almost OK with Sec II
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@ -172,15 +172,15 @@ However, all the ideas presented in this work can be adapted without too much di
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Without entering into the mathematical details (which are presented below), the main ingredient of DMC in order to perform the matrix-vector multiplications probabilistically is the stochastic matrix (or transition probability matrix) that generates stepwise a series of states over which statistical averages are evaluated.
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The critical aspect of any Monte Carlo scheme is the amount of computational effort required to reach a given statistical error.
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Two important avenues to decrease the error are the use of variance reduction techniques (for example, by introducing improved estimators \cite{Assaraf_1999}) or to improve the quality of the sampling (minimization of the correlation time between states).
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Two important avenues to decrease the error are the use of variance reduction techniques (for example, by introducing improved estimators \cite{Assaraf_1999A}) or to improve the quality of the sampling (minimization of the correlation time between states).
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Another possibility, at the heart of the present work, is to integrate out exactly some parts of the dynamics, thus reducing the number of degrees of freedom and, hence, the amount of statistical fluctuations.
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In a previous work,\cite{assaraf_99,caffarel_00} it has been shown that the probability for a walker to stay a certain amount of time in the same state obeys a Poisson law and that the \titou{on-state} dynamics can be integrated out to generate an effective dynamics connecting only different states with some renormalized estimators for the properties.
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In previous works,\cite{Assaraf_1999B,Caffarel_2000} it has been shown that the probability for a walker to stay a certain amount of time in the same state obeys a Poisson law and that the \titou{on-state} dynamics can be integrated out to generate an effective dynamics connecting only different states with some renormalized estimators for the properties.
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Numerical applications have shown that the statistical errors can be very significantly decreased.
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Here, we extend this idea to the general case where a walker remains a certain amount of time within a finite domain no longer restricted to a single state.
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It is shown how to define an effective stochastic dynamics describing walkers moving from one domain to another.
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The equations of the effective dynamics are derived and a numerical application for a model (one-dimensional) problem is presented.
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In particular, it shows that the statistical convergence of the energy can be greatly enhanced when domains associated with large average trapping times are cnosidered.
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In particular, it shows that the statistical convergence of the energy can be greatly enhanced when domains associated with large average trapping times are considered.
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It should be noted that the use of domains in quantum Monte Carlo is not new.
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In their pioneering work, \cite{Kalos_1974} Kalos and collaborators introduced the so-called domain Green's function Monte Carlo approach in continuous space that they applied to a system of bosons with hard-sphere interaction.
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@ -194,7 +194,7 @@ Section \ref{sec:DMC} presents the basic equations and notations of DMC.
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First, the path integral representation of the Green's function is given in Subsec.~\ref{sec:path}.
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The probabilistic framework allowing the Monte Carlo calculation of the Green's function is presented in Subsec.~\ref{sec:proba}.
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Section \ref{sec:DMC_domains} is devoted to the use of domains in DMC.
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First, we recall in Subsec.~\ref{sec:single_domains} the case of a domain consisting of a single state. \cite{Assaraf_1999}
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First, we recall in Subsec.~\ref{sec:single_domains} the case of a domain consisting of a single state. \cite{Assaraf_1999B}
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The general case is then treated in Subsec.~\ref{sec:general_domains}.
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In Subsec.~\ref{sec:Green}, both the time- and energy-dependent Green's function using domains is derived.
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Section \ref{sec:numerical} presents the application of the approach to the one-dimensional Hubbard model.
