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g.tex
@ -232,17 +232,16 @@ For example, in the important case of the energy, one can rely on the following
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\label{eq:E0}
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E_0 = \lim_{N \to \infty } \frac{\mel{\PsiT}{H T^N}{\Psi_0}}{\mel{\PsiT}{T^N}{\Psi_0}},
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\ee
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where $|\PsiT\rangle$ is some trial vector (some approximation of the true ground-state wave function) on which $T^N \ket{\Psi_0}$ is projected out.
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where $\ket{\PsiT}$ is a trial wave function (some approximation of the true ground-state wave function) on which $T^N \ket{\Psi_0}$ is projected out.
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To proceed further we introduce the time-dependent Green's matrix $G^{(N)}$ defined as
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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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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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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 $e^{-N\tau H}$, 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, $\{ \ket{i_k} \}_{1 \le k \le N-1}$, in the $N$th product of $T$,} $G^{(N)}$ can be written in the following expanded form
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Introducing the set of $N-1$ intermediate states, $\{ \ket{i_k} \}_{1 \le k \le N-1}$, in $T^N$, $G^{(N)}$ 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} \cdots \sum_{i_{N-1}} \prod_{k=0}^{N-1} T_{i_{k} i_{k+1}},
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@ -250,8 +249,8 @@ $e^{-N\tau H}$ which is usually referred to as the imaginary-time dependent Gree
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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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The series of states $\ket{i_0}, \ldots,\ket{i_N}$ is interpreted as a ``path'' in the Hilbert space starting at vector $\ket{i_0}$ and ending at vector $\ket{i_N}$ where $k$ plays the role of a time index.
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In quantum physics, Eq.~\eqref{eq:cn} is referred to as the path-integral representation of the Green's matrix (or function).
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The series of states $\ket{i_0}, \ldots,\ket{i_N}$ is interpreted as a ``path'' in the Hilbert space starting at vector $\ket{i_0}$ and ending at vector $\ket{i_N}$, where $k$ plays the role of a time index.
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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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@ -261,7 +260,7 @@ Each path is associated with a weight $\prod_{k=0}^{N-1} T_{i_{k} i_{k+1}}$ and
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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 $i_N$th component of the ground-state wave function (obtained as $\lim_{N \to \infty} G^{(N)}_{i_0 i_N})$ is the weighted sum over all possible paths arriving at vector $\ket{i_N}$.
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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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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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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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@ -271,26 +270,26 @@ This is the central theme of the present work.
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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, $\PsiG_i > 0$, in order to apply a similarity transformation to the operators $G^{(N)}$ and $T$ as follows:
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To derive a probabilistic expression for the Green's matrix, we introduce a guiding wave function, $\ket{\PsiG}$, having strictly positive components, \ie, $\PsiG_i > 0$, in order to perform a similarity transformation to the operators $G^{(N)}$ and $T$, as follows:
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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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\end{align}
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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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Note that, thanks to the properties of similarity transformations, the path integral expression relating $G^{(N)}$ and $T$ [see Eq.~\eqref{eq:G}] remains unchanged for $\bar{G}^{(N)}$ and $\bar{T}$.
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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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\ee
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Here, we recall that a stochastic matrix is defined as a matrix with positive entries and obeying
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Here, we recall that a stochastic matrix is defined as a matrix with positive entries that obeys
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\be
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\label{eq:sumup}
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\sum_j p_{i \to j}=1.
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\ee
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To build the transition probability density the following operator is introduced
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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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@ -330,22 +329,22 @@ We are now in the position to define the stochastic matrix as
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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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As readily seen in Eq.~\eqref{eq:pij}, the off-diagonal terms of the stochastic matrix are positive, while the diagonal terms 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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\ee
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The sum-over-states condition [see Eq.~\eqref{eq:sumup}]
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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{\mel{i}{T^+}{\PsiG}}{\PsiG_{i}} = 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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follows from the fact that $\ket{\PsiG}$ is eigenvector of $T^+$ [see Eq.~\eqref{eq:relT+}].
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follows from the fact that $\ket{\PsiG}$ is eigenvector of $T^+$, as evidenced by 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 rather tight since, for very large matrices, it may impose an extremely small value of the time step.
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At first sight, the condition defining the maximum value of $\tau$ [see 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 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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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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@ -393,7 +392,7 @@ In this framework, the energy defined in Eq.~\eqref{eq:E0} is given by
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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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\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 this population.
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Note that, instead of using a single walker, it is common to introduce a population of independent walkers and to calculate the averages over this population.
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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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We shall not here insist on these practical details that are discussed, for example, in Refs.~\onlinecite{Foulkes_2001,Kolorenc_2011}.
