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RapportStage/Rapport.tex
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@ -825,11 +825,13 @@ For $R=3/2$ the Hamiltonian is block diagonal and this is equivalent to the RHF
The singularity structure in this case is more complex because of the spin contamination of the wave function. We can not use CSFs in this case. So when we compute all the degeneracies using Eqs.~\eqref{eq:PolChar} and \eqref{eq:DPolChar} some correspond to EPs and some correspond to conical intersections. The numerical distinction of those singularities is very difficult so we will first look at the energies $E(\lambda)$ obtained with this basis set.
Figure \ref{fig:UHFMiniBas} is the analog of Fig.~\ref{fig:RHFMiniBas} in the UHF formalism. We see that in this case the sp\textsubscript{z} triplet interacts with the s\textsuperscript{2} and the p\textsubscript{z}\textsuperscript{2} singlets. Those avoided crossings are due to the spin contamination of the wave function. The EPs resulting from those avoided crossings will be discussed in Sec.~\ref{sec:uhfSing}.
In this study we have used spherical harmonics (or combination of spherical harmonics) as basis functions which have a delocalized nature. It would also be interesting to investigate the use of localized basis functions \cite{Seidl_2018} (for example gaussians) because these functions would be more adapted to describe the strongly correlated regime.
\titou{In this study we have used spherical harmonics (or combination of spherical harmonics) as basis functions which have a delocalized nature. It would also be interesting to investigate the use of localized basis functions \cite{Seidl_2018} (for example gaussians) because these functions would be more adapted to describe the strongly correlated regime.}
\titou{T2: Please move to concluding section.}
\subsection{Exceptional points in the UHF formalism}\label{sec:uhfSing}
In the RHF case there are only $\alpha$ singularities and large avoided crossings but we can see in Fig.~\ref{fig:UHFMiniBas} that in the UHF case there are sharp avoided crossings which are known to be connected to $\beta$ singularities. For example at $R=10$ the pair of singularities connected to the avoided crossing between s\textsuperscript{2} and sp\textsubscript{z} $^{3}P$ is $0.999\pm0.014\,i$. And the one between sp\textsubscript{z} $^{3}P$ and p\textsubscript{z}\textsuperscript{2} is connected with the singularities $2.207\pm0.023\,i$. However, in the spherium the electrons can't be ionized so those singularities cannot be the same $\beta$ singularities as the ones highlighted by Sergeev and Goodson \cite{Sergeev_2005}. We can see in Fig.~\ref{fig:UHFEP} that the s\textsuperscript{2} singlet and the sp\textsubscript{z} triplet states are degenerated for $R=3/2$. For $R>3/2$, it becomes an avoided crossing on the real axis and the degeneracies are $moved$ in the complex plane. The wave function is spin contaminated by $Y_1$ for $R>3/2$ this why the s\textsuperscript{2} singlet energy can not cross the sp\textsubscript{z} triplet curves anymore. When $R$ increases this avoided crossing becomes sharper. As presented before $\beta$ singularities are linked to quantum phase transition so it seems that this singularity is linked to the spin symmetry breaking of the UHF wave function. The fact that a similar pair of $\beta$ singularities appears for $R<-75/62$ confirms this assumption. A second pair of $\beta$ singularities appear for $R\gtrsim 2.5$, this is probably due to an excited-state quantum phase transition but this still need to be investigated.
