1st cleanup of Sec III
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g.tex
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g.tex
@ -52,6 +52,7 @@
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\newcommand{\cT}{\mathcal{T}}
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\newcommand{\cC}{\mathcal{C}}
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\newcommand{\cD}{\mathcal{D}}
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\newcommand{\cE}{\mathcal{E}}
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\newcommand{\EPT}{E_{\PT}}
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\newcommand{\laPT}{\lambda_{\PT}}
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@ -443,7 +444,7 @@ and this defines a Poisson law with an average number of trapping events
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\ee
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Introducing the continuous time $t_i = n \tau$, the average trapping time is thus given by
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\be
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\bar{t_i}= \frac{1}{H^+_{ii}-(\EL^+)_{i}},
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\bar{t_i}= \qty[ H^+_{ii}-(\EL^+)_{i} ]^{-1},
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\ee
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and, in the limit $\tau \to 0$, the Poisson probability takes the usual form
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\be
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@ -541,19 +542,19 @@ we have
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\ee
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and the average trapping time is
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\be
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t_{I}={\bar n}_{I} \tau = \frac{1}{\PsiG_{I}} \mel{ I }{ P_{I} \frac{1}{H^+ - \EL^+ \Id} P_{I} }{ \PsiG }.
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t_{I}={\bar n}_{I} \tau = \frac{1}{\PsiG_{I}} \mel{ I }{ P_{I} \qty( H^+ - \EL^+ \Id )^{-1} P_{I} }{ \PsiG }.
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\ee
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In practice, the various quantities restricted to the domain are computed by diagonalizing the matrix $(H^+-\EL^+ \Id)$ in $\cD_{I}$.
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Note that it is possible only if the dimension of the domains is not too large (say, less than a few thousands).
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%=======================================%
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\subsection{Expressing the Green's matrix using domains}
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\subsection{Time-dependent Green's matrix using domains}
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\label{sec:Green}
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%=======================================%
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%--------------------------------------------%
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\subsubsection{Time-dependent Green's matrix}
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\label{sec:time}
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%\subsubsection{Time-dependent Green's matrix}
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%\label{sec:time}
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%--------------------------------------------%
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In this section we generalize the path-integral expression of the Green's matrix, Eqs.~\eqref{eq:G} and \eqref{eq:cn_stoch}, to the case where domains are used.
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For that we introduce the Green's matrix associated with each domain
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@ -588,10 +589,10 @@ where $\delta_{i,j}$ is a Kronecker delta.
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This expression is the path-integral representation of the Green's matrix using only the variables $(\ket{I_k},n_k)$ of the effective dynamics defined over the set of domains.
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The standard formula derived above [see Eq.~\eqref{eq:G}] may be considered as the particular case where the domain associated with each state is empty,
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In that case, $p=N$ and $n_k=1$ for $0 \le k \le N$ and we are left only with the $p$th component of the sum, that is, $G^{(N)}_{I_0 I_N}
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= \prod_{k=0}^{N-1} \mel{ I_k }{ F_{I_k} }{ I_{k+1} } $ where $F=T$.
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In this case, $p=N$ and $n_k=1$ (for $0 \le k \le N$) and we are left only with the $p$th component of the sum, that is, $G^{(N)}_{I_0 I_N}
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= \prod_{k=0}^{N-1} \mel{ I_k }{ F_{I_k} }{ I_{k+1} }$, with $F=T$.
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To express the fundamental equation for $G$ under the form of a probabilistic average, we write the importance-sampled version of the equation
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To express the fundamental equation of $G$ under the form of a probabilistic average, we rely on the importance-sampled version of Eq.~\eqref{eq:G}, \ie,
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\begin{multline}
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\label{eq:Gbart}
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\bar{G}^{(N)}_{I_0 I_N}=\bar{G}^{(N),\cD}_{I_0 I_N} +
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@ -603,9 +604,9 @@ To express the fundamental equation for $G$ under the form of a probabilistic av
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\delta_{\sum_k n_k,N+1} \qty{ \prod_{k=0}^{p-1} \qty[ \frac{\PsiG_{I_{k+1}}}{\PsiG_{I_k}} \mel{ I_k }{ \qty(T_{I_k})^{n_k-1} F_{I_k} }{ I_{k+1} } ] }
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\bar{G}^{(n_p-1),\cD}_{I_p I_N}.