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@ -465,7 +464,7 @@ Details of the implementation of this effective dynamics can be in found in Refs
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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, 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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For example, domains associated with different states may 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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In practice, it is not difficult to impose such a condition.
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@ -525,7 +524,7 @@ corresponding to the last move connecting the inside and outside regions of the
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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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\titou{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 in ${\cal D}_{I}$ 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 }{ \qty(T^+_{I})^{n-1} F^+_{I} }{ \PsiG }.
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@ -578,7 +577,7 @@ It follows that
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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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\sum_{p=1}^{N}
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\sum_{|I_1\rangle \notin {\cal D}_{I_0}, \ldots , |I_p\rangle \notin {\cal D}_{I_{p-1}} }
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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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\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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@ -602,7 +601,7 @@ To express the fundamental equation of $G$ under the form of a probabilistic ave
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\sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}
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\\
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\times
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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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\delta_{\sum_{k=0}^p 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 weights
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@ -622,7 +621,7 @@ where, in this context, the average of a given function $F$ is defined as
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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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\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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\delta_{\sum_{k=0}^p 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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@ -658,14 +657,15 @@ Let us define the energy-dependent Green's matrix
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G^E_{ij}= \tau \sum_{N=0}^{\infty} \mel{ i }{ T^N }{ j} = \mel{i}{ \qty( H-E \Id )^{-1} }{j},
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\ee
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which does not depends on the time step.
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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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Note that, in a continuous space, this quantity is essentially the \titou{Laplace transform of the time-dependent Green's function}.
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We shall use the same denomination in the following.
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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} \cdots \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} \cdots \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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The remarkable property is that, thanks to the summation over $N$ up to 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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\begin{multline}
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\sum_{N=1}^\infty \sum_{p=1}^N \sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1} \delta_{\sum_{k=0}^p n_k,N+1} F(n_0,\ldots,n_N)
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\\
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= \sum_{p=1}^{\infty} \sum_{n_0=1}^{\infty} \cdots \sum_{n_p=1}^{\infty} F(n_0,\ldots,n_N).
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\end{multline}
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It is then trivial to integrate out exactly the $n_k$ variables, leading to
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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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@ -677,7 +677,7 @@ It is then a trivial matter to integrate out exactly the $n_k$ variables, leadin
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\times
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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 two-state system.
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As a didactical example, Appendix \ref{app:A} reports the exact derivation of this formula in the case of a two-state system.
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%----------------------------%
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\subsubsection{Dyson equation}
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@ -694,7 +694,7 @@ where $H_0$ is some arbitrary reference Hamiltonian, we have the Dyson equation
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\ee
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with $G^E_{0,ij} = \mel{i}{ \qty( H_0-E \Id )^{-1} }{j}$.
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Let us choose $H_0$ 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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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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@ -741,24 +741,25 @@ Finally, the probabilistic expression reads
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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 energy estimator
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To calculate the energy, we introduce the following estimator
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\be
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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 $\cE(E)= E$.
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and search for the solution of the non-linear equation $\cE(E)= E$.
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Using the spectral decomposition of $H$, we have
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\be
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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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\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 $c_i = \braket{ I_0 }{ \Phi_i } \braket{ \Phi_i}{ \PsiT }$, \titou{where $\Phi_i$ are eigenstates of $H$}.
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It is easy to check that, in the vicinity of $E = E_0$, $\cE(E)$ is a linear function of $E - E_0$.
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Therefore, in practice, we compute the value of $\cE(E)$ for several values of $E$, and fit these data using a linear or quadratic function of $E$ in order to obtain, via extrapolation, the exact value $E_0$.
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where $c_i = \braket{ I_0 }{ \Phi_i } \braket{ \Phi_i}{ \PsiT }$ and $\Phi_i$ are eigenstates of $H$, \ie, $H \ket{\Phi_i} = E_i \ket{\Phi_i}$.
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It is easy to check that, in the vicinity of the exact energy $E_0$, $\cE(E)$ is a linear function of $E - E_0$.
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Therefore, in practice, we compute the value of $\cE(E)$ for several values of $E$, and fit these data using a linear or quadratic function of $E$ in order to obtain, via extrapolation, an estimate of $E_0$.
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In order to have a precise extrapolation of the energy, it is best to compute $\cE(E)$ for values of $E$ as close as possible to the exact energy.
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However, as $E \to E_0$, both the numerators and denominators of Eq.~\eqref{eq:calE} diverge.
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This is reflected by the fact that one needs to compute more and more $p$-components with an important increase of statistical fluctuations.
