388 lines
13 KiB
TeX
388 lines
13 KiB
TeX
|
\documentclass[final]{beamer}
|
||
|
%% Possible paper sizes: a0, a0b, a1, a2, a3, a4.
|
||
|
%% Possible orientations: portrait, landscape
|
||
|
%% Font sizes can be changed using the scale option.
|
||
|
\usepackage[size=a0,orientation=portrait,scale=1.2]{beamerposter}
|
||
|
\usepackage[backend=biber,style=numeric,sorting=none]{biblatex}
|
||
|
\addbibresource{HKvsLieb.bib}
|
||
|
\usepackage{mathtools}
|
||
|
|
||
|
\newcommand{\red}[1]{\textcolor{red}{#1}}
|
||
|
\newcommand{\blue}[1]{\textcolor{blue}{#1}}
|
||
|
\newcommand{\green}[1]{\textcolor{green}{#1}}
|
||
|
\newcommand{\purple}[1]{\textcolor{purple}{#1}}
|
||
|
\newcommand{\hH}{\Hat{H}}
|
||
|
\newcommand{\hT}{\Hat{T}}
|
||
|
\newcommand{\hV}{\Hat{V}}
|
||
|
\newcommand{\hn}{\Hat{n}}
|
||
|
\newcommand{\cre}[1]{a_{#1}^\dagger}
|
||
|
\newcommand{\ani}[1]{a_{#1}}
|
||
|
|
||
|
\newcommand{\br}{\boldsymbol{r}}
|
||
|
|
||
|
\newcommand{\up}{\uparrow}
|
||
|
\newcommand{\dw}{\downarrow}
|
||
|
\newcommand{\Dv}{\Delta v}
|
||
|
\newcommand{\Dn}{\Delta n}
|
||
|
\newcommand{\ie}{\textit{i.e.}\xspace}
|
||
|
|
||
|
\AtEveryBibitem{%
|
||
|
\ifentrytype{article}{
|
||
|
\clearfield{url}%
|
||
|
\clearfield{urlyear}%
|
||
|
\clearfield{doi}%
|
||
|
\clearfield{issn}%
|
||
|
\clearfield{month}%
|
||
|
\clearfield{day}%
|
||
|
\clearfield{title}%
|
||
|
}{}
|
||
|
}
|
||
|
|
||
|
\DeclareFieldFormat[article]{pages}{#1}
|
||
|
|
||
|
\renewbibmacro{in:}{%
|
||
|
\ifentrytype{article}{}{\printtext{\bibstring{in}\intitlepunct}}}
|
||
|
|
||
|
\usetheme{gemini}
|
||
|
\usecolortheme{gemini}
|
||
|
\useinnertheme{rectangles}
|
||
|
|
||
|
% ====================
|
||
|
% Packages
|
||
|
% ====================
|
||
|
|
||
|
\usepackage[utf8]{inputenc}
|
||
|
\usepackage{graphicx,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,wrapfig,tikz,siunitx,bm}
|
||
|
\usepackage{booktabs}
|
||
|
\usepackage{tikz}
|
||
|
\usepackage{pgfplots}
|
||
|
|
||
|
\newfontfamily\qp{Times New Roman}[
|
||
|
% the font has no small caps, so we use another one for them
|
||
|
SmallCapsFont=TeX Gyre Termes,
|
||
|
SmallCapsFeatures={Letters=SmallCaps}
|
||
|
]
|
||
|
|
||
|
|
||
|
% ====================
|
||
|
% Lengths
|
||
|
% ====================
|
||
|
|
||
|
% If you have N columns, choose \sepwidth and \colwidth such that
|
||
|
% (N+1)*\sepwidth + N*\colwidth = \paperwidth
|
||
|
\newlength{\sepwidth}
|
||
|
\newlength{\colwidth}
|
||
|
\setlength{\sepwidth}{0.03\paperwidth}
|
||
|
\setlength{\colwidth}{0.45\paperwidth}
|
||
|
|
||
|
\newcommand{\separatorcolumn}{\begin{column}{\sepwidth}\end{column}}
|
||
|
|
||
|
|
||
|
% ====================
|
||
|
% Logo (optional)
|
||
|
% ====================
|
||
|
|
||
|
% LaTeX logo taken from https://commons.wikimedia.org/wiki/File:LaTeX_logo.svg
|
||
|
