OK with IIIB
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g.tex
@ -323,9 +323,9 @@ We are now in the position to define the stochastic matrix as
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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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1 - \tau \qty[ H^+_{ii}- (\EL^+)_{i} ], & \qif* 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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\tau \frac{\PsiG_{j}}{\PsiG_{i}} \abs{H_{ij}} \ge 0, & \qif* i\neq j.
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\end{cases}
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\ee
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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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@ -418,7 +418,7 @@ We shall not here insist on these practical details that are discussed, for exam
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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_1999A,Assaraf_1999B,Caffarel_2000} and applied to the SU($N$) one-dimensional Hubbard model.
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fluctuations. This idea was proposed some time ago and applied to the SU($N$) one-dimensional Hubbard model.\cite{Assaraf_1999A,Assaraf_1999B,Caffarel_2000}
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Considering a given state $\ket{i}$, the probability that a walker remains exactly $n$ times in $\ket{i}$ (with $1 \le n < \infty$) and then exits to a different state $j$ (with $j \neq i$) is
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\be
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@ -443,13 +443,13 @@ 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}= \frac{1}{H^+_{ii}-(\EL^+)_{i}},
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\ee
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In the limit $\tau \to 0$, the Poisson probability takes the usual form
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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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P_{i}(t) = \frac{1}{\bar{t}_i} \exp(-\frac{t}{\bar{t}_i}).
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\ee
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The time-averaged contribution of the on-state weight can be easily calculated to be
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The time-averaged contribution of the on-state weight can then be easily calculated to be
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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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@ -461,85 +461,89 @@ Details of the implementation of this effective dynamics can be in found in Refs
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%=======================================%
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Let us now extend the results of Sec.~\ref{sec:single_domains} to a general domain.
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To do so, let us associate to each state $\ket{i}$ a set of states, called the domain of $\ket{i}$ denoted $\cD_i$, consisting of the state $\ket{i}$ plus a certain number of states.
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To do so, we associate to each state $\ket{i}$ a set of states, called the domain of $\ket{i}$ denoted $\cD_i$, consisting of the state $\ket{i}$ plus a certain number of states.
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No particular constraints on the type of domains are imposed.
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For example, domains associated with different states can be identical, and they may or may not have common states.
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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).
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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.
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In practice, it is not difficult to impose such a condition.
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Let us write an arbitrary path of length $N$ as
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\be
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\ket{i_0} \to \ket{i_1} \to \cdots \to \ket{i_N}
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\ket{i_0} \to \ket{i_1} \to \cdots \to \ket{i_N},
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\ee
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where the successive states are drawn using the transition probability matrix, $p_{i \to j}$.
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This series can be rewritten
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This series can be recast
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\be
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\label{eq:eff_series}
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(\ket*{I_0},n_0) \to (\ket*{I_1},n_1) \to \cdots \to (\ket*{I_p},n_p)
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(\ket*{I_0},n_0) \to (\ket*{I_1},n_1) \to \cdots \to (\ket*{I_p},n_p),
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\ee
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where $\ket{I_0}=\ket{i_0}$ is the initial state, $n_0$ is the number of times the walker remains within the domain of $\ket{i_0}$ (with $1 \le n_0 \le N+1$), $\ket{I_1}$ is the first exit state that does not belong to $\cD_{i_0}$, $n_1$ is the number of times the walker remains in $\cD_{i_1}$ (with $1 \le n_1 \le N+1-n_0$), $\ket{I_2}$ is the second exit state, and so on.
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Here, the integer $p$ goes from 0 to $N$ and indicates the number of exit events occurring along the path.
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The two extreme cases, $p=0$ and $p=N$, correspond to the cases where the walker remains in the initial domain during the entire path, and to the case where the walker exits a domain at each step, respectively.