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\end{multline}
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Introducing the weight
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Introducing the weights
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\be
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W_{I_k I_{k+1}} = \frac{\mel{ I_k }{ \qty(T_{I_k})^{n_k-1} F_{I_k} }{ I_{k+1} }}{\mel{ I_k }{ \qty(T^{+}_{I_k})^{n_k-1} F^+_{I_k} }{ I_{k+1} }}
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W_{I_k I_{k+1}} = \frac{\mel{ I_k }{ \qty(T_{I_k})^{n_k-1} F_{I_k} }{ I_{k+1} }}{\mel{ I_k }{ \qty(T^{+}_{I_k})^{n_k-1} F^+_{I_k} }{ I_{k+1} }},
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\ee
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and using the effective transition probability defined in Eq.~\eqref{eq:eq3C}, we get
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\be
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@ -613,10 +614,9 @@ and using the effective transition probability defined in Eq.~\eqref{eq:eq3C}, w
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\bar{G}^{(N)}_{I_0 I_N} = \bar{G}^{(N),\cD}_{I_0 I_N} + \sum_{p=1}^{N}
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\expval{
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\qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} )
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\bar{G}^{(n_p-1), {\cal D}}_{I_p I_N}
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}
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\bar{G}^{(n_p-1), {\cal D}}_{I_p I_N} },
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\ee
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where the average is defined as
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where the average of a given function $F$ is defined as
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\begin{multline}
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\expval{F}
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= \sum_{\ket{I_1} \notin \cD_{I_0}, \ldots , \ket{I_p} \notin \cD_{I_{p-1}}}
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@ -624,7 +624,7 @@ where the average is defined as
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\delta_{\sum_k n_k,N+1}
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\\
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\times
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\prod_{k=0}^{N-1}\cP_{I_k \to I_{k+1}}(n_k-1) F(I_0,n_0;\ldots.;I_N,n_N)
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\prod_{k=0}^{N-1}\cP_{I_k \to I_{k+1}}(n_k-1) F(I_0,n_0;\ldots.;I_N,n_N).
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\end{multline}
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In practice, a schematic DMC algorithm to compute the average is as follows.\\
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i) Choose some initial vector $\ket{I_0}$\\
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@ -637,161 +637,158 @@ iv) Go to step ii) until some maximum number of paths is reached.\\
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At the end of the simulation, an estimate of the average for a few values of $N$ greater but close to $N_\text{max}$ is obtained.
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At large $N_\text{max}$ where the convergence of the average as a function of $p$ is reached, such values can be averaged.
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%--------------------------------------------%
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\subsubsection{Integrating out the trapping times : The Domain Green's Function Monte Carlo approach}
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%==============================================%
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\subsection{Domain Green's Function Monte Carlo}
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\label{sec:energy}
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%--------------------------------------------%
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%==============================================%
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Now, let us show that it is possible to go further by integrating out the trapping times, $n_k$, of the preceding expressions, thus defining a new effective
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stochastic dynamics involving now only the exit states. Physically, it means that we are going to compute exactly within the time-evolution of all
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stochastic paths trapped within each domain. We shall present two different ways to derive the new dynamics and renormalized probabilistic averages.
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The first one, called the pedestrian way, consists in starting from the preceding time-expression for $G$ and make the explicit integration over the
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$n_k$. The second, more direct and elegant, is based on the Dyson equation.\\
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\\
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{\it $\bullet$ The pedestrian way}. Let us define the quantity\\
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The aim of this section is to show that it is possible to go further by integrating out the trapping times, $n_k$, of the preceding expressions, thus defining a new effective stochastic dynamics involving now only the exit states.
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Physically, it means that we are going to compute exactly within the time-evolution of all stochastic paths trapped within each domain.
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We shall present two different ways to derive the new dynamics and renormalized probabilistic averages.
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The first one, called the pedestrian way, consists in starting from the preceding time-expression for $G$ and make the explicit integration over the $n_k$'s.
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The second, more direct and elegant, is based on the Dyson equation.
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%--------------------------------------------%
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\subsubsection{The pedestrian way}
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%--------------------------------------------%
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Let us define the quantity\\
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$$
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G^E_{ij}= \tau \sum_{N=0}^{\infty} \mel{ i }{ T^N }{ j}.