% use this to include logos on the left and/or right side of the header:
|
||
|
\logoright{\vspace{6cm} \includegraphics[width=10cm]{fig/Fermi.jpg}}
|
||
|
\logoleft{\vspace{-6cm} \includegraphics[width=10cm]{fig/ERC.jpg}}% \hspace{1cm} \includegraphics[height=6cm]{fig/CNRS.png} }
|
||
|
\addtobeamertemplate{headline}{}
|
||
|
{
|
||
|
\begin{tikzpicture}[remember picture,overlay]
|
||
|
\node [shift={(7.5 cm,-3.5cm)}] at (current page.north west) {\includegraphics[height=6cm]{fig/CNRS.png}};
|
||
|
\node [shift={(-7.25 cm,-4.5cm)}] at (current page.north east) {\includegraphics[width=10cm]{fig/LCPQ.jpg}};
|
||
|
\end{tikzpicture}
|
||
|
}
|
||
|
|
||
|
% ====================
|
||
|
% Footer (optional)
|
||
|
% ====================
|
||
|
|
||
|
\footercontent{
|
||
|
\href{mailto:sgiarrusso@irsamc.ups-tlse.fr}{\texttt{sgiarrusso@irsamc.ups-tlse.fr}}
|
||
|
\hfill
|
||
|
Sara Giarrusso and Pierre-Fran\c cois Loos, arXiv preprint arXiv:2303.15084
|
||
|
\hfill
|
||
|
\href{mailto:loos@irsamc.ups-tlse.fr}{\texttt{loos@irsamc.ups-tlse.fr}
|
||
|
}
|
||
|
}
|
||
|
% (can be left out to remove footer)
|
||
|
|
||
|
% ====================
|
||
|
% My own customization
|
||
|
% - BibLaTeX
|
||
|
% - Boxes with tcolorbox
|
||
|
% - User-defined commands
|
||
|
% ====================
|
||
|
%\input{custom-defs.tex}
|
||
|
|
||
|
%% Reference Sources
|
||
|
%\addbibresource{refs.bib}
|
||
|
%\renewcommand{\pgfuseimage}[1]{\includegraphics[scale=2.0]{#1}}
|
||
|
|
||
|
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
||
|
|
||
|
\title{Exact Excited-State Functionals of the Asymmetric Hubbard Dimer}
|
||
|
|
||
|
\author{\underline{Sara Giarrusso},\inst{1}, Pierre-Fran\c cois Loos,\inst{1}}
|
||
|
\institute[shortinst]{\inst{1} \LCPQ}
|
||
|
|
||
|
\begin{document}
|
||
|
|
||
|
\begin{frame}[t]
|
||
|
|
||
|
%%%%%%%%%%%%%%%%
|
||
|
%%% Abstract %%%
|
||
|
%%%%%%%%%%%%%%%%
|
||
|
\begin{alertblock}{Abstract}
|
||
|
We derive, for the case of the asymmetric Hubbard dimer at half-filling, the exact functional associated with each singlet ground and excited state, using both Levy's constrained search and Lieb's convex formulation.
|
||
|
While the ground-state functional is, as commonly known, a convex function with respect to the density (or, more precisely, the site occupation), the functional associated with the (highest) doubly-excited state is found to be concave.
|
||
|
Also, we find that, because the density of the first excited state is non-invertible, its ``functional'' is a partial, multi-valued function composed of one concave and one convex branch that correspond to two separate sets of values of the external potential.
|
||
|
These findings offer insight into the challenges of developing state-specific excited-state density functionals for general applications in electronic structure theory.
