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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}
%=======================================%
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 $c_i$.