From f9b9d520ac51fcae286f01146e4936675e04fde4 Mon Sep 17 00:00:00 2001 From: Hugh Burton Date: Sun, 29 Nov 2020 18:48:40 +0000 Subject: [PATCH] Transferred RMP critical point discussion from notebook as skeleton. --- Manuscript/EPAWTFT.tex | 90 +- Manuscript/rmp_critical_point.pdf | Bin 0 -> 18861 bytes Manuscript/rmp_critical_point_surf.pdf | Bin 0 -> 570930 bytes Notebooks/HughHubbarrdDimer.nb | 240005 ++++++++++++++++++++-- 4 files changed, 223221 insertions(+), 16874 deletions(-) create mode 100644 Manuscript/rmp_critical_point.pdf create mode 100644 Manuscript/rmp_critical_point_surf.pdf diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index c3342da..5e3f250 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -9,6 +9,7 @@ \definecolor{hughgreen}{RGB}{0, 128, 0} \newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\hugh}[1]{\textcolor{hughgreen}{#1}} +\newcommand{\hughDraft}[1]{\textcolor{orange}{#1}} \newcommand{\trash}[1]{\textcolor{red}{\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. %======================================= -%\subsection{Critical Point in the Hubbard Dimer} -%\label{sec:critical_point_hubbard} +\subsection{Critical Points in the Hubbard Dimer} +\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} %%==================================================== diff --git a/Manuscript/rmp_critical_point.pdf b/Manuscript/rmp_critical_point.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7215bebc6b76ec741e07bf0ed80b6d4a514c716d GIT binary patch literal 18861 zcma&O1z4Rs(>08{7KZ{GcXxMpcXxO9;!vb`ai@53hvHD&i@UoNE%0x8PS0~*yS|?i zlFURVb7#}Z%Hk$d6cwjqqGy96YdEcXf@1|R0vwF3;dprg46(l#%ZQlirUk?Va+dsp*1Pos91LAHT0|p-+X`Zep2G?J*_;?Lo9g{q&s$U;k zwf&P-E1o%CeA@lJp2sf3yA8k^t1gc-3AF+Hq z&gnStCfe$c8@rTST#=9DVIOlY{Ia*$(^)4o1Jh_d*@ddlgxT>0 zZ(e`sQJ8L9K*;T?{$v0hZ(8uQJ2!rJQ0*JzA=7`**5Qz0)@kdCg?n>*w(#@&u6GW& z1I~PYyYt!kEKnq)qMo?)z%r*rq%|>3}$Iyo7Z=gOe{G zJ!*wOT1R#z)id15ppRA~H?qPEb4^gV+T;)wy$$Aha}M>SO$v_f(7wvmSV1XP3vaZ} 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