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Pierre-Francois Loos 2022-09-23 10:33:30 +02:00
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@ -203,6 +203,8 @@ Section \ref{sec:numerical} presents the application of the approach to the one-
Finally, in Sec.\ref{sec:conclu}, some conclusions and perspectives are given.
Atomic units are used throughout.
\titou{Comment on deterministic vs stochastic (PT2, FCIQMC, etc).}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Diffusion Monte Carlo}
\label{sec:DMC}
@ -1167,10 +1169,10 @@ For each system at hand, one needs to determine, on physical grounds, which \tit
In the first application presented here on the one-dimensional Hubbard model, we exploit the physics of the large-$U$ regime that is known to approach the Heisenberg limit where double occupations have small weights.
This simple example has been chosen to illustrate the various aspects of the approach.
Our goal is, of course, to tackle much larger systems, like those treated by state-of-the-art methods, such as selected CI, \cite{Huron_1973,Harrison_1991,Giner_2013,Holmes_2016,Schriber_2016,Tubman_2020}, FCIQMC, \cite{Booth_2009,Cleland_2010}, AFQMC, \cite{Zhang_2003} or DMRG. \cite{White_1999,Chan_2011}
\titou{Our goal is, of course, to tackle much larger systems, like those treated by state-of-the-art methods, such as selected CI, \cite{Huron_1973,Harrison_1991,Giner_2013,Holmes_2016,Schriber_2016,Tubman_2020}, FCIQMC, \cite{Booth_2009,Cleland_2010}, AFQMC, \cite{Zhang_2003} or DMRG. \cite{White_1999,Chan_2011}
Here, we have mainly focused on the theoretical aspects of the approach.
In order to consider larger systems, an elaborate implementation of the present method is necessary in order to keep under control the cost of the simulation.
This is outside the scope of the present study and will be presented in a forthcoming work.
This is outside the scope of the present study and will be presented in a forthcoming work.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments{

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\documentclass[tikz]{standalone}
\usepackage{physics}
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[]
% frame
\draw[-,thick] (0,0) -- (5.5,0);
\draw[-,dashed,thick] (5.5,0) -- (6.5,0);
\draw[->,thick] (6.5,0) -- (8,0) node[anchor=north west]{$\text{time} = N \tau $};
\draw[->,thick] (0,0) node[anchor=north ]{$0$} -- (0,4) node[anchor=north east]{states};
% vertical lines
\draw[-] (1,0) node[anchor=north]{$1$} -- (1,4);
\draw[-] (2,0) node[anchor=north]{$2$} -- (2,4);
\draw[-] (3,0) node[anchor=north]{$3$} -- (3,4);
\draw[-] (4,0) node[anchor=north]{$4$} -- (4,4);
\draw[-] (5,0) node[anchor=north]{$5$} -- (5,4);
\draw[-] (7,0) node[anchor=north]{$N$} -- (7,4);
% initial and final states
\draw[] (0,2) node[anchor=east]{$\ket{i_0}$};
\draw[] (7,3) node[anchor=west]{$\ket{i}$};
\draw[] (8,3) node[anchor=west]{$\displaystyle \ket{\Phi_0} = \sum_i c_i \ket{i}$};
\draw[] (8,2) node[anchor=west]{$\displaystyle c_i \sim \sum_{\substack{\text{all paths} \\ \text{ending at $\ket{i}$}}} \prod_{k}^{\infty} W_{i_k i_{k+1}}$};
% paths
\draw[thick,magenta] (0,2) node{$\bullet$} -- (1,3) node{$\bullet$} -- (2,2) node{$\bullet$} -- (3,3) node{$\bullet$} -- (4,3.5) node{$\bullet$} -- (5,3.75) node{$\bullet$} -- (7,3) node{$\bullet$};
\draw[thick,red] (0,2) node{$\bullet$} -- (1,2) node{$\bullet$} -- (2,2) node{$\bullet$} -- (3,1.5) node{$\bullet$} -- (4,3) node{$\bullet$} -- (5,3) node{$\bullet$} -- (7,3) node{$\bullet$};
\draw[thick,green] (0,2) node{$\bullet$} -- (1,1) node{$\bullet$} -- (2,2) node{$\bullet$} -- (3,2.5) node{$\bullet$} -- (4,2.5) node{$\bullet$} -- (5,2) node{$\bullet$} -- (7,3) node{$\bullet$};
\draw[thick,blue] (0,2) node{$\bullet$} -- (1,0.25) node{$\bullet$} -- (2,1) node{$\bullet$} -- (3,1) node{$\bullet$} -- (4,2) node{$\bullet$} -- (5,1) node{$\bullet$} -- (7,3) node{$\bullet$};
\end{tikzpicture}
\end{document}