE MODEL \hugh{% The simplified site basis of the Hubbard dimer makes explicilty modelling the ionisation continuum impossible. @@ -1356,6 +1358,87 @@ In the limit $t \rightarrow 0$, these EPs approach the real axis, mirroring Serg set representations of the MP critical point.\cite{Sergeev_2006} } +%------------------------------------------------------------------% +% Figure on the UMP critical point +%------------------------------------------------------------------% +\begin{figure*}[t] + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth]{ump_critical_point} + \subcaption{\label{subfig:ump_cp}} + \end{subfigure} + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth]{ump_ep_to_cp} + \subcaption{\label{subfig:ump_ep_to_cp}} + \end{subfigure} + \caption{% + \hugh{Modelling the RMP critical point using the asymmetric Hubbard dimer. + (\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed). + (\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the + real axis, giving a sharp avoided crossing on the real axis (solid). + } + \label{fig:RMP_cp}} +\end{figure*} +%------------------------------------------------------------------% + +% RELATIONSHIP BETWEEN QPT AND UMP +\hughDraft{% +The ground-state EP observed in the UMP series also has a small imaginary part and falls close to the real axis. +As the correlation strength increases, this EP moves closer to the real axis and closer to the radius of convergence at +$\lambda = 1$. So can we understand this using the arguments related to the critical point? +Closed-shell case: +The work by Stillinger builds an argument around the HF potential $v_{\text{HF}}$ which, by itself, +is repulsive and concentrated around the +occupied orbitals. As $\lambda$ becomes increasingly positive along the real axis, this HF potential +becomes attractive for $\lambda > 1$. +However, the explicit two-electron becomes increasingly repulsive for larger positive $\lambda$, +until eventually single electrons are successively expelled from the molecule. +Effect of symmetry-breaking: +Things become more subtle for the symmetry-broken case because we must now consider the $\alpha$ and $\beta$ HF potentials. +When UHF symmetry breaking occurs in the Hubbard dimer, with the $\alpha$ electron localising on the left site, +the $\alpha$ HF potential +will then be a repulsive interaction localised around the $\beta$ electron, so on the right site. +The same is true for the $\beta$ HF potential. +Therefore, as $\lambda$ becomes greater than 1 and the HF potentials become attractive, +there is a driving force for the $\alpha$ and $\beta$. +electrons to swap sites. If both electrons swap at the same time, then we get a double excitation. Therefore, we expect an +avoided crossing as $\lambda$ is increased beyond 1. +The "sharpness" of this avoided crossing is controlled by the strength of the electron-electron repulsion... see below. +} + +\hughDraft{% +For small $U/t$, the HF potentials will be weak and the electrons will slowly delocalise over +both sites as $\lambda$ increases. The UHF reference itself already has some degree of delocalisation over +both sites as we are only just beyond the CFP. This leads to a +"shallow" avoided crossing, and the corresponding EPs corresponding with have non-zero imaginary parts. +At larger $U/t$, the HF potentials will becomes increasingly repulsive and the "swapping" driving force will become stronger. +Furthermore, the strong on-site repulsion will make delocalisation over both sites increasingly unfavourable. +We therefore see that the electrons swap sites almost instantaneously as $\lambda$ passes through 1, as this is the threshold point +where the HF potential becomes attractive. This very rapid change in the nature of the state corresponds to a "sharp" avoided +crossing, with EPs close to the real axis. +Note that, although this appears to be an avoided crossing with the first-excited state, +by the time we have reached $\lambda \approx 1$, +we are beyond the excited-state avoided crossing and thus the first-excited state qualitatively corresponds to a double +excitation from the reference. This matches our expectation of both electrons swapping sites. +Okay we can't actually do $U/t = \infty$, but we can make it very large. With such a strong attractive HF potential and such strong +on-site repulsion, the electrons swapping process will occur exactly at $\lambda=1$. In this limit, the change is so sudden that it +now corresponds to a QPT in the perturbation approximation, and again the EPs fall on the real axis. +} + +\hughDraft{% +By applying the separation of the Hamiltionian proposed by Stillinger, we have been able to rationalise why the UMP ground-state +occurs so close to radius of convergence and increasingly close to the real axis for large $U/t$. The key feature here is the fact that the +HF potential is different for the $\alpha$ and $\beta$ electrons, and by extension the potential felt by +an electron is not strictly localised around +that electrons orbitals. This is completely different to the symmetric structure in the closed-shell case. Because of these different +potentials, there can be a large driving force for spatial rearrangement occurring as soon as the HF potential becomes positive +(at $\lambda=1$). +This sudden change in electron distribution is made more extreme if the HF potential and electron repulsion dominate over the +one-electron terms, which also corresponds to the regime of strong wave function symmetry breaking. +We have therefore provided a physical motivation behind why this EP can behave like a QPT for strong symmetry breaking. +This supports the conclusions in Antoine's report that the symmetry-broken EP behaves as a class $\beta$ singularity. +} + + %%==================================================== %\subsection{The physics of quantum phase transitions} %%==================================================== diff --git a/Manuscript/fig1a.pdf b/Manuscript/fig1a.pdf index bb1ba34..0c07b48 100644 Binary files a/Manuscript/fig1a.pdf and b/Manuscript/fig1a.pdf differ diff --git a/Manuscript/rmp_critical_point.pdf b/Manuscript/rmp_critical_point.pdf index 97da554..e2fe6c2 100644 Binary files a/Manuscript/rmp_critical_point.pdf and b/Manuscript/rmp_critical_point.pdf differ diff --git a/Manuscript/ump_critical_point.pdf b/Manuscript/ump_critical_point.pdf new file mode 100644 index 0000000..e3aa9b8 Binary files /dev/null and b/Manuscript/ump_critical_point.pdf differ diff --git a/Manuscript/ump_ep_to_cp.pdf b/Manuscript/ump_ep_to_cp.pdf new file mode 100644 index 0000000..02cf529 Binary files /dev/null and b/Manuscript/ump_ep_to_cp.pdf differ diff --git a/Notebooks/HughHubbarrdDimer.nb b/Notebooks/HughHubbarrdDimer.nb index bec10a6..e682e61 100644 --- a/Notebooks/HughHubbarrdDimer.nb +++ b/Notebooks/HughHubbarrdDimer.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 24583778, 414938] -NotebookOptionsPosition[ 24559535, 414545] -NotebookOutlinePosition[ 24560064, 414566] 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