From dc167624d182e12f5197f3d9b49aac2a09cc4c71 Mon Sep 17 00:00:00 2001 From: Hugh Burton Date: Thu, 19 Nov 2020 12:40:15 +0000 Subject: [PATCH] comment on complex adiabatic connection, added figure legend --- Manuscript/EPAWTFT.tex | 68 ++++++++++++++++++++++++++---------------- 1 file changed, 42 insertions(+), 26 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 1f0823e..dabb340 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -499,6 +499,28 @@ and the ground-state RHF energy (Fig.~\ref{fig:HF_real}) However, in the strongly correlated regime (large $U$), the closed-shell restriction on the orbitals prevents RHF from correctly modelling the physics of the system with the two electrons on opposing sites. +%%% FIG 3 (?) %%% +% Analytic Continuation of HF +%%%%%%%%%%%%%%%%% +\begin{figure*}[t] + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -35pt},clip]{HF_cplx_angle} + \subcaption{\label{subfig:UHF_cplx_angle}} + \end{subfigure} + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth]{HF_cplx_energy} + \subcaption{\label{subfig:UHF_cplx_energy}} + \end{subfigure} + \caption{% + \hugh{(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$. + Symmetry-broken solutions correspond to an individual sheets and become equivalent at the quasi-EP (black dot). + The RHF solution is independent of $\lambda$, giving constant plane at $\pi/2$. + (\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic + point at the quasi-exceptional point.} + \label{fig:HF_cplx}} +\end{figure*} +%%%%%%%%%%%%%%%%% + As the on-site repulsion is increased from 0, the HF approximation reaches a critical value at $U=2t$ where a symmetry-broken UHF solution appears with a lower energy than the RHF one. This critical point is analogous to the infamous Coulson--Fischer point identified in the hydrogen dimer.\cite{Coulson_1949} @@ -523,24 +545,6 @@ of the HF energy rather than a minimum. \subsection{Self-consistency as a perturbation} %OR {Complex adiabatic connection} %============================================================% -%%% FIG 3 (?) %%% -% Analytic Continuation of HF -%%%%%%%%%%%%%%%%% -\begin{figure*}[t] - \begin{subfigure}{0.49\textwidth} - \includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -30pt},clip]{HF_cplx_angle} - \subcaption{\label{subfig:UHF_cplx_angle}} - \end{subfigure} - \begin{subfigure}{0.49\textwidth} - \includegraphics[height=0.65\textwidth]{HF_cplx_energy} - \subcaption{\label{subfig:UHF_cplx_energy}} - \end{subfigure} - \caption{% - Analytic continuation of HF into the complex $\lambda$ plane. - \label{fig:HF_cplx}} -\end{figure*} -%%%%%%%%%%%%%%%%% - % INTRODUCE PARAMETRISED FOCK HAMILTONIAN \hugh{The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency in the HF approximation, and is usually solved through an iterative approach.\cite{SzaboBook} @@ -559,6 +563,7 @@ in the Hubbard dimer directly mirror the energies shown in Fig.~\ref{fig:HF_real with coalesence points at \begin{equation} \lambda_{\text{c}} = \pm \frac{2t}{U}. + \label{eq:scaled_fock} \end{equation} In contrast, when $\lambda$ becomes complex, the HF equations become non-Hermitian and each HF solutions can be analytically continued for all $\lambda$ values using @@ -573,16 +578,27 @@ accurate representation for the true HF ground state at $\lambda = 1$. For example, in the Hubbard dimer with $U > 2t$, one finds $\lambda_{\text{c}} < 1$ and the symmetry-pure orbitals do not provide a good representation of the HF ground state. In contrast, $U < 2t$ yields $\lambda_{\text{c}} > 1$ and corresponds to -the regime where the HF ground state is correctly represented by symmetry-pure orbitals.} +the regime where the HF ground state is correctly represented by symmetry-pure orbitals. +} % COMPLEX ADIABATIC CONNECTION -\titou{In a recent paper, \cite{Burton_2019} using holomorphic Hartree-Fock (h-HF) \cite{Hiscock_2014,Burton_2018,Burton_2016} as an analytic continuation of conventional HF theory, we have demonstrated, on a simple model, that one can interconvert states of different symmetries and natures by following well-defined contours in the complex $\lambda$-plane, where $\lambda$ is the strength of the electron-electron interaction (see Fig.~\ref{fig:iAC}). -In particular, by slowly varying $\lambda$ in a similar (yet different) manner to an adiabatic connection in density-functional theory, \cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} one can ``morph'' a ground-state wave function into an excited-state wave function via a stationary path of HF solutions. \cite{Seidl_2018} -In such a way, we could obtain a doubly-excited state wave function starting from the ground state wave function, a process which is not as easy as one might think. \cite{Gilbert_2008,Thom_2008,Shea_2018} -One of the fundamental discovery we made was that Coulson-Fisher points (where multiple symmetry-broken solutions coalesce) play a central role and can be classified as \textit{quasi}-exceptional points, as the wave functions do not become self-orthogonal. -The findings reported in Ref.~\onlinecite{Burton_2019} represent the very first study of non-Hermitian quantum mechanics for the exploration of multiple solutions at the HF level. -It perfectly illustrates the deeper topology of electronic states revealed using a complex-scaled electron-electron interaction. -Through the introduction of non-Hermiticity, we have provided a more general framework in which the complex and diverse characteristics of multiple solutions can be explored and understood.} +\hugh{We have recently shown that the complex scaled Fock operator Eq.~\eqref{eq:scaled_fock} +also allows states of different symmetries to be interconverted by following a well-defined +contour in the complex $\lambda$-plane.\cite{Burton_2019} +In particular, by slowly varying $\lambda$ in a similar (yet different) manner +to an adiabatic connection in density-functional theory,\cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} +a ground-state wave function can be ``morphed'' into an excited-state wave function +via a stationary path of HF solutions. +This novel approach to identifying excited-state wave functions demonstrates the fundamental +role of quasi-EPs in determining the behaviour of the HF approximation.} + +%\titou{In a recent paper, \cite{Burton_2019} using holomorphic Hartree-Fock (h-HF) \cite{Hiscock_2014,Burton_2018,Burton_2016} as an analytic continuation of conventional HF theory, we have demonstrated, on a simple model, that one can interconvert states of different symmetries and natures by following well-defined contours in the complex $\lambda$-plane, where $\lambda$ is the strength of the electron-electron interaction (see Fig.~\ref{fig:iAC}). +%In particular, by slowly varying $\lambda$ in a similar (yet different) manner to an adiabatic connection in density-functional theory, \cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} one can ``morph'' a ground-state wave function into an excited-state wave function via a stationary path of HF solutions. \cite{Seidl_2018} +%In such a way, we could obtain a doubly-excited state wave function starting from the ground state wave function, a process which is not as easy as one might think. \cite{Gilbert_2008,Thom_2008,Shea_2018} +%One of the fundamental discovery we made was that Coulson-Fisher points (where multiple symmetry-broken solutions coalesce) play a central role and can be classified as \textit{quasi}-exceptional points, as the wave functions do not become self-orthogonal. +%The findings reported in Ref.~\onlinecite{Burton_2019} represent the very first study of non-Hermitian quantum mechanics for the exploration of multiple solutions at the HF level. +%It perfectly illustrates the deeper topology of electronic states revealed using a complex-scaled electron-electron interaction. +%Through the introduction of non-Hermiticity, we have provided a more general framework in which the complex and diverse characteristics of multiple solutions can be explored and understood.} %=====================================================% \subsection{M{\o}ller-Plesset perturbation theory}