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where $\ket{I_0}=\ket{i_0}$ is the initial state, $n_0$ is the number of times the walker remains in the domain of $\ket{i_0}$ (with $1 \le n_0 \le N+1$), $\ket{I_1}$ is the first exit state that does not belong to $\cD_{i_0}$, $n_1$ is the number of times the walker remains in $\cD_{i_1}$ (with $1 \le n_1 \le N+1-n_0$), $\ket{I_2}$ is the second exit state, and so on.
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Here, the integer $p$ (with $0 \le p \le N$) indicates the number of exit events occurring along the path.
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The two extreme values, $p=0$ and $p=N$, correspond to the cases where the walker remains in the initial domain during the entire path, and where the walker exits a domain at each step, respectively.
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\titou{In what follows, we shall systematically write the integers representing the exit states in capital letter.}
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%Generalizing what has been done for domains consisting of only one single state, the general idea here is to integrate out exactly the stochastic dynamics over the
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%set of all paths having the same representation, Eq.(\ref{eff_series}). As a consequence, an effective Monte Carlo dynamics including only exit states
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%averages for renormalized quantities will be defined.\\
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Let us define the probability of being $n$ times within the domain of $\ket{I_0}$ and, then, to exit at $\ket{I} \notin \cD_{I_0}$.
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It is given by
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Let us define the probability of remaining $n$ times in the domain of $\ket{I_0}$ and to exit at $\ket{I} \notin \cD_{I_0}$ as
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\be
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\label{eq:eq1C}
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\cP_{I_0 \to I}(n)
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= \sum_{\ket{i_1} \in \cD_{I_0}} \cdots \sum_{\ket{i_{n-1}} \in \cD_{I_0}}
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p_{I_0 \to i_1} \ldots p_{i_{n-2} \to i_{n-1}} p_{i_{n-1} \to I}
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p_{I_0 \to i_1} \ldots p_{i_{n-2} \to i_{n-1}} p_{i_{n-1} \to I}.
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\ee
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To proceed we need to introduce the projector associated with each domain
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\titou{To proceed} we must introduce the projector associated with each domain
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\be
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\label{eq:pi}
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P_I = \sum_{\ket{k} \in \cD_I} \dyad{k}{k}
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P_I = \sum_{\ket{i} \in \cD_I} \dyad{i}{i}
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\ee
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and to define the restriction of the operator $T^+$ to the domain
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and the projection of the operator $T^+$ to the domain that governs the dynamics of the walkers moving in $\cD_{I}$, \ie,
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\be
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T^+_I= P_I T^+ P_I.
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\ee
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$T^+_I$ is the operator governing the dynamics of the walkers moving within $\cD_{I}$.
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Using Eqs.~\eqref{eq:pij} and \eqref{eq:eq1C}, the probability can be rewritten as
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\be
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\label{eq:eq3C}
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\cP+{I_0 \to I}(n) = \frac{1}{\PsiG_{I_0}} \mel{I_0}{\qty(T^+_{I_0})^{n-1} F^+_{I_0}}{I} \PsiG_{I},
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\cP_{I_0 \to I}(n) = \frac{1}{\PsiG_{I_0}} \mel{I_0}{\qty(T^+_{I_0})^{n-1} F^+_{I_0}}{I} \PsiG_{I},
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\ee
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where the operator $F$, corresponding to the last move connecting the inside and outside regions of the
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domain, is given by
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where the operator
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\be
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\label{eq:Fi}
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F^+_I = P_I T^+ (1-P_I),
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\ee
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that is, $(F^+_I)_{ij}= T^+_{ij}$ when $(\ket{i} \in \cD_{I}, \ket{j} \notin \cD_{I})$, and zero
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otherwise.
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corresponding to the last move connecting the inside and outside regions of the domain, has the following matrix elements:
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\be
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(F^+_I)_{ij} =
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\begin{cases}
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T^+_{ij}, & \qif* \ket{i} \in \cD_{I} \titou{\land} \ket{j} \notin \cD_{I},
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\\
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0, & \text{otherwise}.
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\end{cases}
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\ee
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Physically, $F$ may be seen as a flux operator through the boundary of ${\cal D}_{I}$.