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@ -215,14 +215,14 @@ As previously mentioned, DMC is a stochastic implementation of the power method
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\be
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T = \Id -\tau (H-E\Id),
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\ee
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where $\Id$ is the identity operator, $\tau$ a small positive parameter playing the role of a time-step, $E$ some arbitrary reference energy, and $H$ the Hamiltonian operator. Starting from some initial vector, $\ket{\Psi_0}$, we have
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where $\Id$ is the identity operator, $\tau$ a small positive parameter playing the role of a time step, $E$ some arbitrary reference energy, and $H$ the Hamiltonian operator. For any initial vector $\ket{\Psi_0}$ provided that $\braket{\Phi_0}{\Psi_0} \ne 0$ and for $\tau$ sufficiently small, we have
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\be
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\label{eq:limTN}
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\lim_{N \to \infty} T^N \ket{\Psi_0} = \ket{\Phi_0},
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\ee
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where $\ket{\Phi_0}$ is the ground-state wave function.
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The equality is up to a global phase factor playing no role in physical quantum averages.
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This result is true for any $\ket{\Psi_0}$ provided that $\braket{\Phi_0}{\Psi_0} \ne 0$ and for $\tau$ sufficiently small.
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At large but finite $N$, the vector $T^N \ket{\Psi_0}$ differs from $\ket{\Phi_0}$ only by an exponentially small correction, making easy to extrapolate the finite-$N$ results to $N \to \infty$.
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where $\ket{\Phi_0}$ is the ground-state wave function, \ie, $H \ket{\Phi_0} = E_0 \ket{\Phi_0}$.
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The equality in Eq.~\eqref{eq:limTN} holds up to a global phase factor playing no role in physical quantum averages.
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At large but finite $N$, the vector $T^N \ket{\Psi_0}$ differs from $\ket{\Phi_0}$ only by an exponentially small correction, making it straightforward to extrapolate the finite-$N$ results to $N \to \infty$.
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Likewise, ground-state properties may be obtained at large $N$.
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For example, in the important case of the energy, one can rely on the following formula
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@ -236,15 +236,16 @@ To proceed further we introduce the time-dependent Green's matrix $G^{(N)}$ defi
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\be
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G^{(N)}_{ij}=\mel{j}{T^N}{i}.
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\ee
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\titou{where $\ket{i}$ and $\ket{j}$ are basis vectors.}
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The denomination ``time-dependent Green's matrix'' is used here since $G$ may be viewed as a short-time approximation of the (time-imaginary) evolution operator,
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$e^{-N\tau H}$ which is usually referred to as the imaginary-time dependent Green's function.
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\titou{Introducing the set of $N-1$ intermediate states, $\{ i_k \}_{1 \le k \le N-1}$, in the $N$th product of $T$,} $G^{(N)}$ can be written in the following expanded form
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\titou{Introducing the set of $N-1$ intermediate states, $\{ \ket{i_k} \}_{1 \le k \le N-1}$, in the $N$th product of $T$,} $G^{(N)}$ can be written in the following expanded form
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\be
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\label{eq:cn}
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G^{(N)}_{i_0 i_N} = \sum_{i_1} \sum_{i_2} ... \sum_{i_{N-1}} \prod_{k=0}^{N-1} T_{i_{k} i_{k+1}},
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\ee
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\titou{where $T_{i_{k} i_{k+1}} = ??$}.
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\titou{where $T_{ij} =\mel{i}{T}{j}$}.
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Here, each index $i_k$ runs over all basis vectors.
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In quantum physics, Eq.~\eqref{eq:cn} is referred to as the path-integral representation of the Green's matrix (function).
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@ -252,31 +253,32 @@ The series of states $\ket{i_0}, \ldots,\ket{i_N}$ is interpreted as a ``path''
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Each path is associated with a weight $\prod_{k=0}^{N-1} T_{i_{k} i_{k+1}}$ and the path integral expression of $G$ can be recast in the more suggestive form as follows:
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\be
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\label{eq:G}
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G^{(N)}_{i_0 i_N}= \sum_{\text{all paths $\ket{i_1},\ldots,\ket{i_{N-1}}$}} \prod_{k=0}^{N-1} T_{i_{k} i_{k+1}}
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G^{(N)}_{i_0 i_N}= \sum_{\text{all paths $\ket{i_1},\ldots,\ket{i_{N-1}}$}} \prod_{k=0}^{N-1} T_{i_{k} i_{k+1}}.