\titou{T2: please modify structure of this section.}
Within the RHF formalism, we have observed only $\alpha$ singularities and large avoided crossings but one can see in Fig.~\ref{fig:UHFMiniBas} that in the UHF case there are sharp avoided crossings which are known to be connected to $\beta$ singularities. For example at $R=10$ the pair of singularities connected to the avoided crossing between s\textsuperscript{2} and sp\textsubscript{z} $^{3}P$ is $0.999\pm0.014\,i$. And the one between sp\textsubscript{z} $^{3}P$ and p\textsubscript{z}\textsuperscript{2} is connected with the singularities $2.207\pm0.023\,i$. However, in spherium, the electrons cannot be ionized so those singularities cannot be the same $\beta$ singularities as the ones highlighted by Sergeev and Goodson \cite{Sergeev_2005}. We can see in Fig.~\ref{fig:UHFEP} that the s\textsuperscript{2} singlet and the sp\textsubscript{z} triplet states are degenerated for $R=3/2$. For $R>3/2$, it becomes an avoided crossing on the real axis and the degeneracies are $moved$ in the complex plane. The wave function is spin contaminated by $Y_1$ for $R>3/2$ this is why the s\textsuperscript{2} singlet energy cannot cross the sp\textsubscript{z} triplet curves anymore. When $R$ increases this avoided crossing becomes sharper. As presented before $\beta$ singularities are linked to quantum phase transition so it seems that this singularity is linked to the spin symmetry breaking of the UHF wave function. The fact that a similar pair of $\beta$ singularities appears for $R<-75/62$ confirms this assumption. A second pair of $\beta$ singularities appear for $R\gtrsim 2.5$, this is probably due to an excited-state quantum phase transition but this still need to be investigated.
\begin{figure}
\centering
@ -839,9 +841,9 @@ In the RHF case there are only $\alpha$ singularities and large avoided crossing
\label{fig:UHFEP}
\end{figure}
As shown before, some matrix elements of the Hamiltonian become complex in the holomorphic domain. Therefore the Hamiltonian becomes non-Hermitian for those value of $R$. In \cite{Burton_2019a} Burton \textit{et al.}~proved that for the \ce{H_2} molecule the unrestricted Hamiltonian is not \pt -symmetric in the holomorphic domain. An analog reasoning can be done with the spherium model to prove the same result. The \pt -symmetry (invariance with respect to combined space reflection $\mathcal{P}$ and time reversal $\mathcal{T}$) is a property which ensures that a non-Hermitian Hamiltonian has a real energy spectrum. Thus \pt -symmetric Hamiltonian can be seen as an intermediate between Hermitian and non-Hermitian.
As shown before, some matrix elements of the Hamiltonian become complex in the holomorphic domain. Therefore the Hamiltonian becomes non-Hermitian for these values of $R$. In Ref.~\cite{Burton_2019a}, Burton \textit{et al.}~proved that for the \ce{H_2} molecule the unrestricted Hamiltonian is not \pt -symmetric in the holomorphic domain. An analog reasoning can be done with the spherium model to prove the same result. The \pt -symmetry (invariance with respect to combined space reflection $\mathcal{P}$ and time reversal $\mathcal{T}$) is a property which ensures that a non-Hermitian Hamiltonian has a real energy spectrum \cite{BenderBook}. Thus \pt -symmetric Hamiltonians can be seen as an intermediate class between Hermitian and non-Hermitian Hamiltonians.
Figure \ref{fig:UHFPT} shows that for the spherium model a part of the energy spectrum becomes complex when R is in the holomorphic domain. The parameter domain of value where the energy becomes complex is called the broken \pt symmetry region. This is consistent with the fact that in the holomorphic domain the Hamiltonian is no more \pt -symmetric.
Figure \ref{fig:UHFPT} shows that for the spherium model a part of the energy spectrum becomes complex when $R$ is in the holomorphic domain. The domain of values where the energy becomes complex is called the broken \pt-symmetry region. This is consistent with the fact that in the holomorphic domain the Hamiltonian is no more \pt -symmetric.
\begin{figure}
\centering
@ -851,7 +853,7 @@ Figure \ref{fig:UHFPT} shows that for the spherium model a part of the energy sp
\label{fig:UHFPT}
\end{figure}
For a non-Hermitian Hamiltonian the EPs can lie on the real axis. In particular, at the point of PT transition (the point where the energies become complex) the two energies are degenerate resulting in such an EP on the real axis. This degeneracy can be seen in Fig.~\ref{fig:UHFPT}.
For a non-Hermitian Hamiltonian the EPs can lie on the real axis. In particular, at the point of {\pt} transition (the point where the energies become complex) the two energies are degenerate resulting in such an EP on the real axis. This degeneracy can be seen in Fig.~\ref{fig:UHFPT}.
\section{Conclusion}