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$$
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By summing over $N$ we obtain
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\be
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G^E_{ij}= \mel{i}{\frac{1}{H-E \Id}}{j}.
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G^E_{ij}= \mel{i}{ \qty( H-E \Id )^{-1} }{j}.
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\ee
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This quantity, which no longer depends on the time-step, is referred to as the energy-dependent Green's matrix. Note that in the continuum this quantity is
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essentially the Laplace transform of the time-dependent Green's function. Here, we then use the same denomination. The remarkable property
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is that, thanks to the summation over $N$ up to the
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infinity the constrained multiple sums appearing in Eq.(\ref{Gt}) can be factorized in terms of a product of unconstrained single sums as follows
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This quantity, which no longer depends on the time step, is referred to as the energy-dependent Green's matrix.
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Note that, \titou{in a continuous space}, this quantity is essentially the \titou{Laplace transform of the time-dependent Green's function}.
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Here, we then use the same denomination.
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The remarkable property is that, thanks to the summation over $N$ up to the infinity the constrained multiple sums appearing in Eq.~\eqref{eq:Gt} can be factorized in terms of a product of unconstrained single sums as follows
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\be
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\sum_{N=1}^\infty \sum_{p=1}^N \sum_{n_0 \ge 1} ...\sum_{n_p \ge 1} \delta_{n_0+...+n_p,N+1}
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= \sum_{p=1}^{\infty} \sum_{n_0=1}^{\infty} ...\sum_{n_p=1}^{\infty}.
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\sum_{N=1}^\infty \sum_{p=1}^N \sum_{n_0 \ge 1} ...\sum_{n_p \ge 1} \delta_{n_0+...+n_p,N+1} \titou{??}
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= \sum_{p=1}^{\infty} \sum_{n_0=1}^{\infty} ...\sum_{n_p=1}^{\infty} \titou{??}.
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\ee
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It is then a trivial matter to integrate out exactly the $n_k$ variables, leading to
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$$
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\langle I_0|\frac{1}{H-E}|I_N\rangle = \langle I_0|P_0\frac{1}{H-E} P_0|I_N\rangle
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+ \sum_{p=1}^{\infty}
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\sum_{I_1 \notin {\cal D}_0, \hdots , I_p \notin {\cal D}_{p-1}}
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$$
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\begin{multline}
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\label{eq:eqfond}
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\mel{ I_0 }{ \qty(H-E \Id)^{-1} }{ I_N }
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= \mel{ I_0 }{ P_0 \qty(H-E \Id)^{-1} P_0 }{ I_N}
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\\
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+ \sum_{p=1}^{\infty} \sum_{I_1 \notin \cD_0, \hdots , I_p \notin \cD_{p-1}}
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\qty[ \prod_{k=0}^{p-1} \mel{ I_k }{ P_k \qty( H-E \Id )^{-1} P_k (-H)(1-P_k) }{ I_{k+1} } ]
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\mel{ I_p }{ P_p \qty( H-E \Id)^{-1} P_p }{ I_N }
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\end{multline}
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As an illustration, Appendix \ref{app:A} reports the exact derivation of this formula in the case of a \titou{two-level} system.
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%----------------------------%
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\subsubsection{Dyson equation}
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%----------------------------%
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In fact, there is a more direct way to derive the same equation by resorting to the Dyson equation. Starting from the well-known equality
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\be
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\Big[ \prod_{k=0}^{p-1} \langle I_k| P_k \frac{1}{H-E} P_k (-H)(1-P_k)|I_{k+1} \rangle \Big]
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\langle I_p| P_p \frac{1} {H-E} P_p|I_N\rangle
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\label{eqfond}
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\qty(H-E\Id)^{-1} = \qty(H_0-E\Id)^{-1} + \qty(H_0-E\Id)^{-1} (H_0-H) \qty(H-E\Id)^{-1},
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\ee
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\noindent
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As an illustration, Appendix \ref{A} reports the exact derivation of this formula in the case of a two-state system.\\
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\\
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{\it $\bullet$ Dyson equation.} In fact, there is a more direct way to derive the same equation by resorting to the Dyson equation. Starting from the
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well-known equality
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where $H_0$ is some arbitrary reference Hamiltonian, we have the Dyson equation
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\be
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\frac{1}{H-E} = \frac{1}{H_0-E}
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+ \frac{1}{H_0-E} (H_0-H)\frac{1}{H-E}
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\ee
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where $H_0$ is some arbitrary reference Hamiltonian, we have
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the Dyson equation
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$$
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G^E_{ij}= G^E_{0,ij} + \sum_{k,l} G^{E}_{0,ik} (H_0-H)_{kl} G^E_{lj}
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$$
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Let us choose as $H_0$
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$$
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\langle i |H_0|j\rangle= \langle i|P_i H P_i|j\rangle \;\;\; {\rm for \; all \; i \;j}.