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\ee
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This expression allows a simple and vivid interpretation of the solution.
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In the limit $N \to \infty$, the $i$th component of the ground state wave funciton (obtained as $\lim_{N \to \infty} G^{(N)}_{\titou{i_0 i_N}})$ is the weighted sum over all possible paths arriving at vector \titou{$\ket{i_N}$}.
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In the limit $N \to \infty$, the $i$th component of the ground-state wave function (obtained as $\lim_{N \to \infty} G^{(N)}_{\titou{i_0 i_N}})$ is the weighted sum over all possible paths arriving at vector \titou{$\ket{i_N}$}.
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This result is independent of the initial vector $\ket{i_0}$, apart from some irrelevant global phase factor.
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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.
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We are in the realm of what one would call the ``deterministic'' power methods, such as the Lancz\`os or Davidson approaches.
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If not, probabilistic techniques for generating only the paths contributing significantly to the sums are to be prefered.
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This is the
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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.
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If not, probabilistic techniques for generating only the paths contributing significantly to the sums are to be preferred.
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This is the central theme of the present work.
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%=======================================%
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\subsection{Probabilistic framework}
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\label{sec:proba}
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%=======================================%
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In order to derive a probabilistic expression for the Green's matrix we introduce a so-called guiding vector, $\ket{\PsiG}$, having strictly positive components, \ie, $\PsiG_i > 0$, and apply a similarity transformation to the operators $G^{(N)}$ and $T$
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In order to derive a probabilistic expression for the Green's matrix we introduce a so-called guiding vector, $\ket{\PsiG}$, having strictly positive components, \ie, $\PsiG_i > 0$, in order to apply a similarity transformation to the operators $G^{(N)}$ and $T$ as follows:
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\begin{align}
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\label{eq:defT}
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\bar{T}_{ij} & = \frac{\PsiG_j}{\PsiG_i} T_{ij}
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\\
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\bar{G}^{(N)}_{ij}& = \frac{\PsiG_j}{\PsiG_i} G^{(N)}_{ij}
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\bar{T}_{ij} & = \frac{\PsiG_j}{\PsiG_i} T_{ij},
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&
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\bar{G}^{(N)}_{ij}& = \frac{\PsiG_j}{\PsiG_i} G^{(N)}_{ij}.
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\end{align}
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Note that under the similarity transformation the path integral expression, Eq.~\eqref{eq:G}, relating $G^{(N)}$ and $T$ remains unchanged for the similarity-transformed operators, $\bar{G}^{(N)}$ and $\bar{T}$.
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Note that, under this similarity transformation, the path integral expression relating $G^{(N)}$ and $T$ [see Eq.~\eqref{eq:G}] remains unchanged for $\bar{G}^{(N)}$ and $\bar{T}$.
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Next, the matrix elements of $\bar{T}$ are expressed as those of a stochastic matrix multiplied by some residual weight, namely
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Next, the matrix elements of $\bar{T}$ are expressed as those of a stochastic matrix multiplied by some residual weight, namely,
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\be
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\label{eq:defTij}
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\bar{T}_{ij}= p_{i \to j} w_{ij}.
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@ -290,6 +292,7 @@ To build the transition probability density the following operator is introduced
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%As known, there is a natural way of associating a stochastic matrix to a matrix having a positive ground-state vector (here, a positive vector is defined here as
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%a vector with all components positive).
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\be
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\label{eq:T+}
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T^+= \Id - \tau \qty( H^+ - \EL^+ \Id ),
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\ee
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where
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@ -297,9 +300,9 @@ $H^+$ is the matrix obtained from $H$ by imposing the off-diagonal elements to b
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\be
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H^+_{ij}=
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\begin{cases}
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\phantom{-}H_{ij}, & \text{if $i=j$}.
