From 0af98b9eaaf26529c30bdc2542886f8f8ba5f0ab Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 3 Oct 2022 11:30:09 +0200 Subject: [PATCH] OK with Sec IIB --- g.tex | 32 +++++++++++++++----------------- 1 file changed, 15 insertions(+), 17 deletions(-) diff --git a/g.tex b/g.tex index 1777eab..31e9c1a 100644 --- a/g.tex +++ b/g.tex @@ -289,7 +289,7 @@ Here, only four paths of infinite length have been represented. \label{sec:proba} %=======================================% -To derive a probabilistic expression for the Green's matrix, we introduce a guiding wave function, $\ket{\PsiG}$, having strictly positive components, \ie, $\PsiG_i > 0$, in order to perform a similarity transformation to the operators $G^{(N)}$ and $T$, as follows: +To derive a probabilistic expression for the Green's matrix, we introduce a guiding wave function $\ket{\PsiG}$ having strictly positive components, \ie, $\PsiG_i > 0$, in order to perform a similarity transformation of the operators $G^{(N)}$ and $T$, \begin{align} \label{eq:defT} \bar{T}_{ij} & = \frac{\PsiG_j}{\PsiG_i} T_{ij}, @@ -298,7 +298,7 @@ To derive a probabilistic expression for the Green's matrix, we introduce a guid \end{align} Note that, thanks to the properties of similarity transformations, the path integral expression relating $G^{(N)}$ and $T$ [see Eq.~\eqref{eq:G}] remains unchanged for $\bar{G}^{(N)}$ and $\bar{T}$. -Next, the matrix elements of $\bar{T}$ are expressed as those of a stochastic matrix multiplied by some residual weight, namely, +Next, the matrix elements of $\bar{T}$ are expressed as those of a stochastic matrix multiplied by some residual weights, namely, \be \label{eq:defTij} \bar{T}_{ij}= p_{i \to j} w_{ij}. @@ -330,7 +330,7 @@ Here, $\EL^+ \Id$ is the diagonal matrix whose diagonal elements are defined as (\EL^+)_{i}= \frac{\sum_j H^+_{ij}\PsiG_j}{\PsiG_i}. \ee The vector $\EL^+$ is known as the local energy vector associated with $\PsiG$. -By construction, the operator $H^+ - \EL^+ \Id$ in the definition of the operator $T^+$ [see Eq.~\eqref{eq:T+}] has been chosen to admit $\ket{\PsiG}$ as a ground-state wave function with zero eigenvalue, \ie, $\qty(H^+ - E_L^+ \Id) \ket{\PsiG} = 0$, leading to the relation +By construction, the operator $H^+ - \EL^+ \Id$ in the definition of $T^+$ [see Eq.~\eqref{eq:T+}] has been chosen to admit $\ket{\PsiG}$ as a ground-state wave function with zero eigenvalue, \ie, $\qty(H^+ - E_L^+ \Id) \ket{\PsiG} = 0$, leading to the relation \be \label{eq:relT+} T^+ \ket{\PsiG} = \ket{\PsiG}. @@ -373,23 +373,21 @@ Note that one can eschew this condition via a simple generalization of the trans \ee This new transition probability matrix with positive entries reduces to Eq.~\eqref{eq:pij} when $T^+_{ij}$ is positive as $\sum_j \PsiG_{j} T^+_{ij} = 1$. -Now, we need to make the connection between the transition probability matrix, $p_{i \to j}$, just defined from the operator $T^+$ corresponding -to the approximate Hamiltonian -$H^{+}$ and the operator $T$ associated with the exact Hamiltonian $H$. -This is done thanks to the relation, Eq.(\ref{eq:defTij}), connecting $p_{i \to j}$ and $T_{ij}$ through a weight. -Using Eqs.~\eqref{eq:defT} and \eqref{eq:pij}, the weights read +Now, we need to make the connection between the transition probability matrix, $p_{i \to j}$, defined from the approximate Hamiltonian $H^{+}$ via $T^+$ and the operator $T$ associated with the exact Hamiltonian $H$. +This is done thanks to Eq.~\eqref{eq:defTij} that connects $p_{i \to j}$ and $T_{ij}$ through the weight \be - w_{ij}=\frac{T_{ij}}{T^+_{ij}}. + w_{ij}=\frac{T_{ij}}{T^+_{ij}}, \ee +derived from Eqs.