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OK with Sec IIB

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Pierre-Francois Loos 2022-10-03 11:30:09 +02:00
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@ -289,7 +289,7 @@ Here, only four paths of infinite length have been represented.
\label{sec:proba} \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} \begin{align}
\label{eq:defT} \label{eq:defT}
\bar{T}_{ij} & = \frac{\PsiG_j}{\PsiG_i} T_{ij}, \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} \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 $\bar{T}$. 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 \be
\label{eq:defTij} \label{eq:defTij}
\bar{T}_{ij}= p_{i \to j} w_{ij}. \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}. (\EL^+)_{i}= \frac{\sum_j H^+_{ij}\PsiG_j}{\PsiG_i}.
\ee \ee
The vector $\EL^+$ is known as the local energy vector associated with $\PsiG$. 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 \be
\label{eq:relT+} \label{eq:relT+}
T^+ \ket{\PsiG} = \ket{\PsiG}. T^+ \ket{\PsiG} = \ket{\PsiG}.
@ -373,23 +373,21 @@ Note that one can eschew this condition via a simple generalization of the trans
\ee \ee
This new transition probability matrix with positive entries reduces to Eq.~\eqref{eq:pij} when $T^+_{ij}$ is positive as $\sum_j \PsiG_{j} T^+_{ij} = 1$. 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 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 $H$.
to the approximate Hamiltonian This is done thanks to Eq.~\eqref{eq:defTij} that connects $p_{i \to j}$ and $T_{ij}$ through the weight
$H^{+}$ 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
\be \be
w_{ij}=\frac{T_{ij}}{T^+_{ij}}. w_{ij}=\frac{T_{ij}}{T^+_{ij}},
\ee \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 Using these notations the similarity-transformed Green's matrix components can be rewritten as
\be \be
\label{eq:GN_simple} \label{eq:GN_simple}
\bar{G}^{(N)}_{i_0 i_N} = \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}}, \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 \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 \be
E_0 = \lim_{N \to \infty } E_0 = \lim_{N \to \infty }
\frac{ \sum_{i_N} G^{(N)}_{i_0 i_N} {(H\PsiT)}_{i_N} } \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 \be
E_0 = \lim_{N \to \infty } 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} }}} \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 \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 \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 \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 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 \be
@ -421,8 +419,8 @@ as it should.
%\label{eq:cn_stoch} %\label{eq:cn_stoch}
% \bar{G}^{(N)}_{i_0 i_N}= \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}}}. % \bar{G}^{(N)}_{i_0 i_N}= \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}}}.
%\ee %\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 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 $\text{Prob}_{i_0}$. 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 %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}. %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. 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. 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 $|i\rangle$. %{\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 $c_i$. %To each state is associated the (positive or negative) quantity $c_i$.