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Now, the probability of being trapped $n$ times within ${\cal D}_{I}$ is given by
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\titou{Now}, the probability of being trapped $n$ times within ${\cal D}_{I}$ is given by
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\be
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\label{eq:PiN}
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P_{I}(n) = \frac{1}{\PsiG_{I}} \mel{ I }{ {T^+_{I}}^{n-1} F^+_{I} }{ \PsiG }.
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P_{I}(n) = \frac{1}{\PsiG_{I}} \mel{ I }{ \qty(T^+_{I})^{n-1} F^+_{I} }{ \PsiG }.
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\ee
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Using the fact that
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\be
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\label{eq:relation}
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{T^+_I}^{n-1} F^+_I = {T^+_I}^{n-1} T^+ - {T^+_I}^n,
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\qty(T^+_{I})^{n-1} F^+_I = \qty(T^+_{I})^{n-1} T^+ - \qty(T^+_I)^n,
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\ee
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we have
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\be
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\sum_{n=0}^{\infty} P_{I}(n)
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= \frac{1}{\PsiG_{I}} \sum_{n=1}^{\infty} \qty( \mel{ I }{ {T^+_{I}}^{n-1} }{ \PsiG }
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- \mel{ I }{ {T^+_{I}}^{n} }{ \PsiG } ) = 1
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= \frac{1}{\PsiG_{I}} \sum_{n=1}^{\infty} \qty[ \mel{ I }{ \qty(T^+_{I})^{n-1} }{ \PsiG }
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- \mel{ I }{ \qty(T^+_{I})^{n} }{ \PsiG } ] = 1,
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\ee
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and the average trapping time
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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^+} P_{I} }{ \PsiG }.
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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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\ee
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In practice, the various quantities restricted to the domain are computed by diagonalizing the matrix $(H^+-\EL^+)$ in $\cD_{I}$.
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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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@ -556,85 +560,76 @@ For that we introduce the Green's matrix associated with each domain
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\be
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G^{(N),\cD}_{IJ}= \mel{ J }{ T_I^N }{ I }.
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\ee
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Starting from Eq.~\eqref{eq:cn}
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\be
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G^{(N)}_{i_0 i_N}= \sum_{i_1,...,i_{N-1}} \prod_{k=0}^{N-1} \langle i_k| T |i_{k+1} \rangle.
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\ee
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and using the representation of the paths in terms of exit states and trapping times we write
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%Starting from Eq.~\eqref{eq:cn}
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%\be
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%G^{(N)}_{i_0 i_N}= \sum_{i_1,...,i_{N-1}} \prod_{k=0}^{N-1} \langle i_k| T |i_{k+1} \rangle.
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%\ee
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Starting from Eq.~\eqref{eq:cn} and using the representation of the paths in terms of exit states and trapping times, we get
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\be
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G^{(N)}_{I_0 I_N} = \sum_{p=0}^N
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\sum_{{\cal C}_p} \sum_{(i_1,...,i_{N-1}) \in \;{\cal C}_p}
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\prod_{k=0}^{N-1} \langle i_k|T |i_{k+1} \rangle
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\sum_{\cC_p} \sum_{(i_1,...,i_{N-1}) \in \cC_p}
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\prod_{k=0}^{N-1} \mel{ i_k }{ T }{ i_{k+1} }
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\ee
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where ${\cal C}_p$ is the set of paths with $p$ exit states, $|I_k\rangle$, and trapping times $n_k$ with the
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constraints that $|I_k\rangle \notin {\cal D}_{k-1}$, $1 \le n_k \le N+1$ and $\sum_{k=0}^p n_k= N+1$.
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where $\cC_p$ is the set of paths with $p$ exit states, $\ket{I_k}$, and trapping times $n_k$ with the constraints that $\ket{I_k} \notin \cD_{k-1}$ (with $1 \le n_k \le N+1$ and $\sum_{k=0}^p n_k= N+1$).