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$$
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The Dyson equation becomes
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$$
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\langle i| \frac{1}{H-E}|j\rangle
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= \langle i| P_i \frac{1}{H-E} P_i|j\rangle
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$$
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\be
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+ \sum_k \langle i| P_i \frac{1}{H-E} P_i(H_0-H)|k\rangle \langle k|\frac{1}{H-E}|j\rangle
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\label{eq:GE}
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G^E_{ij}= \titou{G^E_{0,ij}} + \sum_{kl} G^{E}_{0,ik} (H_0-H)_{kl} G^E_{lj}
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\ee
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Let us choose as $H_0$ such that $\mel{ i }{ H_0 }{ j } = \mel{ i }{ P_i H P_i }{ j }$ for all $i$ and $j$.
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Then, The Dyson equation \eqref{eq:GE} becomes
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\begin{multline}
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\mel{ i }{ \qty(H-E \Id)^{-1} }{ j }
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= \mel{ i }{ P_i \qty(H-E \Id)^{-1} P_i }{ j }
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\\
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+ \sum_k \mel{ i }{ P_i \qty(H-E \Id)^{-1} P_i(H_0-H) }{ k } \mel{ k }{ \qty(H-E \Id)^{-1} }{ j }.
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\end{multline}
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Now, we have
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$$
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P_i \frac{1}{H-E} P_i(H_0-H) = P_i \frac{1}{H-E} P_i (P_i H P_i - H)
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$$
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$$
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= P_i \frac{1}{H-E} P_i (-H) (1-P_i)
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$$
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\be
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\begin{split}
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P_i \qty(H-E \Id)^{-1} P_i(H_0-H)
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& = P_i \qty(H-E \Id)^{-1} P_i (P_i H P_i - H)
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\\
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& = P_i \qty(H-E \Id)^{-1} P_i (-H) (1-P_i)
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\end{split}
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\ee
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and the Dyson equation may be written under the form
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$$
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\langle i| \frac{1}{H-E}|j\rangle
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= \langle i| P_i \frac{1}{H-E} P_i|j\rangle
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$$
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$$
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+ \sum_{k \notin {\cal D}_i} \langle i| P_i \frac{1}{H-E} P_i (-H)(1-P_i)|k\rangle \langle k|\frac{1}{H-E}|j\rangle
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$$
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which is identical to Eq.(\ref{eqfond}) when $G$ is expanded iteratively.\\
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\begin{multline}
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\mel{ i }{ \qty(H-E \Id)^{-1} }{ j }
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= \mel{ i }{ P_i \qty(H-E \Id)^{-1} P_i }{ j}
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\\
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+ \sum_{k \notin \cD_i} \mel{ i }{ P_i \qty(H-E \Id)^{-1} P_i (-H)(1-P_i) }{ k } \mel{k}{\qty(H-E \Id)^{-1}}{j}
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\end{multline}
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which is identical to Eq.~\eqref{eq:eqfond} when $G$ is expanded iteratively.
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\\
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Let us use as effective transition probability density
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\be
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P(I \to J) = \frac{1} {\PsiG(I)} \langle I| P_I \frac{1}{H^+-E^+_L} P_I (-H^+) (1-P_I)|J\rangle \PsiG(J)
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P_{I \to J} = \frac{1} {\PsiG(I)} \mel{ I }{ P_I \qty(H^+ - \EL^+ \Id)^{-1} P_I (-H^+) (1-P_I) }{ J } \PsiG(J)
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\ee
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and the weight
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\be
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W^E_{IJ} =
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\frac{\langle I|\frac{1}{H-E} P_I (-H)(1-P_I) |J\rangle }{\langle I|\frac{1}{H^+-E^+_L} P_I (-H^+)(1-P_I) |J\rangle}
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W^E_{IJ} =
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\frac{ \mel{ I }{ \qty(H-E \Id)^{-1} P_I (-H)(1-P_I) }{ J} }
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{\mel{ I }{ \qty(H^+ - \EL^+ \Id)^{-1} P_I (-H^+)(1-P_I) }{ J} }
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\ee
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Using Eqs.~\eqref{eq:eq1C}, \eqref{eq:eq3C} and \eqref{eq:relation}, we verify that $P_{I \to J} \ge 0$ and $\sum_J P_{I \to J}=1$.