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Thus, from a practical point of view, a trade-off has to be found between the \titou{quality} of the extrapolation and the amount of statistical fluctuations.
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\titou{This is reflected by the fact that one needs to compute more and more $p$-components with an important increase of statistical fluctuations.}
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Thus, from a practical point of view, a trade-off has to be found between the quality of the extrapolation and the amount of statistical fluctuations.
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%Let us describe the 3 functions used here for the fit.
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@ -800,7 +801,7 @@ Let us consider the one-dimensional Hubbard Hamiltonian for a chain of $N$ sites
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H= -t \sum_{\expval{ i j } \sigma} \hat{a}^+_{i\sigma} \hat{a}_{j\sigma}
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+ U \sum_i \hat{n}_{i\uparrow} \hat{n}_{i\downarrow},
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\ee
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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 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.
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where $\expval{ i j }$ denotes the summation over two neighboring sites, $\hat{a}_{i\sigma}$ ($\hat{a}_{i\sigma}$) is the fermionic creation (annihilation) operator of 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.
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We consider a chain with an even number of sites and open boundary conditions at half-filling, that is, $N_{\uparrow}=N_{\downarrow}=N/2$.
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In the site representation, a general vector of the Hilbert space can be written as
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\be
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@ -910,7 +911,7 @@ Let us begin with a small chain of 4 sites with $U=12$.
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From now on, we shall take $t=1$.
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The size of the linear space is ${\binom{4}{2}}^2= 36$ and the ground-state energy obtained by exact diagonalization is $E_0=-0.768068...$.
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The two variational parameters of the trial vector have been optimized and fixed at the values of $\alpha=1.292$, and $\beta=0.552$ with a variational energy of $E_\text{T}=-0.495361...$.
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In what follows $|I_0\rangle$ will be systematically chosen as one of the two N\'eel states, {\it e.g.} $|I_0\rangle =|\uparrow,\downarrow, \uparrow,...\rangle$.
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In what follows $\ket{I_0}$ will be systematically chosen as one of the two N\'eel states, {\it e.g.} $\ket{I_0} = \ket{\uparrow,\downarrow, \uparrow,\ldots}$.
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Figure \ref{fig1} shows the convergence of $H_p$ as a function of $p$ for different values of the reference energy $E$.
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We consider the simplest case where a single-state domain is associated to each state.
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@ -949,7 +950,7 @@ Table \ref{tab1} illustrates the dependence of the Monte Carlo results upon the
|
||||
The reference energy is $E=-1$.
|
||||
The first column indicates the various domains consisting of the union of some elementary domains as explained above.
|
||||
The first line of the table gives the results when using a minimal single-state domain for all states, and the last one for the maximal domain containing the full linear space.
|
||||
The size of the various domains is given in column 2, the average trapping time for the state $|I_0\rangle$ in the third column, and an estimate of the speed of convergence of the $p$-expansion for the energy in the fourth column.
|
||||
The size of the various domains is given in column 2, the average trapping time for the state $\ket{I_0}$ in the third column, and an estimate of the speed of convergence of the $p$-expansion for the energy in the fourth column.
|
||||
To quantify the rate of convergence, we report the quantity, $p_{conv}$, defined as the smallest value of $p$ for which the energy is converged with six decimal places.
|
||||
The smaller $p_{conv}$, the better the convergence is.
|
||||
Although this is a rough estimate, it is sufficient here for our purpose.
|
||||
@ -1160,15 +1161,12 @@ This project has received funding from the European Union's Horizon 2020 --- Res
|
||||
\label{app:A}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
For the simplest case of a two-state system, $\ket{1}$ and $\ket{2}$, the fundamental equation given in Eq.~\eqref{eq:eqfond} simplifies to
|
||||
For the simplest case of a system containing only two states, $\ket{1}$ and $\ket{2}$, the fundamental equation given in Eq.~\eqref{eq:eqfond} simplifies to
|
||||
\be
|
||||
\begin{split}
|
||||
\cI
|
||||
& = \mel{ I_0 }{ \qty(H-E \Id)^{-1} }{ \Psi }
|
||||
\\
|
||||
& = \mel{ I_0 }{ P_0 \qty(H-E \Id)^{-1} P_0 }{ \Psi }
|
||||
= \mel{ I_0 }{ \qty(H-E \Id)^{-1} }{ \Psi }
|
||||
= \mel{ I_0 }{ P_0 \qty(H-E \Id)^{-1} P_0 }{ \Psi }
|
||||
+ \sum_{p=1}^{\infty} \cI_p,
|
||||
\end{split}
|
||||
\ee
|
||||
with
|
||||
\begin{multline}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user