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We then have
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$$
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\begin{multline}
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\label{eq:Gt}
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G^{(N)}_{I_0 I_N}= G^{(N),{\cal D}}_{I_0 I_N} +
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$$
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\be
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\sum_{p=1}^{N}
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\sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}} }
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\sum_{n_0 \ge 1} ... \sum_{n_p \ge 1}
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\ee
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\be
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\label{eq:Gt}
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\delta(\sum_{k=0}^p n_k=N+1) \Big[ \prod_{k=0}^{p-1} \langle I_k|T^{n_k-1}_{I_k} F_{I_k} |I_{k+1} \rangle \Big]
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\sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}
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\delta_{\sum_{k=0}^p n_k,N+1}
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\\
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\times
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\qty[ \prod_{k=0}^{p-1} \mel{ I_k }{ \qty(T_{I_k})^{n_k-1} F_{I_k} }{ I_{k+1} } ]
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G^{(n_p-1),{\cal D}}_{I_p I_N}
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\ee
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This expression is the path-integral representation of the Green's matrix using only the variables $(|I_k\rangle,n_k)$ of the effective dynamics defined over the set
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of domains. The standard formula derived above, Eq.~\eqref{eq:G} may be considered as the particular case where the domain associated with each state is empty,
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\end{multline}
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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, 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 $k=0$ to $N$ 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} \langle I_k|F_{I_k}|I_{k+1} \rangle $
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where $F=T$.
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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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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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$$
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{\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N} +
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$$
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\be
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\label{eq:Gbart}
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{\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N} +
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\sum_{p=1}^{N}
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\sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}}}
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\sum_{n_0 \ge 1} ... \sum_{n_p \ge 1}
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\ee
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\be
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\delta(\sum_k n_k=N+1) \Big[ \prod_{k=0}^{p-1} [\frac{\PsiG_{I_{k+1}}}{\PsiG_{I_k}} \langle I_k| T^{n_k-1}_{I_k} F_{I_k} |I_{k+1} \rangle \Big]
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{\bar G}^{(n_p-1),{\cal D}}_{I_p I_N}.
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\label{Gbart}
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\ee
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Introducing the weight
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\be
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W_{I_k I_{k+1}}=\frac{\langle I_k|T^{n_k-1}_{I_k} F_{I_k} |I_{k+1}\rangle}{\langle I_k|T^{+\;n_k-1}_{I_k} F^+_{I_k} |I_{k+1} \rangle}
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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, Eq.(\ref{eq3C}), we get
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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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{\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N}+ \sum_{p=1}^{N}
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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}+ \sum_{p=1}^{N}
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\bigg \langle
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\Big( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} \Big)
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{\bar G}^{(n_p-1), {\cal D}}_{I_p I_N}
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\bigg \rangle
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\label{Gbart}
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\ee
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where the average is defined as
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$$
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\bigg \langle F \bigg \rangle
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= \sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}}}
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\sum_{n_0 \ge 1} ... \sum_{n_p \ge 1}
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\delta(\sum_k n_k=N+1)
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$$
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\be
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\prod_{k=0}^{N-1}{\cal P}(I_k \to I_{k+1},n_k-1) F(I_0,n_0;...;I_N,n_N)
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\expval{F}
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= \sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}}}
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\sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}
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\delta_{\sum_k n_k,N+1}
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\prod_{k=0}^{N-1}\cP_{I_k \to I_{k+1}}(n_k-1) F(I_0,n_0;...;I_N,n_N)
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\ee
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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 $|I_0\rangle$\\
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i) Choose some initial vector $\ket{I_0}$\\
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ii) Generate a stochastic path by running over $k$ (starting at $k=0$) as follows.\\
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$\;\;\;\bullet$ Draw $n_k$ using the probability $P_{I_k}(n)$, Eq.(\ref{PiN})\\
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$\;\;\;\bullet$ Draw the exit state, $|I_{k+1}\rangle$, using the conditional probability $$\frac{{\cal P}(I_k \to I_{k+1},n_k)}{P_{I_k}(n_k)}$$\\
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$\;\;\;\bullet$ Draw $n_k$ using the probability $P_{I_k}(n)$ [see Eq.~\eqref{eq:PiN}]\\
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$\;\;\;\bullet$ Draw the exit state, $\ket{I_{k+1}}$, using the conditional probability $$\frac{\cP_{I_k \to I_{k+1}}(n_k)}{P_{I_k}(n_k)}$$\\
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iii) Terminate the path when $\sum_k n_k=N$ is greater than some target value $N_{\rm max}$ and compute $F(I_0,n_0;...;I_N,n_N)$\\
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iv) Go to step ii) until some maximum number of paths is reached.\\
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\\
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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 $N_{max}$ is obtained. At large $N_{max}$ where the
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convergence of the average as a function of $p$ is reached, such values can be averaged.