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Using Eqs.~\eqref{eq:eq1C}, \eqref{eq:eq3C} and \eqref{eq:relation}, one can easily verify that $P_{I \to J} \ge 0$ and $\sum_J P_{I \to J}=1$.
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Finally, the probabilistic expression writes
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$$
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\langle I_0| \frac{1}{H-E}|I_N\rangle
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= \langle I_0| P_{I_0} \frac{1}{H-E} P_{I_0}|I_N\rangle
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$$
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\begin{multline}
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\label{eq:final_E}
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\mel{ I_0 }{ \qty(H-E \Id)^{-1} }{ I_N }
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= \mel{ I_0 }{ P_{I_0} \qty(H-E \Id)^{-1} P_{I_0} }{ I_N }
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\\
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+ \sum_{p=0}^{\infty} \expval{ \qty( \prod_{k=0}^{p-1} W^E_{I_k I_{k+1}} ) \mel{ I_p }{ P_{I_p} \qty(H-E \Id)^{-1} P_{I_p} }{ I_N } }.
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\end{multline}
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%----------------------------%
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\subsubsection{Energy estimator}
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%----------------------------%
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To calculate the energy we introduce the following quantity
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\be
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+ \sum_{p=0}^{\infty} \Bigg \langle \Big( \prod_{k=0}^{p-1} W^E_{I_k I_{k+1}} \Big) \langle I_p| P_{I_p} \frac{1}{H-E} P_{I_p}|I_N\rangle \Bigg \rangle
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\label{final_E}
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\ee
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{\it Energy estimator.} To calculate the energy we introduce the following quantity
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\be
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{\cal E}(E) = \frac{ \langle I_0|\frac{1}{H-E}|H\PsiT\rangle} {\langle I_0|\frac{1}{H-E}|\PsiT\rangle}.
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\cE(E) = \frac{ \mel{ I_0 }{ \qty(H-E \Id)^{-1} }{ H\PsiT } } {\mel{ I_0 }{ \qty(H-E \Id)^{-1} }{ \PsiT } }.
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\label{calE}
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\ee
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and search for the solution $E=E_0$ of
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\be
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{\cal E}(E)= E
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\ee
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and search for the solution $E=E_0$ of $\cE(E)= E$.
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Using the spectral decomposition of $H$ we have
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\be
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{\cal E}(E) = \frac{ \sum_i \frac{E_i c_i}{E_i-E}}{\sum_i \frac{c_i}{E_i-E}}
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\label{calE}
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\label{eq:calE}
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\cE(E) = \frac{ \sum_i \frac{E_i c_i}{E_i-E}}{\sum_i \frac{c_i}{E_i-E}}
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\ee
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with
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\be
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c_i = \langle I_0| \Phi_0\rangle \langle \Phi_0| \PsiT\rangle
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\ee
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It is easy to check that in the vicinity of $E=E_0$, ${\cal E}(E)$ is a linear function of $E-E_0$.
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So, in practice we compute a few values of ${\cal E}(E^{k})$ and fit the data using some function of $E$ close to the linearity
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to extrapolate the exact value of $E_0$. Let us describe the 3 functions used here for the fit.\\
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with \titou{$c_i = \braket{ I_0 }{ \Phi_0 } \braket{ \Phi_0}{ \PsiT }$}.
|
||||
It is easy to check that, in the vicinity of $E = E_0$, $\cE(E)$ is a linear function of $E - E_0$.
|
||||
Therefore, in practice, we compute several values of \titou{$\cE(E^{k})$} and fit the data using some function of $E$ close to the linearity to extrapolate the exact value of $E_0$.
|
||||
%Let us describe the 3 functions used here for the fit.
|
||||
|
||||
i) Linear fit\\
|
||||
We write
|
||||
\be
|
||||
{\cal E}(E)= a_0 + a_1 E
|
||||
\ee
|
||||
and search the best value of $(a_0,a_1)$ by fitting the data.
