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Added discussion on UMP relationship to critical point

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Hugh Burton 2020-11-30 16:19:24 +00:00
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@ -117,12 +117,14 @@
\newcommand{\bbR}{\mathbb{R}} \newcommand{\bbR}{\mathbb{R}}
\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbC}{\mathbb{C}}
% Making life easier
\newcommand{\Lup}{\mathcal{L}^{\uparrow}} \newcommand{\Lup}{\mathcal{L}^{\uparrow}}
\newcommand{\Ldown}{\mathcal{L}^{\downarrow}} \newcommand{\Ldown}{\mathcal{L}^{\downarrow}}
\newcommand{\Lsi}{\mathcal{L}^{\sigma}} \newcommand{\Lsi}{\mathcal{L}^{\sigma}}
\newcommand{\Rup}{\mathcal{R}^{\uparrow}} \newcommand{\Rup}{\mathcal{R}^{\uparrow}}
\newcommand{\Rdown}{\mathcal{R}^{\downarrow}} \newcommand{\Rdown}{\mathcal{R}^{\downarrow}}
\newcommand{\Rsi}{\mathcal{R}^{\sigma}} \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.} \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} \subsection{Spin-Contamination in the Hubbard Dimer}
\label{sec:spin_cont}
%==========================================% %==========================================%
%%% FIG 2 %%% %%% FIG 2 %%%
@ -1362,83 +1365,75 @@ set representations of the MP critical point.\cite{Sergeev_2006}
% Figure on the UMP critical point % Figure on the UMP critical point
%------------------------------------------------------------------% %------------------------------------------------------------------%
\begin{figure*}[t] \begin{figure*}[t]
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
\includegraphics[height=0.65\textwidth]{ump_critical_point} \includegraphics[height=0.65\textwidth]{ump_critical_point}
\subcaption{\label{subfig:ump_cp}} \subcaption{\label{subfig:ump_cp}}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
\includegraphics[height=0.65\textwidth]{ump_ep_to_cp} \includegraphics[height=0.65\textwidth]{ump_ep_to_cp}
\subcaption{\label{subfig:ump_ep_to_cp}} \subcaption{\label{subfig:ump_ep_to_cp}}
\end{subfigure} \end{subfigure}
\caption{% \caption{%
\hugh{Modelling the RMP critical point using the asymmetric Hubbard dimer. \hugh{%
(\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed). The UMP ground-state EP becomes a critical point in the strong correlation limit (large $U/t$).
(\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the (\subref{subfig:ump_cp}) As the $U/t$ increases, the avoided crossing on the real $\lambda$ axis
real axis, giving a sharp avoided crossing on the real axis (solid). becomes increasingly sharp.
} (\subref{subfig:ump_ep_to_cp}) The convergence of the EPs onto the real axis for $U/t \rightarrow \infty$
\label{fig:RMP_cp}} mirrors the formation of the RMP critical point and other QPTs in the infinite basis set limit.
}
\label{fig:UMP_cp}}
\end{figure*} \end{figure*}
%------------------------------------------------------------------% %------------------------------------------------------------------%
% RELATIONSHIP BETWEEN QPT AND UMP % RELATIONSHIP BETWEEN QPT AND UMP
\hughDraft{% \hugh{%
The ground-state EP observed in the UMP series also has a small imaginary part and falls close to the real axis. In Section~\ref{sec:spin_cont} we showed that the slow convergence of the UMP series for the strongly correlated
As the correlation strength increases, this EP moves closer to the real axis and closer to the radius of convergence at Hubbard dimer was due to a complex-conjguate pair of EPs just outside the radius of convergence.
$\lambda = 1$. So can we understand this using the arguments related to the critical point? These EPs have positive real components and small imaginary components (Fig.~\ref{fig:UMP}), suggesting a potential
Closed-shell case: connection to MP critical points and QPTs.
The work by Stillinger builds an argument around the HF potential $v_{\text{HF}}$ which, by itself, For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's
is repulsive and concentrated around the Hamiltonian [Eq.~\eqref{eq:HamiltonianStillinger}], while the explicit electron-electron interaction
occupied orbitals. As $\lambda$ becomes increasingly positive along the real axis, this HF potential becomes increasingly repulsive.
becomes attractive for $\lambda > 1$. Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent
However, the explicit two-electron becomes increasingly repulsive for larger positive $\lambda$, points were the two-electron repulsion overcomes the attractive HF potential and an electron
until eventually single electrons are successively expelled from the molecule. are successively expelled from the molecule.\cite{Sergeev_2006}
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{% \hugh{%
For small $U/t$, the HF potentials will be weak and the electrons will slowly delocalise over Symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
both sites as $\lambda$ increases. The UHF reference itself already has some degree of delocalisation over Consider the reference UHF solution where the spin-up and spin-down electrons are localised on the left and
both sites as we are only just beyond the CFP. This leads to a right sites respectively.
"shallow" avoided crossing, and the corresponding EPs corresponding with have non-zero imaginary parts. The spin-up HF potential will then be a repulsive interaction from the spin-down electron
At larger $U/t$, the HF potentials will becomes increasingly repulsive and the "swapping" driving force will become stronger. density that is centred around the right site (and vice-versa).
Furthermore, the strong on-site repulsion will make delocalisation over both sites increasingly unfavourable. As $\lambda$ becomes greater than 1 and the HF potentials become attractive, there will be a sudden
We therefore see that the electrons swap sites almost instantaneously as $\lambda$ passes through 1, as this is the threshold point driving force for the electrons to swap sites.
where the HF potential becomes attractive. This very rapid change in the nature of the state corresponds to a "sharp" avoided This swapping process can also be represented as a double excitation, and thus an avoided crossing will occur
crossing, with EPs close to the real axis. for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
Note that, although this appears to be an avoided crossing with the first-excited state, Note that, although this appears to be an avoided crossing between the ground and first-excited state,
by the time we have reached $\lambda \approx 1$, the earlier excited-state avoided crossing means that the first-excited state qualitatively
we are beyond the excited-state avoided crossing and thus the first-excited state qualitatively corresponds to a double represents the double excitation for $\lambda > 0.5.$
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{% % SHARPNESS AND QPT
By applying the separation of the Hamiltionian proposed by Stillinger, we have been able to rationalise why the UMP ground-state \hugh{%
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 The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
HF potential is different for the $\alpha$ and $\beta$ electrons, and by extension the potential felt by For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites,
an electron is not strictly localised around both in the UHF reference and as $\lambda$ increases.
that electrons orbitals. This is completely different to the symmetric structure in the closed-shell case. Because of these different This delocalisataion dampens the electron swapping process and leads to a ``shallow'' avoided crossing
potentials, there can be a large driving force for spatial rearrangement occurring as soon as the HF potential becomes positive that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}).
(at $\lambda=1$). As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping
This sudden change in electron distribution is made more extreme if the HF potential and electron repulsion dominate over the term to make electron delocalisation less favourable.
one-electron terms, which also corresponds to the regime of strong wave function symmetry breaking. These effects create a stronger driving force for the electrons to swap sites until eventually this swapping
We have therefore provided a physical motivation behind why this EP can behave like a QPT for strong symmetry breaking. occurs exactly at $\lambda = 1$.
This supports the conclusions in Antoine's report that the symmetry-broken EP behaves as a class $\beta$ singularity. 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} %\subsection{The physics of quantum phase transitions}
%%==================================================== %%====================================================