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starting MP

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Pierre-Francois Loos 2020-11-17 22:32:00 +01:00
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Manuscript/EPAWTFT.tex
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@ -174,6 +174,7 @@ More importantly here, although EPs usually lie off the real axis, these singula
%===================================% %===================================%
\subsection{Illustrative Example} \subsection{Illustrative Example}
\label{sec:example}
%===================================% %===================================%
%%% FIG 1 %%% %%% FIG 1 %%%
@ -201,7 +202,7 @@ Using the (localised) site basis, the (singlet) Hilbert space of the Hubbard dim
& \ket{\Lup \Ldown} & & \ket{\Lup\Rdown} & & \ket{\Rup\Ldown} & & \ket{\Rup\Rdown} & \ket{\Lup \Ldown} & & \ket{\Lup\Rdown} & & \ket{\Rup\Ldown} & & \ket{\Rup\Rdown}
\end{align} \end{align}
where $\Lsi$ ($\Rsi$) denotes an electron with spin $\sigma$ on the left (right) site. where $\Lsi$ ($\Rsi$) denotes an electron with spin $\sigma$ on the left (right) site.
The exact Hamiltonian is then The exact [or full configuration interaction (FCI)] Hamiltonian is then
\begin{equation} \begin{equation}
\label{eq:H_FCI} \label{eq:H_FCI}
\bH = \bH =
@ -315,7 +316,7 @@ This function is smooth for $x \in \mathbb{R}$ and infinitely differentiable in
In the Hartree-Fock (HF) approximation, the many-electron wave function is approximated as a single Slater determinant $\Psi^{\text{HF}}(\vb{x}_1,\ldots,\vb{x}_N)$ [where $\vb{x} = (\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates] defined as an antisymmetric combination of $N$ (real-valued) one-electron spin-orbitals $\phi_p(\vb{x})$, which are, by definition, eigenfunctions of the one-electron Fock operator In the Hartree-Fock (HF) approximation, the many-electron wave function is approximated as a single Slater determinant $\Psi^{\text{HF}}(\vb{x}_1,\ldots,\vb{x}_N)$ [where $\vb{x} = (\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates] defined as an antisymmetric combination of $N$ (real-valued) one-electron spin-orbitals $\phi_p(\vb{x})$, which are, by definition, eigenfunctions of the one-electron Fock operator
\begin{equation}\label{eq:FockOp} \begin{equation}\label{eq:FockOp}
\Hat{f}(\vb{x}) \phi_p(\vb{x}) = [ \Hat{h}(\vb{x}) + \Hat{v}^\text{HF}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}), \Hat{f}(\vb{x}) \phi_p(\vb{x}) = [ \Hat{h}(\vb{x}) + \Hat{v}_\text{HF}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}),
\end{equation} \end{equation}
where where
\begin{equation} \begin{equation}
@ -323,7 +324,7 @@ where
\end{equation} \end{equation}
is the core Hamiltonian and is the core Hamiltonian and
\begin{equation} \begin{equation}
\Hat{v}^\text{HF}(\vb{x}) = \sum_i^{N} \qty[ \Hat{J}_i(\vb{x}) - \Hat{K}_i(\vb{x}) ] \Hat{v}_\text{HF}(\vb{x}) = \sum_i^{N} \qty[ \Hat{J}_i(\vb{x}) - \Hat{K}_i(\vb{x}) ]
\end{equation} \end{equation}
is the HF mean-field potential with is the HF mean-field potential with
\begin{subequations} \begin{subequations}
@ -354,7 +355,7 @@ From hereon, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied (or v
Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational principle to find an approximation of $\Psi$ as a single Slater determinant. Hence a Slater determinant is not an eigenfunction of the exact Hamiltonian $\hH$. However, it is, by definition, an eigenfunction of the so-called (approximated) HF many-electron Hamiltonian defined as the sum of the one-electron Fock operators Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational principle to find an approximation of $\Psi$ as a single Slater determinant. Hence a Slater determinant is not an eigenfunction of the exact Hamiltonian $\hH$. However, it is, by definition, an eigenfunction of the so-called (approximated) HF many-electron Hamiltonian defined as the sum of the one-electron Fock operators
\begin{equation}\label{eq:HFHamiltonian} \begin{equation}\label{eq:HFHamiltonian}
\hH^{\text{HF}} = \sum_{i} f(\vb{x}_i). \hH_{\text{HF}} = \sum_{i} f(\vb{x}_i).
\end{equation} \end{equation}
% %
@ -502,8 +503,9 @@ However, the choice of the functional form of the fit remains a subtle task.
This technique was first generalized by using complex scaling parameters and applying analytic continuation by solving the Laplace equation, \cite{Surjan_2018} and then further improved thanks to Cauchy's integral formula \cite{Mihalka_2019} This technique was first generalized by using complex scaling parameters and applying analytic continuation by solving the Laplace equation, \cite{Surjan_2018} and then further improved thanks to Cauchy's integral formula \cite{Mihalka_2019}
\begin{equation} \begin{equation}
\label{eq:Cauchy} \label{eq:Cauchy}
\frac{1}{2\pi i} \oint_{\gamma} \frac{E(z)}{z - a} = E(a). \frac{1}{2\pi i} \oint_{\gamma} \frac{E(z)}{z - a} = E(a),
\end{equation} \end{equation}
which states that the value of the energy can be computed at $z=a$ inside the contour $\gamma$ only by the knowledge of its values on the same contour.
