Transferred RMP critical point discussion from notebook as skeleton.

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Hugh Burton 2020-11-29 18:48:40 +00:00
parent 1900836b7f
commit f9b9d520ac
4 changed files with 223221 additions and 16874 deletions

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@ -9,6 +9,7 @@
\definecolor{hughgreen}{RGB}{0, 128, 0} \definecolor{hughgreen}{RGB}{0, 128, 0}
\newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\hugh}[1]{\textcolor{hughgreen}{#1}} \newcommand{\hugh}[1]{\textcolor{hughgreen}{#1}}
\newcommand{\hughDraft}[1]{\textcolor{orange}{#1}}
\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}} \newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashHB}[1]{\textcolor{orange}{\sout{#1}}} \newcommand{\trashHB}[1]{\textcolor{orange}{\sout{#1}}}
@ -1239,10 +1240,95 @@ In contrast, $\alpha$ singularities correspond to large avoided crossings that a
states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT. states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT.
%======================================= %=======================================
%\subsection{Critical Point in the Hubbard Dimer} \subsection{Critical Points in the Hubbard Dimer}
%\label{sec:critical_point_hubbard} \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{RMP critical point in the Hubbard dimer.}
\label{fig:RMP_cp}}
\end{figure*}
%------------------------------------------------------------------%
\hughDraft{%
The simplified site basis of the Hubbard dimer makes explicitly modelling the ionisation continuum
impossible.
To model the auto-ionisation in the Hubbard dimer, we need to turn one of the sites into a ghost atom that will act as
a destination for ionised electrons. We do this by considering an asymmetric Hubbard dimer where the "atomic" site
has a negative diagonal term in the one-electron Hamiltonian, representing the nuclear attraction. This term is already
encoded in the Initialisation section such that the core Hamiltonian is
To model the doubly-occupied atom, we can define our reference HF state as the configuration with $\theta = 0$ and energy
\begin{equation}
\frac{1}{2} (2 U - 4 \epsilon)
\end{equation}
The RMP Hamiltonian then becomes
...
}
\hughDraft{%
Now let's think about the physics of the problem...
For the ghost site to truly represent ionised electrons, we need the hopping term to vanish (or become very small).
$U$ controls the strength of the HF repulsive potential. A stronger repulsion will encourage the electrons to be forced
away from the "atomic" site at a less negative value of $\lambda$.
$\epsilon$ controls the strength of attraction to the atom. A stronger attraction to the nucleus will mean that the electrons
are more tightly bound to the atom and a more negative $\lambda$ will be needed for auto-ionisation.
We therefore expect that the position of the RMP critical point will be controlled by the ratio $\epsilon / U$, with smaller ratios
making ionisation occur closer to $\lambda$ origin, and making the divergence in these cases more likely.
}
\hughDraft{%
Taking the exact case with $t=0$, the RMP energies becomes
\begin{subequations}
\begin{align}
E_{-} &= 2U - 2 \epsilon - U \lambda
\\
E_{\text{S}} &= U - \epsilon - U \lambda
\\
E_{+} &= U \lambda
\end{align}
\end{subequations}
By comparison with Fig.~\ref{fig:RMP_cp}, the critical point can be identified as by solving $E_{-} = E_{+}$,
giving
\begin{equation}
\lc = 1 - \epsilon / U.
\end{equation}
Thus, as expected, the critical point lies on the real axis and moves closer to the origin for larger
$U / \epsilon$.
}
\hughDraft{%
The position of the critical point is controlled by the ratio $\epsilon / U$.
If the magnitude of the ratio becomes greater than one, then the series diverges.
We can interpret large $U$ as strong electron repulsion effects in electron dense molecules,
such as \ce{F-}. A small $\epsilon$ is also likely to correspond to
strong nuclear screening by the core and valence electrons. Both of these factors are
common in atoms on the right-hand-side of periodic table,
\eg\ \ce{F}, \ce{O}, \ce{Ne}, etc, as well as negatively-charged species, and so we recover the
class $\beta$ system classification.
}
\hughDraft{%
In practical calculations, one considers the perturbation energy in a finite basis and the critical point is modelled
as a cluster of branch points close to the real axis. While we cannot change the size of our basis, we can adjust
the extent to which the second site behaves as a ghost by making the hopping term larger.
When $t$ is (slightly) non-zero, our modelled critical point becomes an EP point close to the real axis with a sharp
associated avoided crossing for real $\lambda$. If $t$ becomes larger, this avoided crossing becomes less sharp and the EPs
move away from the real axis. This mirrors the discussion of EPs approaching the real axis in the exact basis.
This effect is shown by the dashed lines in Fig.~\ref{fig:RMP_cp}.
}
%%==================================================== %%====================================================
%\subsection{The physics of quantum phase transitions} %\subsection{The physics of quantum phase transitions}
%%==================================================== %%====================================================

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