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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.
|
||||
|
||||
%--------------------------------------------%
|
||||
\subsubsection{Integrating out the trapping times : The Domain Green's Function Monte Carlo approach}
|
||||
@ -649,22 +644,20 @@ $n_k$. The second, more direct and elegant, is based on the Dyson equation.\\
|
||||
\\
|
||||
{\it $\bullet$ The pedestrian way}. Let us define the quantity\\
|
||||
$$
|
||||
G^E_{ij}= \tau \sum_{N=0}^{\infty} \langle i|T^N|j\rangle.
|
||||
G^E_{ij}= \tau \sum_{N=0}^{\infty} \mel{ i }{ T^N }{ j}.
|
||||
$$
|
||||
By summing over $N$ we obtain
|
||||
\be
|
||||
G^E_{ij}= \mel{i}{\frac{1}{H-E}}{j}.
|
||||
G^E_{ij}= \mel{i}{\frac{1}{H-E \Id}}{j}.
|
||||
\ee
|
||||
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
|
||||
essentially the Laplace transform of the time-dependent Green's function. Here, we then use the same denomination. The remarkable property
|
||||
is that, thanks to the summation over $N$ up to the
|
||||
infinity the constrained multiple sums appearing in Eq.(\ref{Gt}) can be factorized in terms of a product of unconstrained single sums as follows
|
||||
$$
|
||||
\sum_{N=1}^\infty \sum_{p=1}^N \sum_{n_0 \ge 1} ...\sum_{n_p \ge 1} \delta(n_0+...+n_p=N+1)
|
||||
$$
|
||||
$$
|
||||
\be
|
||||
\sum_{N=1}^\infty \sum_{p=1}^N \sum_{n_0 \ge 1} ...\sum_{n_p \ge 1} \delta_{n_0+...+n_p,N+1}
|
||||
= \sum_{p=1}^{\infty} \sum_{n_0=1}^{\infty} ...\sum_{n_p=1}^{\infty}.
|
||||
$$
|
||||
\ee
|
||||
It is then a trivial matter to integrate out exactly the $n_k$ variables, leading to
|
||||
$$
|
||||
\langle I_0|\frac{1}{H-E}|I_N\rangle = \langle I_0|P_0\frac{1}{H-E} P_0|I_N\rangle
|
||||
@ -728,7 +721,7 @@ and the weight
|
||||
W^E_{IJ} =
|
||||
\frac{\langle I|\frac{1}{H-E} P_I (-H)(1-P_I) |J\rangle }{\langle I|\frac{1}{H^+-E^+_L} P_I (-H^+)(1-P_I) |J\rangle}
|
||||
\ee
|
||||
Using Eqs.(\ref{eq1C},\ref{eq3C},\ref{relation}) we verify that $P(I \to J) \ge 0$ and $\sum_J P(I \to J)=1$.
|
||||
Using Eqs.~\eqref{eq:eq1C}, \eqref{eq:eq3C} and \eqref{eq:relation}, we verify that $P_{I \to J} \ge 0$ and $\sum_J P_{I \to J}=1$.
|
||||
Finally, the probabilistic expression writes
|
||||
$$
|
||||
\langle I_0| \frac{1}{H-E}|I_N\rangle
|
||||
|
||||
Loading…
Reference in New Issue
Block a user