|
||
|
\end{alertblock}
|
||
|
%%%%%%%%%%%%%%%%
|
||
|
|
||
|
\vspace{-1.3cm}
|
||
|
\begin{columns}[t]
|
||
|
\separatorcolumn
|
||
|
|
||
|
\begin{column}{\colwidth}
|
||
|
|
||
|
%\vspace{-2cm}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
\begin{block}{Theoretical background}
|
||
|
The variational principle can be reformulated in terms of the Hohenberg-Kohn functional as
|
||
|
\begin{equation}
|
||
|
\label{eq:HKvar}
|
||
|
E_0[v] = \min_{\rho}\qty{ F[\rho] + \int v(\br) \rho(\br) d\br } \Leftrightarrow \fdv{F[\rho(\br)]}{\rho(\br)} = - v(\br)
|
||
|
\end{equation}
|
||
|
$ F[\rho]$ is the ``Fenchel conjugate" of $E_0[v]$ and can be obtained from the Lieb variational principle (or convex formulation) as:
|
||
|
\begin{equation}
|
||
|
\label{eq:Lvar}
|
||
|
F[\rho] = \max_{v}\qty{ E[v] - \int v(\br) \rho(\br) d\br } \Leftrightarrow \fdv{E[v(\br)]}{v(\br)} = \rho(\br)
|
||
|
\end{equation}
|
||
|
Levy's constrained search requires $\rho$ to be only $N$-representable rather than $v$-representable, defining the Levy functional as
|
||
|
\begin{equation}
|
||
|
\label{eq:FLL}
|
||
|
F[\rho]
|
||
|
= \min_{\Psi \leadsto \rho} \mel{\Psi}{\hT + \hV_{ee}}{\Psi}
|
||
|
\end{equation}
|
||
|
\end{block}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
%\vspace{-1.cm}
|
||
|
\begin{block}{The model}
|
||
|
The Hamiltonian of the asymmetric Hubbard dimer is
|
||
|
\begin{equation}
|
||
|
\hH =
|
||
|
- t \sum_{\sigma=\up,\dw} \qty( \cre{0\sigma} \ani{1\sigma} + \text{h.c.} )
|
||
|
+ U \sum_{i=0}^{1} \hat{n}_{i\up} \hat{n}_{i\dw}
|
||
|
+ \Dv \frac{\hn_{1} - \hn_{0}}{2}
|
||
|
\end{equation}
|
||
|
A generic singlet wave function can be written as
|
||
|
\begin{equation}
|
||
|
\ket{\Psi} = x \ket{0_\up0_\dw} + y \frac{\ket{0_\up1_\dw} - \ket{0_\dw1_\up}}{\sqrt{2}} + z \ket{1_\up1_\dw}
|
||
|
\end{equation}
|
||
|
with $-1 \le x,y,z \le 1$ and the normalization condition
|
||
|
\begin{equation}
|
||
|
\label{eq:normalization}
|
||
|
x^2 + y^2 + z^2 = 1
|
||
|
\end{equation}
|
||
|
\end{block}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
% \vspace{-0.5cm}
|
||
|
\begin{block}{Ground and excited-state energies and densities}
|
||
|
|
||
|
The total energy, $E$, is given by the sum of
|
||
|
\begin{subequations}
|
||
|
\begin{align}
|
||
|
\label{eq:T}
|
||
|
&T = - 2\sqrt{2} t y \qty(x + z)
|
||
|
\\
|
||
|
\label{eq:Vee}
|
||
|
&V_{ee} = U \qty(x^2 + z^2)
|
||
|
\\
|
||
|
\label{eq:V}
|
||
|
&V = \rho \cdot \Dv \quad \text{with} \quad \rho=\mel{\Psi}{\frac{\hn_{1} - \hn_{0}}{2}}{\Psi} = (z^2 - x^2)
|
||
|
\end{align}
|
||
|
\end{subequations}
|
||
|
\includegraphics[scale=1.7]{fig/fig1}
|
||
|
|
||
|
\begin{columns}[T]
|
||
|
\begin{column}{.45\textwidth}
|
||
|
\includegraphics[scale=1.4]{fig/fig2}
|
||
|
\end{column}
|
||
|
\begin{column}{.45\textwidth}
|
||
|
In agreement with Eq.~\eqref{eq:Lvar}, one finds
|
||
|
\begin{equation}
|
||
|
\dv{E_0(\Dv)}{\Dv} =2\, \rho_0(\Dv)
|
||
|
\end{equation}
|
||
|
However, the same holds for the excited states
|
||
|
\begin{subequations}
|
||
|
\begin{align}
|
||
|
& \dv{E_1(\Dv)}{\Dv} =2\, \rho_1(\Dv)\\
|
||
|
& \dv{E_2(\Dv)}{\Dv} =2\, \rho_2(\Dv)