|
||||
Then, ${\cal E}(E)=E$ leads to
|
||||
$$
|
||||
E_0= \frac{a_0}{1-a_1}
|
||||
$$
|
||||
ii) Quadratic fit\\
|
||||
At the quadratic level we write
|
||||
$$
|
||||
{\cal E}(E)= a_0 + a_1 E + a_2 E^2
|
||||
$$
|
||||
leading to
|
||||
$$
|
||||
E_0 = \frac{1 - a_1 \pm \sqrt{ (a_1 -1)^2 - 4 a_0 a_2}}{2 a_2}
|
||||
$$
|
||||
iii) Two-component fit\\
|
||||
We take the advantage that the exact expression of ${\cal E}(E)$ is known as an infinite series, Eq.(\ref{calE}).
|
||||
If we limit ourselves to the first two-component, we write
|
||||
\be
|
||||
{\cal E}(E) = \frac{ \frac{E_0 c_0}{E_0-E} + \frac{E_1 c_1}{E_1-E}}{\frac{c_0}{E_0-E} + \frac{c_1}{E_1-E} }
|
||||
\ee
|
||||
Here, the variational parameters used for the fit are ($c_0,E_0,c_1,E_1)$.\\
|
||||
%i) Linear fit\\
|
||||
%We write
|
||||
%\be
|
||||
%\cE(E)= a_0 + a_1 E
|
||||
%\ee
|
||||
%and search the best value of $(a_0,a_1)$ by fitting the data.
|
||||
%Then, $\cE(E)=E$ leads to
|
||||
%$$
|
||||
%E_0= \frac{a_0}{1-a_1}
|
||||
%$$
|
||||
%ii) Quadratic fit\\
|
||||
%At the quadratic level we write
|
||||
%$$
|
||||
%\cE(E)= a_0 + a_1 E + a_2 E^2
|
||||
%$$
|
||||
%leading to
|
||||
%$$
|
||||
%E_0 = \frac{1 - a_1 \pm \sqrt{ (a_1 -1)^2 - 4 a_0 a_2}}{2 a_2}
|
||||
%$$
|
||||
%iii) Two-component fit\\
|
||||
%We take the advantage that the exact expression of $\cE(E)$ is known as an infinite series, Eq.(\ref{calE}).
|
||||
%If we limit ourselves to the first two-component, we write
|
||||
%\be
|
||||
%\cE(E) = \frac{ \frac{E_0 c_0}{E_0-E} + \frac{E_1 c_1}{E_1-E}}{\frac{c_0}{E_0-E} + \frac{c_1}{E_1-E} }
|
||||
%\ee
|
||||
%Here, the variational parameters used for the fit are ($c_0,E_0,c_1,E_1)$.\\
|
||||
|
||||
In order to have a precise extrapolation of the energy, it is interesting to compute the ratio
|
||||
${\cal E}(E)$ for values of $E$ as close as possible to the exact energy. However, in that case
|
||||
the numerators and denominators computed diverge. This is reflected by the fact that we need to compute
|
||||
more and more $p$-components with an important increase of statistical fluctuations. So, in practice
|
||||
a tradoff has to be found between the possible bias in the extrapolation and the amount of simulation time
|
||||
required.
|
||||
In order to have a precise extrapolation of the energy, it is interesting to compute the ratio $\cE(E)$ for values of $E$ as close as possible to the exact energy.
|
||||
However, in that case the numerators and denominators computed diverge.
|
||||
This is reflected by the fact that we need to compute more and more $p$-components with an important increase of statistical fluctuations.
|
||||
So, in practice a trade-off has to be found between the possible bias in the extrapolation and the amount of simulation time required.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Numerical application to the Hubbard model}
|
||||
@ -805,17 +802,17 @@ Let us consider the one-dimensional Hubbard Hamiltonian for a chain of $N$ sites
|
||||
\ee
|
||||
where $\langle i j\rangle$ denotes the summation over two neighboring sites,
|
||||
$\hat{a}_{i\sigma} (\hat{a}_{i\sigma})$ is the fermionic creation (annihilation) operator of
|
||||
an electron of spin $\sigma$ ($=\uparrow$ or $\downarrow$) at site $i$, $\hat{n}_{i\sigma} = \hat{a}^+_{i\sigma} \hat{a}_{i\sigma}$ the
|
||||
an spin-$\sigma$ electron (with $\sigma$ = $\uparrow$ or $\downarrow$) on site $i$, $\hat{n}_{i\sigma} = \hat{a}^+_{i\sigma} \hat{a}_{i\sigma}$ the
|
||||
number operator, $t$ the hopping amplitude and $U$ the on-site Coulomb repulsion.
