diff --git a/g.tex b/g.tex index e80fc0b..1777eab 100644 --- a/g.tex +++ b/g.tex @@ -206,6 +206,7 @@ In Subsec.~\ref{sec:Green}, both the time- and energy-dependent Green's function Section \ref{sec:numerical} presents the application of the approach to the one-dimensional Hubbard model. Finally, in Sec.\ref{sec:conclu}, some conclusions and perspectives are given. Atomic units are used throughout. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Diffusion Monte Carlo} \label{sec:DMC} @@ -262,25 +263,25 @@ Each path is associated with a weight $\prod_{k=0}^{N-1} T_{i_{k} i_{k+1}}$ and This expression allows a simple and vivid interpretation of the solution. In the limit $N \to \infty$, the $i_N$th component of the ground-state wave function (obtained as $\lim_{N \to \infty} G^{(N)}_{i_0 i_N})$ is the weighted sum over all possible paths arriving at vector $\ket{i_N}$. -This result is independent of the initial vector $\ket{i_0}$, apart from some irrelevant global phase factor. We illustrate this fundamental property pictorially in -Fig.\ref{fig1}. +This result is independent of the initial vector $\ket{i_0}$, apart from some irrelevant global phase factor. +We illustrate this fundamental property pictorially in Fig.~\ref{fig:paths}. When the size of the linear space is small, the explicit calculation of the full sums involving $M^N$ terms (where $M$ is the size of the Hilbert space) can be performed. -In such a case, we are in the realm of what one would call the ``deterministic'' power methods, such as the Lancz\`os or Davidson approaches. +In such a case, we are in the realm of what one would call ``deterministic'' power methods, such as the Lancz\`os or Davidson approaches. If not, probabilistic techniques for generating only the paths contributing significantly to the sums are to be preferred. This is the central theme of the present work. -%%% FIG 0 %%% -\begin{figure} -\includegraphics[width=\columnwidth]{fig1} +%%% FIG 1 %%% +\begin{figure*} +\includegraphics[width=0.7\textwidth]{fig1} +\label{fig:paths} \caption{ -\titou{$\Psi$ or $\Phi$?} -Path integral representation of the exact coefficient $c_i=\braket{i}{\Psi_0}$ of the ground-state wave function $\ket{\Psi_0}$ obtained as an infinite sum of paths starting -from $\ket{i_0}$ and ending at $\ket{i}$ [see Eq.\eqref{eq:G}]. Each path carries a weight $\prod_k T_{i_{k} i_{k+1}}$ computed along the path. -The result is independent of the choice of the initial state $\ket{i_0}$, provided that $\braket{i_0}{\Psi_0}$ is non-zero. +\titou{$\Psi_0$ or $\Phi_0$?} +Path integral representation of the exact coefficient $c_i=\braket{i}{\Psi_0}$ of the ground-state wave function $\ket{\Psi_0}$ obtained as an infinite sum of paths starting from $\ket{i_0}$ and ending at $\ket{i}$ [see Eq.\eqref{eq:G}]. +Each path carries a weight $\prod_k T_{i_{k} i_{k+1}}$ computed along it. +The result is independent of the choice of the initial state $\ket{i_0}$, provided that $\braket{i_0}{\Psi_0} \neq 0$. Here, only four paths of infinite length have been represented. } -\label{fig1} -\end{figure} +\end{figure*} %=======================================%