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\phantom{-}H_{ij}, & \text{if $i=j$},
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\\
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-\abs{H_{ij}}, & \text{if $i\neq j$}.
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-\abs{H_{ij}}, & \text{if $i\neq j$}.
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\end{cases}
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\ee
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Here, $\EL^+ \Id$ is the diagonal matrix whose diagonal elements are defined as
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@ -307,55 +310,46 @@ Here, $\EL^+ \Id$ is the diagonal matrix whose diagonal elements are defined as
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(\EL^+)_{i}= \frac{\sum_j H^+_{ij}\PsiG_j}{\PsiG_i}.
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\ee
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The vector $\EL^+$ is known as the local energy vector associated with $\PsiG$.
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Actually, the operator $H^+ - \EL^+ \Id$ in the definition of the operator $T^+$ has been chosen to admit by construction $\ket{\PsiG}$ as ground-state with zero eigenvalue
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\be
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\label{eq:defTplus}
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\qty(H^+ - E_L^+ \Id) \ket{\PsiG} = 0,
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\ee
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leading to the relation
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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
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\be
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\label{eq:relT+}
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T^+ \ket{\PsiG} = \ket{\PsiG}.
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\ee
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The stochastic matrix is now defined as
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We are now in the position to define the stochastic matrix as
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\be
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\label{eq:pij}
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p_{i \to j} = \frac{\PsiG_j}{\PsiG_i} T^+_{ij}.
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p_{i \to j}
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= \frac{\PsiG_j}{\PsiG_i} T^+_{ij}
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=
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\begin{cases}
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1 - \tau \qty[ H^+_{ii}- (\EL^+)_{i} ], & \text{if $i=j$},
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\\
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\tau \frac{\PsiG_{j}}{\PsiG_{i}} \abs{H_{ij}} \ge 0, & \text{if $i\neq j$}.
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\end{cases}
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\ee
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The diagonal matrix elements of the stochastic matrix write
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\be
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p_{i \to i} = 1 - \tau \qty[ H^+_{ii}- (\EL^+)_{i} ]
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\ee
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while, for $i \ne j$,
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\be
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p_{i \to j} = \tau \frac{\PsiG_{j}}{\PsiG_{i}} \abs{H_{ij}} \ge 0
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\ee
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As seen, the off-diagonal terms, $p_{i \to j}$ are positive while the diagonal ones, $p_{i \to i}$, can be made positive if $\tau$ is chosen sufficiently small.
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More precisely, the condition writes
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As readily seen in Eq.~\eqref{eq:pij}, the off-diagonal terms of the stochastic matrix are positive, while the diagonal ones can be made positive if $\tau$ is chosen sufficiently small via the condition
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\be
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\label{eq:cond}
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\tau \leq \frac{1}{\max_i\abs{H^+_{ii}-(\EL^+)_{i}}}
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\tau^{-1} \geq \max_i\abs{H^+_{ii}-(\EL^+)_{i}}.
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\ee
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The sum-over-states condition, Eq.~\eqref{eq:sumup}, follows from the fact that $|\PsiG\rangle$ is eigenvector of $T^+$, Eq.~\eqref{eq:relT+}
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The sum-over-states condition [see Eq.~\eqref{eq:sumup}]
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\be
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\sum_j p_{i \to j}= \frac{1}{\PsiG_{i}} \mel{i}{T^+}{\PsiG} = 1.
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\sum_j p_{i \to j}= \frac{\mel{i}{T^+}{\PsiG}}{\PsiG_{i}} = 1.
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\ee
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We have then verified that $p_{i \to j}$ is indeed a stochastic matrix.
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follows from the fact that $\ket{\PsiG}$ is eigenvector of $T^+$ [see Eq.~\eqref{eq:relT+}].
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This ensures that $p_{i \to j}$ is indeed a stochastic matrix.