~\eqref{eq:defT} and \eqref{eq:pij}. Using these notations the similarity-transformed Green's matrix components can be rewritten as \be \label{eq:GN_simple} \bar{G}^{(N)}_{i_0 i_N} = \sum_{i_1,\ldots,i_{N-1}} \qty( \prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}} ) \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}}, \ee -which is amenable to Monte Carlo calculations by generating paths using the transition probability matrix, $p_{i \to j}$. +which is amenable to Monte Carlo calculations by generating paths using the transition probability matrix $p_{i \to j}$. -Let us illustrate this in the case of the energy as given by Eq.~\eqref{eq:E0}. Taking $\ket{\Psi_0}=\ket{i_0}$ as initial condition, we have +Let us illustrate this in the case of the energy as given by Eq.~\eqref{eq:E0}. Taking $\ket{\Psi_0}=\ket{i_0}$ as initial state, we have \be E_0 = \lim_{N \to \infty } \frac{ \sum_{i_N} G^{(N)}_{i_0 i_N} {(H\PsiT)}_{i_N} } @@ -399,11 +397,11 @@ which can be rewritten probabilistically as \be E_0 = \lim_{N \to \infty } \frac{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \frac{ {(H\PsiT)}_{i_N} }{ \PsiG_{i_N} }}} - { \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \frac{ {\PsiT}_{i_N} } {\PsiG_{i_N}} }}. + { \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \frac{ {\PsiT}_{i_N} } {\PsiG_{i_N}} }}, \ee -where $\expval{...}$ is the probabilistic average defined over the set of paths $\ket{i_1},\ldots,\ket{i_N}$ occuring with probability +where $\expval{...}$ is the probabilistic average defined over the set of paths $\ket{i_1},\ldots,\ket{i_N}$ occurring with probability \be - \text{Prob}_{i_0}(i_1,\ldots,i_{N}) = \prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}} + \text{Prob}_{i_0}(i_1,\ldots,i_{N}) = \prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}}. \ee Using Eq.~\eqref{eq:sumup} and the fact that $p_{i \to j} \ge 0$, one can easily verify that $\text{Prob}_{i_0}$ is positive and obeys \be @@ -421,8 +419,8 @@ as it should. %\label{eq:cn_stoch} % \bar{G}^{(N)}_{i_0 i_N}= \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}}}. %\ee -To calculate the probabilistic averages -an artificial (mathematical) ``particle'' called walker (or psi-particle) is introduced. + +To calculate the probabilistic averages, an artificial (mathematical) ``particle'' called walker (or psi-particle) is introduced. During the Monte Carlo simulation, the walker moves in configuration space by drawing new states with probability $p_{i_k \to i_{k+1}}$, thus realizing the path of probability $\text{Prob}_{i_0}$. %In this framework, the energy defined in Eq.~\eqref{eq:E0} is given by @@ -434,7 +432,7 @@ probability $p_{i_k \to i_{k+1}}$, thus realizing the path of probability $\tex %A schematic algorithm is presented in Fig.\ref{scheme1B}. Note that, instead of using a single walker, it is common to introduce a population of independent walkers and to calculate the averages over this population. In addition, thanks to the ergodic property of the stochastic matrix (see, for example, Ref.~\onlinecite{Caffarel_1988}), a unique path of infinite length from which sub-paths of length $N$ can be extracted may also be used. -We shall not here insist on these practical details that are discussed, for example, in Refs.~\onlinecite{Foulkes_2001,Kolorenc_2011}. +We shall not insist here on these practical details that are discussed, for example, in Refs.~\onlinecite{Foulkes_2001,Kolorenc_2011}. %{\it Spawner representation} In this representation, we no longer consider moving particles but occupied or non-occupied states $|i\rangle$. %To each state is associated the (positive or negative) quantity $c_i$.