|
||
|
\end{align}
|
||
|
\end{subequations}
|
||
|
\end{column}
|
||
|
\begin{column}{.1\textwidth}
|
||
|
|
||
|
\end{column}
|
||
|
\end{columns}
|
||
|
\end{block}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
%\vspace{-0.5cm}
|
||
|
\begin{block}{Levy's constrained search}
|
||
|
\begin{columns}[T]
|
||
|
\begin{column}{.35\textwidth}
|
||
|
\hspace{-3.5cm}
|
||
|
$\phantom{0\,\,}$\includegraphics[scale=1.2]{fig/fig5}
|
||
|
\end{column}
|
||
|
\begin{column}{.5\textwidth}
|
||
|
\vspace{1cm}
|
||
|
Substituting $x$ and $z$, we obtain the four-branch function
|
||
|
\hspace{-3cm}
|
||
|
{\small{\begin{align}
|
||
|
\label{eq:fpm}
|
||
|
f_{\pm\pm}(\rho,y)
|
||
|
= & - 2 t y \qty (\pm\sqrt{1 - y^2 - \rho}
|
||
|
\pm \sqrt{1 - y^2 + \rho}) \notag
|
||
|
\\
|
||
|
&+ U \qty(1 - y^2)
|
||
|
\end{align}} }
|
||
|
|
||
|
Rather than only minimizing $f_{\pm\pm}(\rho,y) $
|
||
|
for a given $\rho$, we seek \textit{all} its stationary points with respect to $y$:
|
||
|
\begin{equation}
|
||
|
\pdv{f_{\pm\pm}(\rho,y)}{y} = 0
|
||
|
\end{equation}
|
||
|
\end{column}
|
||
|
\begin{column}{.1\textwidth}
|
||
|
\end{column}
|
||
|
\end{columns}
|
||
|
\end{block}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
||
|
\end{column}
|
||
|
|
||
|
\separatorcolumn
|
||
|
|
||
|
\begin{column}{\colwidth}
|
||
|
|
||
|
\vspace{3cm}
|
||
|
%
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
% \begin{block}{}
|
||
|
|
||
|
\includegraphics[scale=1.35]{fig/fig3}
|
||
|
$f_{--}(\rho,y)$ (red) and $f_{++}(\rho,y)$ (green) have each one stationary point for $y \geq 0$, while $f_{+-}(\rho,y)$ (yellow) has two stationary points that disappear for $\rho > \rho_c$.
|
||
|
% \end{block}
|
||
|
\vspace{0.5cm}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
\begin{block}{Ground and excited-state ``universal" function(al)s}
|
||
|
%
|
||
|
\begin{columns}[T]
|
||
|
\begin{column}{.45\textwidth}
|
||
|
\includegraphics[scale=1.5]{fig/fig4} \\
|
||
|
\includegraphics[scale=1.5]{fig/Vvsrho}
|
||
|
\end{column}
|
||
|
\begin{column}{.45\textwidth}
|
||
|
\vspace{0.5cm}
|
||
|
\begin{subequations}
|
||
|
\begin{align}
|
||
|
& F_0(\rho)=f_{++}(\rho,y_0)\\
|
||
|
& F_2(\rho)=f_{--}(\rho,y_2)\\
|
||
|
& \begin{rcases} F_1^{\cup}(\rho) = f_{+-}(\rho,y_1^{\cup}))\\
|
||
|
F_1^{\cap}(\rho) = f_{+-}(\rho,y_1^{\cap}
|
||
|
\end{rcases} \quad \rho < \rho_\text{c}
|
||
|
\end{align}
|
||
|
\end{subequations}
|
||
|
|
||
|
\vspace{1.5cm}
|
||
|
%\dv{F_2(\rho)}{\rho} = -\Dv_2 (\rho)
|
||
|
In agreement with Eq.~\eqref{eq:HKvar} and beyond the ground state, one finds
|
||
|
\begin{subequations}
|
||
|
\begin{align}
|
||
|
& \dv{F_0(\rho)}{\rho} = -\Dv_0 (\rho)\\
|
||
|
& \dv{F_2(\rho)}{\rho} = -\Dv_2 (\rho)\\
|
||
|
& \begin{rcases} \dv{F_1^{\cup}(\rho)}{\rho} = - \Dv_1^{\cup}(\rho)\\
|
||
|
\dv{F_1^{\cap}(\rho)}{\rho} = -\Dv_1^{\cap}(\rho) \end{rcases} \quad \rho < \rho_\text{c}
|
||
|
\end{align}
|
||
|
\end{subequations}
|
||
|
\end{column}
|
||
|
\end{columns}
|
||
|
\end{block}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
\begin{block}{Lieb's convex formulation}
|
||
|
\vspace{0.5cm}