Their method consists in refining self-consistently the values of $E(z)$ computed on a contour going through the physical point at $z = 1$ and encloses points of the ``trusted'' region (where the MP series is convergent). The shape of this contour is arbitrary but no singularities are allowed inside the contour to ensure $E(z)$ is analytic. Their method consists in refining self-consistently the values of $E(z)$ computed on a contour going through the physical point at $z = 1$ and encloses points of the ``trusted'' region (where the MP series is convergent). The shape of this contour is arbitrary but no singularities are allowed inside the contour to ensure $E(z)$ is analytic.
When the values of $E(z)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(z=1)$ which corresponds to the final estimate of the FCI energy. When the values of $E(z)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(z=1)$ which corresponds to the final estimate of the FCI energy.
The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019} The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019}
@ -521,11 +523,11 @@ The authors illustrate this protocol on the dissociation curve of \ce{LiH} and t
\label{fig:RMP}} \label{fig:RMP}}
\end{figure*} \end{figure*}
Let us illustrate the behavior of the RMP and UMP series on the Hubbard dimer (see Fig.~\ref{fig:RMP}). Let us illustrate the behaviour of the RMP and UMP series on the Hubbard dimer (see Fig.~\ref{fig:RMP}).
Within the RMP parition thecnique, we have Within the RMP partition technique, we have
\begin{equation} \begin{equation}
\label{eq:H_RMP} \label{eq:H_RMP}
\bH^\text{RMP} = \bH_\text{RMP} =
\begin{pmatrix} \begin{pmatrix}
-2t + U - \lambda U/2 & 0 & 0 & \lambda U/2 \\ -2t + U - \lambda U/2 & 0 & 0 & \lambda U/2 \\
0 & U - \lambda U/2 & \lambda U/2 & 0 \\ 0 & U - \lambda U/2 & \lambda U/2 & 0 \\
@ -536,25 +538,24 @@ Within the RMP parition thecnique, we have
which yields the ground-state energy which yields the ground-state energy
\begin{equation} \begin{equation}
\label{eq:E0MP} \label{eq:E0MP}
E_0(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2} E_{-}(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2}.
\end{equation} \end{equation}
From this expression, it is easy to identify that the radius of convergence is defined by the exceptional points at $\lambda = \pm i 4t / U$ (which is similar to the FCI exceptional point). From this expression, it is easy to identify that the radius of convergence is defined by the EPs at $\lambda = \pm i 4t / U$ (which are actually located at the same position than the exact EPs discussed in Sec.~\ref{sec:example}).
Equation \eqref{eq:E0MP} can be Taylor expanded in terms of $\lambda$ to obtain the $n$th-order MP correction Equation \eqref{eq:E0MP} can be Taylor expanded in terms of $\lambda$ to obtain the $n$th-order MP correction
\begin{equation} \begin{equation}
E_0^{(n)} = U \delta_{0,n} - \frac{1}{2} \frac{U^n}{(4t)^{n-1}} \mqty( n/2 \\ 1/2) E_\text{MP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2),
\end{equation} \end{equation}
with with
\begin{equation} \begin{equation}
E_0(\lambda) = \sum_{n=0}^\infty E_0^{(n)} \lambda^n E_{-}(\lambda) = \sum_{k=0}^\infty E_\text{MP}^{(k)} \lambda^k.
\end{equation} \end{equation}
We illustrate this by plotting the energy expansions at orders of $\lambda$ just below and above the radius of convergence in Fig.~\ref{fig:RMP}. We illustrate this by plotting the energy expansions at orders of $\lambda$ just below and above the radius of convergence in Fig.~\ref{fig:RMP}.
\titou{T2 is going to add a discussion about this figure.}
At the UMP level now, we have At the UMP level now, we have
\begin{widetext} \begin{widetext}
\begin{equation} \begin{equation}
\label{eq:H_UMP} \label{eq:H_UMP}
\bH^\text{UMP} = \bH_\text{UMP} =
\begin{pmatrix} \begin{pmatrix}
-2t^2 \lambda/U & 0 & 0 & +2t^2 \lambda/U \\ -2t^2 \lambda/U & 0 & 0 & +2t^2 \lambda/U \\
0 & U - 2t^2 \lambda/U & +2t^2\lambda/U & +2t \sqrt{U^2 - (2t)^2} \lambda/U \\ 0 & U - 2t^2 \lambda/U & +2t^2\lambda/U & +2t \sqrt{U^2 - (2t)^2} \lambda/U \\
@ -564,7 +565,7 @@ At the UMP level now, we have
\end{equation} \end{equation}
\end{widetext} \end{widetext}
A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting its expression. A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting its expression.
The convergence of the UMP as a function of the ratio $U/t$ is depicted in Fig.~\ref{fig:UMP}.
%%% FIG 3 %%% %%% FIG 3 %%%
\begin{figure*} \begin{figure*}