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At first sight, the condition defining the maximum value of $\tau$ allowed, Eq.~\eqref{eq:cond}, may appear as rather tight
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since for very large matrices it may impose an extremely small value for the time step. However, in practice during the simulation only a (tiny)
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fraction of the linear space is sampled, and the maximum value of $\abs{H^+_{ii} - E^+_{Li}}$ for the sampled states turns out to be not too large, so reasonable values of $\tau$
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can be used without violating the positivity of the transition probability matrix.
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Note that we can even escape from this condition by slightly generalizing the transition probability
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matrix used as follows
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At first sight, the condition defining the maximum value of $\tau$ allowed, Eq.~\eqref{eq:cond}, may appear rather tight since, for very large matrices, it may impose an extremely small value of the time step.
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However, in practice, during the simulation only a (tiny) fraction of the linear space is sampled, and the maximum absolute value of $H^+_{ii}-(\EL^+)_{i}$ for the sampled states turns out to be \titou{not too large}.
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Hence, reasonable values of $\tau$ can be selected without violating the positivity of the transition probability matrix.
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\titou{Note that one can eschew this condition via a simple generalization of the transition probability matrix:}
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\be
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p_{i \to j}
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= \frac{ \frac{\PsiG_{j}}{\PsiG_{i}} \abs{\mel{i}{T^+}{j}} }
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{ \sum_j \frac{\PsiG_{j}}{\PsiG_{i}} \abs{\mel{i}{T^+}{j}} }
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= \frac{ \PsiG_{j} \abs{\mel{i}{T^+}{j}} }
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{ \sum_j \PsiG_{j} \abs{\mel{i}{T^+}{j}} }
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= \frac{ \PsiG_{j} \abs*{T^+_{ij}} }
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{ \sum_j \PsiG_{j} \abs*{T^+_{ij}} }.
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\ee
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This new transition probability matrix with positive entries reduces to Eq.~\eqref{eq:pij} when $T^+_{ij}$ is positive.
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@ -370,34 +364,33 @@ Using these notations the Green's matrix components can be rewritten as
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\ee
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\titou{where $i$ is identified to $i_N$.}
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The product $\prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}}$ is the probability, denoted $\text{Prob}_{i_0 \to i_N}(i_1,...,i_{N-1})$,
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The product $\prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}}$ is the probability, denoted $\text{Prob}_{i_0 \to i_N}(i_1,\ldots,i_{N-1})$,
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for the path starting at $\ket{i_0}$ and ending at $\ket{i_N}$ to occur.
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Using the fact that $p_{i \to j} \ge 0$ and Eq.~\eqref{eq:sumup} we verify that $\text{Prob}_{i_0 \to i_N}$ is positive and obeys
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Using Eq.~\eqref{eq:sumup} and the fact that $p_{i \to j} \ge 0$, one can easily verify that $\text{Prob}_{i_0 \to i_N}$ is positive and obeys
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\be
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\sum_{i_1,\ldots,i_{N-1}} \text{Prob}_{i_0 \to i_N}(i_1,\ldots,i_{N-1}) = 1
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\sum_{i_1,\ldots,i_{N-1}} \text{Prob}_{i_0 \to i_N}(i_1,\ldots,i_{N-1}) = 1,
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\ee
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as it should be.
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The probabilistic average associated with this probability for the path, denoted here as, $ \expval{\cdots}$ is then defined as
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as it should.
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For a given path $i_1,\ldots,i_{N-1}$, the probabilistic average associated with this probability, denoted here as $\expval{\cdots}$, is then defined as
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\be
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\label{eq:average}
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\expval{F} = \sum_{i_1,\ldots,i_{N-1}} F(i_0,\ldots,i_N) \text{Prob}_{i_0 \to i_N}(i_1,\ldots,i_{N-1}),
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\label{average}
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\ee
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where $F$ is an arbitrary function.