|
||||
We consider a chain with an even number of sites and open boundary conditions (OBC)
|
||||
at half-filling, that is, $N_{\uparrow}=N_{\downarrow}=\frac{N}{2}$.
|
||||
We consider a chain with an even number of sites and open boundary conditions
|
||||
at half-filling, that is, $N_{\uparrow}=N_{\downarrow}=N/2$.
|
||||
In the site-representation, a general vector of the Hilbert space will be written as
|
||||
\be
|
||||
|n\rangle = |n_{1 \uparrow},...,n_{N \uparrow},n_{1 \downarrow},...,n_{N \downarrow}\rangle
|
||||
\ee
|
||||
where $n_{i \sigma}=0,1$ is the number of electrons of spin $\sigma$ at site $i$.
|
||||
|
||||
For the 1D Hubbard model and OBC the components of the ground-state vector have the same sign (say, $c_i \ge 0$).
|
||||
For the one-dimensional Hubbard model with open boundary conditions, the components of the ground-state vector have the same sign (say, $c_i \ge 0$).
|
||||
It is then possible to identify the guiding and trial vectors, that is, $|c^+\rangle=|c_T\rangle$.
|
||||
As trial wave function we shall employ a generalization of the Gutzwiller wave function\cite{Gutzwiller_1963} written under the form
|
||||
\be
|
||||
@ -915,7 +912,7 @@ we shall restrict ourselves to the case of the Green's Function Monte Carlo appr
|
||||
|
||||
Following Eqs.(\ref{final_E},\ref{calE}), the practical formula used for computing the QMC energy is written as
|
||||
\be
|
||||
{\cal E}_{QMC}(E,p_{ex},p_{max})= \frac{H_0 +...+ H_{p_{ex}} + \sum_{p=p_{ex}+1}^{p_{max}} H^{QMC}_p }
|
||||
\cE_{QMC}(E,p_{ex},p_{max})= \frac{H_0 +...+ H_{p_{ex}} + \sum_{p=p_{ex}+1}^{p_{max}} H^{QMC}_p }
|
||||
{S_0 +...+ S_{p_{ex}} + \sum_{p=p_{ex}+1}^{p_{max}} S^{QMC}_p }
|
||||
\label{calE}
|
||||
\ee
|
||||
@ -946,9 +943,9 @@ a slower convergence is observed, as expected from the divergence of the matrix
|
||||
Green's matrix at $E=E_0$ where the expansion does not converge at all. Note the oscillations of the curves as a function of $p$ due to a
|
||||
parity effect specific to this system. In practice, it is not too much of a problem since
|
||||
a smoothly convergent behavior is nevertheless observed for the even- and odd-parity curves.
|
||||
The ratio, ${\cal E}_{QMC}(E,p_{ex}=1,p_{max})$ as a function of $E$ is presented in figure \ref{fig2}. Here, $p_{max}$ is taken sufficiently large
|
||||
The ratio, $\cE_{QMC}(E,p_{ex}=1,p_{max})$ as a function of $E$ is presented in figure \ref{fig2}. Here, $p_{max}$ is taken sufficiently large
|
||||
so that the convergence at large $p$ is reached. The values of $E$ are $-0.780$, $-0.790$, $-0,785$, $-0,780$, and $-0.775$. For smaller $E$ the curve is extrapolated using
|
||||
the two-component expression. The estimate of the energy obtained from ${\cal E}(E)=E$ is $-0.76807(5)$ in full agreement with the exact value of $-0.768068...$.
|
||||
the two-component expression. The estimate of the energy obtained from $\cE(E)=E$ is $-0.76807(5)$ in full agreement with the exact value of $-0.768068...$.
|
||||
|
||||
\begin{figure}[h!]
