diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index bdcbec7..f28c7cb 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -101,10 +101,12 @@ \newcommand{\bbR}{\mathbb{R}} \newcommand{\bbC}{\mathbb{C}} -\newcommand{\Lup}{\text{L}^{\uparrow}} -\newcommand{\Ldown}{\text{L}^{\downarrow}} -\newcommand{\Rup}{\text{R}^{\uparrow}} -\newcommand{\Rdown}{\text{R}^{\downarrow}} +\newcommand{\Lup}{\mathcal{L}^{\uparrow}} +\newcommand{\Ldown}{\mathcal{L}^{\downarrow}} +\newcommand{\Lsi}{\mathcal{L}^{\sigma}} +\newcommand{\Rup}{\mathcal{R}^{\uparrow}} +\newcommand{\Rdown}{\mathcal{R}^{\downarrow}} +\newcommand{\Rsi}{\mathcal{R}^{\sigma}} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.} @@ -189,38 +191,32 @@ More importantly here, although EPs usually lie off the real axis, these singula \subcaption{\label{subfig:FCI_cplx}} \end{subfigure} \caption{% - \hugh{Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}). - Only the interacting closed-shell singlets are plotted in the complex plane, becoming degenerate at the EP (black dot).} + Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}). + Only the interacting closed-shell singlets are plotted in the complex plane, becoming degenerate at the EP (black dot). \label{fig:FCI}} \end{figure*} -\hugh{To illustrate the concepts discussed throughout this article, we will consider the symmetric Hubbard dimer at half filling, \ie\ with two opposite-spin fermions. -Using the localised Wannier basis, the Hilbert space for this system comprises the four configurations -\begin{equation} -\ket{\Lup \Ldown} \qquad \ket{\Lup\Rdown} \qquad \ket{\Rup\Ldown} \qquad \ket{\Rup\Rdown} -\end{equation} -%\begin{tabularx}{\linewidth}{YYYY} -%$\ket{\Lup \Ldown}$ & $\ket{\Lup\Rdown}$ & $\ket{\Rup\Ldown}$ & $\ket{\Rup\Rdown}$ -%\\ -%$\uddot \quad \vac$ & $\updot \quad \dwdot$ & $\dwdot \quad \updot$ & $\vac \quad \uddot$ -%\end{tabularx} -where $\text{L}^{\sigma}$ ($\text{R}^{\sigma}$) denotes an electron with spin $\sigma$ on the left (right) site. +To illustrate the concepts discussed throughout this article, we will consider the symmetric Hubbard dimer at half filling, \ie\ with two opposite-spin fermions. +Using the localised site basis, the Hilbert space for this system comprises the four configurations +\begin{align} +& \ket{\Lup \Ldown} & & \ket{\Lup\Rdown} & & \ket{\Rup\Ldown} & & \ket{\Rup\Rdown} +\end{align} +where $\Lsi$ ($\Rsi$) denotes an electron with spin $\sigma$ on the left (right) site. The exact Hamiltonian is then \begin{equation} \label{eq:H_FCI} \bH = \begin{pmatrix} - U & - t & - t & 0 \\ + U & - t & + t & 0 \\ - t & 0 & 0 & - t \\ - - t & 0 & 0 & - t \\ - 0 & - t & - t & U \\ + + t & 0 & 0 & + t \\ + 0 & - t & + t & U \\ \end{pmatrix}, \end{equation} -where $t$ is the hopping parameter and $U$ is the on-site Coulomb repulsion.} +where $t$ is the hopping parameter and $U$ is the on-site Coulomb repulsion. We refer the interested reader to Refs.~\onlinecite{Carrascal_2015,Carrascal_2018} for more details about this system. -\hugh{% To illustrate the formation of an exceptional points, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$. When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) eigenvalues \begin{subequations} @@ -233,10 +229,10 @@ E_{\text{T}} &= 0, E_{\text{S}} &= U. \end{align} \end{subequations} -While the open-shell triplet ($E_{\text{T}}$) and singlet ($E_{\text{S}}$) are independent of $\lambda$, the closed-shell singlet ground state ($E_{-}$) and doubly-excited state ($E_{+}$) couple strongly to form an avoided crossing at $\lambda=0$ (see Figure~\ref{subfig:FCI_real}). +While the open-shell triplet ($E_{\text{T}}$) and singlet ($E_{\text{S}}$) are independent of $\lambda$, the closed-shell singlet ground state ($E_{-}$) and doubly-excited state ($E_{+}$) couple strongly to form an avoided crossing at $\lambda=0$ (see Fig.