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Finally, the path-integral expressed as a probabilistic average reads
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\be
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\bar{G}^{(N)}_{ii_0}= \Big \langle \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \Big \rangle
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\bar{G}^{(N)}_{\titou{ii_0}}= \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}}}
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\label{cn_stoch}
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\ee
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To calculate the probabilistic average, Eq.(\ref{average}),
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To calculate the probabilistic average, Eq.~\eqref{eq:average},
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an artificial (mathematical) ``particle'' called walker (or psi-particle) is introduced.
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During the Monte Carlo simulation the walker moves in configuration space by drawing new states with
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probability $p(i_k \to i_{k+1})$, thus realizing the path of probability
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${\rm Prob}(i_0 \to i_n)$.
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probability $p_{i_k \to i_{k+1}}$, thus realizing the path of probability $\text{Prob}(i_0 \to i_n)$.
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The energy, Eq.~\eqref{eq:E0} is given as
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\be
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E_0 = \lim_{N \to \infty }
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\frac{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} {(H\PsiT)}_{i_N}} }
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{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} {\PsiT}_{i_N} }}
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{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} {\PsiT}_{i_N} }}.
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\ee
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Note that, instead of using a single walker, it is possible to introduce a population of independent walkers and to calculate the averages over the population.
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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.
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@ -426,7 +419,7 @@ We shall not here insist on these practical details that can be found, for examp
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During the simulation, walkers move from state to state with the possibility of being trapped a certain number of times on the same state before
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exiting to a different state. This fact can be exploited in order to integrate out some part of the dynamics, thus leading to a reduction of the statistical
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fluctuations. This idea was proposed some time ago\cite{assaraf_99,Assaraf_1999B,caffarel_00} and applied to the SU(N) one-dimensional Hubbard model.
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fluctuations. This idea was proposed some time ago \cite{Assaraf_1999A,Assaraf_1999B,Caffarel_2000} and applied to the SU(N) one-dimensional Hubbard model.
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Let us consider a given state $|i\rangle$. The probability that the walker remains exactly $n$ times on $|i\rangle$ ($n$ from
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1 to $\infty$) and then exits to a different state $j$ is
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@ -462,7 +455,7 @@ The time-averaged contribution of the \titou{on-state} weight can be easily calc
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\be
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\bar{w}_i= \sum_{n=1}^{\infty} w^n_{ii} P_i(n)= \frac{T_{ii}}{T^+_{ii}} \frac{1-T^+_{ii}}{1-T_{ii}}
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\ee
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Details of the implementation of the effective dynamics can be in found in Refs.~\onlinecite{assaraf_99} and \onlinecite{caffarel_00}.
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Details of the implementation of the effective dynamics can be in found in Refs.~\onlinecite{Assaraf_1999B} and \onlinecite{Caffarel_2000}.
|
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|
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%=======================================%
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\subsection{General domains}
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@ -470,8 +463,8 @@ Details of the implementation of the effective dynamics can be in found in Refs.
|
||||
%=======================================%
|
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|
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Let us now extend the results of the preceding section to a general domain. For that,
|
||||
let us associate to each state $|i\rangle$ a set of states, called the domain of $|i\rangle$ and
|
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denoted ${\cal D}_i$, consisting of the state $|i\rangle$ plus a certain number of states. No particular constraints on the type of domains
|
||||
let us associate to each state $\ket{i}$ a set of states, called the domain of $\ket{i}$ and
|
||||
denoted $\cD_i$, consisting of the state $\ket{i}$ plus a certain number of states. No particular constraints on the type of domains
|
||||
are imposed, for example domains associated with different states can be identical, or may have or not common states. The only important condition is
|
||||
that the set of all domains ensures the ergodicity property of the effective stochastic dynamics (that is, starting from any state there is a
|
||||
non-zero-probability to reach any other state in a finite number of steps). In practice, it is not difficult to impose such a condition.
|
||||
|
||||
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