|
||
|
\begin{columns}
|
||
|
\begin{column}{.45\textwidth}
|
||
|
\includegraphics[scale=1.5]{fig/fig6}
|
||
|
\end{column}
|
||
|
\begin{column}{.5\textwidth}
|
||
|
\begin{equation}
|
||
|
f_{n}(\rho,\Dv) = E_n - \Dv \rho
|
||
|
\end{equation}
|
||
|
Rather than only maximizing $ f_{n}(\rho,\Dv) $ for a given $\rho$, we seek \textit{all} its stationary points with respect to $\Dv$ for each $n$ value,
|
||
|
\begin{equation}
|
||
|
\pdv{f_{n}(\rho,\Dv)}{\Dv} = 0
|
||
|
\end{equation}
|
||
|
\end{column}
|
||
|
\end{columns}
|
||
|
\end{block}
|
||
|
\vspace{-1.3cm}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%
|
||
|
\begin{block}{Conclusions}
|
||
|
While the ground-state functional is well-known to be a convex function with respect to the difference in site occupations, the functional associated with the highest doubly-excited state is found to be concave.
|
||
|
Additionally, and more importantly, we discovered that the ``functional'' for the first excited state is a partial, multi-valued function of the density that is constructed from one concave and one convex branch associated with two separate sets of values of the external potential.
|
||
|
Finally, we find that Levy's constrained search and Lieb's convex formulation are entirely consistent for all the states of the model, producing the same landscape of state-specific functionals.
|
||
|
\end{block}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
\vspace{-0.4cm}
|
||
|
%%% Acknowledgement %%%
|
||
|
%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
\begin{block}{Acknowledgement}
|
||
|
\textit{This project has received funding from the European Research Council (ERC)
|
||
|
under the European Union's Horizon 2020 research and innovation programme,
|
||
|
grant agreement No.~863481.}
|
||
|
\end{block}
|
||
|
%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
\vspace{-1.cm}
|
||
|
%%%%%%%%%%%%%%%%%%
|
||
|
%%% References %%%
|
||
|
%%%%%%%%%%%%%%%%%%
|
||
|
\begin{block}{References}
|
||
|
|
||
|
%\printbibliography
|
||
|
{\footnotesize{
|
||
|
\parbox[b]{\hsize}{[1] P. Hohenberg, and W. Kohn, \textit{Inhomogeneous electron gas}, Phys. rev. 136.3B (1964): B864.}\\
|
||
|
|
||
|
\parbox[b]{\hsize}{[2] M. Levy, \textit{Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem} Proc. Nat. Ac. Sci 76.12 (1979): 6062-6065.}\\
|
||
|
|
||
|
\parbox[b]{\hsize}{[3] E. H. Lieb, \textit{Density functionals for Coulomb systems} Int. J. Quantum Chem. \textbf{24}, 243 (1983).}\\
|
||
|
|
||
|
\parbox[b]{\hsize}{[4] D. J. Carrascal, J. Ferrer, J. C. Smith, and K. Burke, \textit{The Hubbard dimer: a density functional case study of a many-body problem}, J.Phys. Condens. Matter \textbf{27}, 393001 (2015).}\\
|
||
|
|
||
|
\parbox[b]{\hsize}{[5] J. P. Perdew, and M. Levy, \textit{Extrema of the density functional for the energy: Excited states from the ground-state theory}, Phys. Rev. B 31.10 (1985): 6264. \\
|
||
|
}
|
||
|
}}
|
||
|
|
||
|
\end{block}
|
||
|
%%%%%%%%%%%%%%%%%%
|
||
|
|
||
|
\end{column}
|
||
|
|
||
|
\separatorcolumn
|
||
|
\end{columns}
|
||
|
|
||
|
\end{frame}
|
||
|
|
||
|
\end{document}
|