|
||||
\includegraphics[width=\columnwidth]{fig1}
|
||||
@ -981,7 +978,7 @@ the average trapping time becomes infinite. Let us emphasize that the rate of co
|
||||
For example, the domain ${\cal D}(0,3) \cup {\cal D}(1,0)$ has a trapping time for the N\'eel state of 6.2, while
|
||||
the domain ${\cal D}(0,3) \cup {\cal D}(1,1)$ having almost the same number of states (28 states), has an average trapping time about 6 times longer. Finally, the last column gives the energy obtained for
|
||||
$E=-1$. The energy is expected to be independent of the domain and to converge to a common value, which is indeed the case here.
|
||||
The exact value, ${\cal E}(E=-1)=-0.75272390...$, can be found at the last row of the Table for the case of a domain corresponding to the full space.
|
||||
The exact value, $\cE(E=-1)=-0.75272390...$, can be found at the last row of the Table for the case of a domain corresponding to the full space.
|
||||
In sharp contrast, the statistical error depends strongly on the type of domains used. As expected, the largest error of $3 \times 10^{-5}$ is obtained in the case of
|
||||
a single-state domain for all states. The smallest statistical error is obtained for the "best" domain having the largest average
|
||||
trapping time. Using this domain leads to a reduction in the statistical error as large as about three orders of magnitude, nicely illustrating the
|
||||
@ -992,11 +989,11 @@ critical importance of the domains employed.
|
||||
\caption{$N$=4, $U$=12, $E$=-1, $\alpha=1.292$, $\beta=0.552$,$p_{ex}=4$. Simulation with 20 independent blocks and $10^5$ stochastic paths
|
||||
starting from the N\'eel state. $\bar{t}_{I_0}$
|
||||
is the average trapping time for the
|
||||
N\'eel state. $p_{\rm conv}$ is a measure of the convergence of ${\cal E}_{QMC}(p)$ as a function of $p$, see text.}
|
||||
N\'eel state. $p_{\rm conv}$ is a measure of the convergence of $\cE_{QMC}(p)$ as a function of $p$, see text.}
|
||||
\label{tab1}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lcccl}
|
||||
Domain & Size & $\bar{t}_{I_0}$ & $p_{\rm conv}$ & $\;\;\;\;\;\;{\cal E}_{QMC}$ \\
|
||||
Domain & Size & $\bar{t}_{I_0}$ & $p_{\rm conv}$ & $\;\;\;\;\;\;\cE_{QMC}$ \\
|
||||
\hline
|
||||
Single & 1 & 0.026 & 88 &$\;\;\;\;$-0.75276(3)\\
|
||||
${\cal D}(0,3)$ & 2 & 2.1 & 110 &$\;\;\;\;$-0.75276(3)\\
|
||||
@ -1028,7 +1025,7 @@ case where eight components for $H_p$ and $S_p$ are exactly computed.
|
||||
For the best domain, the impact is much more important with a huge reduction of about three orders of
|
||||
magnitude in the statistical error. Table \ref{tab3} reports
|
||||
the energies converged as a function of $p$ with their statistical error on the last digit for $E=
|
||||
-0.8, -0.795, -0.79, -0.785$, and $-0.78$. The values are displayed on Fig.\ref{fig2}. As seen on the figure the behavior of ${\cal E}$ as a function of
|
||||
-0.8, -0.795, -0.79, -0.785$, and $-0.78$. The values are displayed on Fig.\ref{fig2}. As seen on the figure the behavior of $\cE$ as a function of
|
||||
$E$ is very close to the linearity. The extrapolated values obtained from the five values of the energy with the three fitting functions are reported.
|
||||
Using the linear fitting function
|
||||
leads to an energy of -0.7680282(5) to compare with the exact value of -0.768068... A small bias of about $4 \times 10^{-5}$ is observed. This bias vanishes within the
|
||||
@ -1196,7 +1193,6 @@ $$
|
||||
\be
|
||||
\Big[ \prod_{k=0}^{p-1} \langle I_k| P_k \frac{1}{H-E} P_k (-H)(1-P_k)|I_{k+1} \rangle \Big]
|
||||
\langle I_p| P_p \frac{1} {H-E} P_p|\Psi\rangle
|
||||
\label{eqfond}
|
||||
\ee
|
||||
To treat simultaneously the two possible cases for the final state, $|I_N\rangle =|1\rangle$ or $|2\rangle$,
|
||||
the equation has been slightly generalized to the case of a general vector for the final state written as
|
||||
|
||||
Loading…
Reference in New Issue
Block a user