~\ref{subfig:FCI_real}). At non-zero values of $U$ and $t$, these closed-shell singlets can only become degenerate at a pair of complex conjugate points in the complex $\lambda$ plane \begin{equation} -\lambda_{\text{EP}} = \pm \frac{U}{4t} \i, +\lambda_{\text{EP}} = \pm \i \frac{U}{4t}, \end{equation} with energy \begin{equation} @@ -247,9 +243,7 @@ These $\lambda$ values correspond to so-called EPs and connect the ground and ex Crucially, the energy surface becomes non-analytic at $\lambda_{\text{EP}}$ and a square-root singularity forms with two branch cuts running along the imaginary axis from $\lambda_{\text{EP}}$ to $\pm \i \infty$ (see Fig.~\ref{subfig:FCI_cplx}). On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda = \Re(\lambda_{\text{EP}})$. The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction. -} -\hugh{ Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states. This behaviour can be seen by expanding the radicand in Eq.~\eqref{eq:singletE} as a Taylor series around $\lambda_{\text{EP}}$ to give \begin{equation} @@ -259,7 +253,7 @@ Parametrising the complex contour as $\lambda(\theta) = \lambda_{\text{EP}} + R \begin{equation} E_{\pm} \qty(\theta) \approx E_{\text{EP}} \pm \sqrt{32t^2 \lambda_{\text{EP}} R}\, \exp(\i \theta/2) \end{equation} -such that} +such that \begin{align} E_{\pm}(2\pi) & = E_{\mp}(0), & @@ -394,13 +388,28 @@ is the HF mean-field potential with \begin{subequations} \begin{gather} \label{eq:CoulOp} - J_i(\vb{x})\phi_p(\vb{x})=\qty[\int\dd\vb{x}'\phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') ] \phi_p(\vb{x}) + J_i(\vb{x})\phi_p(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') \dd\vb{x}' ] \phi_p(\vb{x}) \\ \label{eq:ExcOp} - K_i(\vb{x})\phi_p(\vb{x})=\qty[\int\dd\vb{x}'\phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_p(\vb{x}') ] \phi_i(\vb{x}) + K_i(\vb{x})\phi_p(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_p(\vb{x}') \dd\vb{x}'] \phi_i(\vb{x}) \end{gather} \end{subequations} being the Coulomb and exchange operators (respectively) in the spin-orbital basis. \cite{SzaboBook} +The HF energy is then defined as +\begin{equation} +\label{eq:E_HF} + E_\text{HF} = \sum_i h_i + \frac{1}{2} \sum_{ij} \qty( J_{ij} - K_{ij} ) +\end{equation} +with +\begin{subequations} +\begin{gather} + h_i = \int \phi_i(\vb{x}) h(\vb{x}) \phi_i(\vb{x}) \dd\vb{x} + \\ + J_{ij} = \int \phi_i(\vb{x}) J_j(\vb{x}) \phi_i(\vb{x}) \dd\vb{x} + \\ + K_{ij} = \int \phi_i(\vb{x}) K_j(\vb{x}) \phi_i(\vb{x}) \dd\vb{x} +\end{gather} +\end{subequations} If the spatial part of the spin-orbitals are restricted to be the same for spin-up and spin-down electrons, one talks about restricted HF (RHF) theory, whereas if one does not enforce this constrain it leads to the so-called unrestricted HF (UHF) theory. From hereon, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied (or virtual) orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. @@ -409,17 +418,17 @@ Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational princi \hH^{\text{HF}} = \sum_{i} f(\vb{x}_i). \end{equation} -Coming back to the Hubbard dimer, the HF energy is +Coming back to the Hubbard dimer, the HF energy is [see Eq.~\eqref{eq:E_HF}] \begin{equation} E_\text{HF} = -t \qty[ \sin \theta_\alpha + \sin \theta_\beta ] + \frac{U}{2} \qty[ 1 + \cos \theta_\alpha \cos \theta_\beta ] \end{equation} where \begin{align} - \psi_{1\sigma} & = \cos(\frac{\theta_\sigma}{2}) s_1 - \sin(\frac{\theta_\sigma}{2})s_2 + \psi_{1\sigma} & = \cos(\frac{\theta_\sigma}{2}) \Lsi - \sin(\frac{\theta_\sigma}{2}) \Rsi \\ - \psi_{2\sigma} & = \sin(\frac{\theta_\sigma}{2}) s_1 + \cos(\frac{\theta_\sigma}{2})s_2 + \psi_{2\sigma} & = \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi \end{align} - +are the one-electron molecule orbitals for the spin-$\sigma$ electrons and the angles which makes the energy stationnary, \ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$ are given by \begin{align} \theta_\alpha & = \arctan (-\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U}) \\