From 921e63f69aeaa44926f67c7fc0e0951c06c73c6d Mon Sep 17 00:00:00 2001 From: Hugh Burton Date: Mon, 30 Nov 2020 16:19:24 +0000 Subject: [PATCH] Added discussion on UMP relationship to critical point --- Manuscript/EPAWTFT.tex | 129 ++++++++++++++++++++--------------------- 1 file changed, 62 insertions(+), 67 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 5d2887a..bf73c65 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -117,12 +117,14 @@ \newcommand{\bbR}{\mathbb{R}} \newcommand{\bbC}{\mathbb{C}} +% Making life easier \newcommand{\Lup}{\mathcal{L}^{\uparrow}} \newcommand{\Ldown}{\mathcal{L}^{\downarrow}} \newcommand{\Lsi}{\mathcal{L}^{\sigma}} \newcommand{\Rup}{\mathcal{R}^{\uparrow}} \newcommand{\Rdown}{\mathcal{R}^{\downarrow}} \newcommand{\Rsi}{\mathcal{R}^{\sigma}} +\newcommand{\vhf}{v_{\text{HF}}} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.} @@ -840,6 +842,7 @@ gradient discontinuities or spurious minima. %==========================================% \subsection{Spin-Contamination in the Hubbard Dimer} +\label{sec:spin_cont} %==========================================% %%% FIG 2 %%% @@ -1362,83 +1365,75 @@ 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}} +\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{% + The UMP ground-state EP becomes a critical point in the strong correlation limit (large $U/t$). + (\subref{subfig:ump_cp}) As the $U/t$ increases, the avoided crossing on the real $\lambda$ axis + becomes increasingly sharp. + (\subref{subfig:ump_ep_to_cp}) The convergence of the EPs onto the real axis for $U/t \rightarrow \infty$ + mirrors the formation of the RMP critical point and other QPTs in the infinite basis set limit. +} +\label{fig:UMP_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. +\hugh{% +In Section~\ref{sec:spin_cont} we showed that the slow convergence of the UMP series for the strongly correlated +Hubbard dimer was due to a complex-conjguate pair of EPs just outside the radius of convergence. +These EPs have positive real components and small imaginary components (Fig.~\ref{fig:UMP}), suggesting a potential +connection to MP critical points and QPTs. +For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's +Hamiltonian [Eq.~\eqref{eq:HamiltonianStillinger}], while the explicit electron-electron interaction +becomes increasingly repulsive. +Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent +points were the two-electron repulsion overcomes the attractive HF potential and an electron +are successively expelled from the molecule.\cite{Sergeev_2006} } -\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. +\hugh{% +Symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons. +Consider the reference UHF solution where the spin-up and spin-down electrons are localised on the left and +right sites respectively. +The spin-up HF potential will then be a repulsive interaction from the spin-down electron +density that is centred around the right site (and vice-versa). +As $\lambda$ becomes greater than 1 and the HF potentials become attractive, there will be a sudden +driving force for the electrons to swap sites. +This swapping process can also be represented as a double excitation, and thus an avoided crossing will occur +for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}). +Note that, although this appears to be an avoided crossing between the ground and first-excited state, +the earlier excited-state avoided crossing means that the first-excited state qualitatively +represents the double excitation for $\lambda > 0.5.$ } -\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. +% SHARPNESS AND QPT +\hugh{% +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. +This delocalisataion 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 +term to make electron delocalisation less favourable. +These effects create a stronger driving force for the electrons to swap sites until eventually this swapping +occurs exactly at $\lambda = 1$. +In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided +crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}). +We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes becomes +a new type of MP critical and represents a QPT in the perturbation approximation. +Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the +radius of convergence (see Fig.~\ref{fig:RadConv}). } - %%==================================================== %\subsection{The physics of quantum phase transitions} %%====================================================