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@ -1,72 +1,79 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-10-05 14:42:12 +0200
%% Created for Pierre-Francois Loos at 2022-09-30 16:13:18 +0200
%% Saved with string encoding Unicode (UTF-8)
%% Saved with string encoding Unicode (UTF-8) *
@article{Ceperley_1983,
abstract = {Random walks with branching have been used to calculate exact properties of the ground state of quantum many-body systems. In this paper, a more general Green's function identity is derived which relates the potential energy, a trial wavefunction, and a trial density matrix to the rules of a branched random walk. It is shown that an efficient algorithm requires a good trial wavefunction, a good trial density matrix, and a good sampling of this density matrix. An accurate density matrix is constructed for Coulomb systems using the path integral formula. The random walks from this new algorithm diffuse through phase space an order of magnitude faster than the previous Green's Function Monte Carlo method. In contrast to the simple diffusion Monte Carlo algorithm, it is an exact method. Representative results are presented for several molecules.},
author = {D Ceperley},
date-modified = {2022-10-05 14:41:32 +0200},
doi = {https://doi.org/10.1016/0021-9991(83)90161-4},
issn = {0021-9991},
journal = {J. Comput. Phys.},
title = {The simulation of quantum systems with random walks: A new algorithm for charged systems},
journal = {Journal of Computational Physics},
volume = {51},
number = {3},
pages = {404-422},
title = {The simulation of quantum systems with random walks: A new algorithm for charged systems},
url = {https://www.sciencedirect.com/science/article/pii/0021999183901614},
volume = {51},
year = {1983},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0021999183901614},
bdsk-url-2 = {https://doi.org/10.1016/0021-9991(83)90161-4}}
issn = {0021-9991},
doi = {https://doi.org/10.1016/0021-9991(83)90161-4},
url = {https://www.sciencedirect.com/science/article/pii/0021999183901614},
author = {D Ceperley},
abstract = {Random walks with branching have been used to calculate exact properties of the ground state of quantum many-body systems. In this paper, a more general Green's function identity is derived which relates the potential energy, a trial wavefunction, and a trial density matrix to the rules of a branched random walk. It is shown that an efficient algorithm requires a good trial wavefunction, a good trial density matrix, and a good sampling of this density matrix. An accurate density matrix is constructed for Coulomb systems using the path integral formula. The random walks from this new algorithm diffuse through phase space an order of magnitude faster than the previous Green's Function Monte Carlo method. In contrast to the simple diffusion Monte Carlo algorithm, it is an exact method. Representative results are presented for several molecules.}
}
@article{Kalos_1962,
author = {Kalos, M. H.},
doi = {10.1103/PhysRev.128.1791},
issue = {4},
journal = {Phys. Rev.},
month = {Nov},
numpages = {0},
pages = {1791--1795},
publisher = {American Physical Society},
title = {Monte Carlo Calculations of the Ground State of Three- and Four-Body Nuclei},
url = {https://link.aps.org/doi/10.1103/PhysRev.128.1791},
author = {Kalos, M. H.},
journal = {Phys. Rev.},
volume = {128},
issue = {4},
pages = {1791--1795},
numpages = {0},
year = {1962},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.128.1791},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.128.1791}}
month = {Nov},
publisher = {American Physical Society},
doi = {10.1103/PhysRev.128.1791},
url = {https://link.aps.org/doi/10.1103/PhysRev.128.1791}
}
@article{Kalos_1970,
author = {Kalos, M. H.},
doi = {10.1103/PhysRevA.2.250},
issue = {1},
journal = {Phys. Rev. A},
month = {Jul},
numpages = {0},
pages = {250--255},
publisher = {American Physical Society},
title = {Energy of a Boson Fluid with Lennard-Jones Potentials},
url = {https://link.aps.org/doi/10.1103/PhysRevA.2.250},
author = {Kalos, M. H.},
journal = {Phys. Rev. A},
volume = {2},
issue = {1},
pages = {250--255},
numpages = {0},
year = {1970},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.2.250},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevA.2.250}}
month = {Jul},
publisher = {American Physical Society},
doi = {10.1103/PhysRevA.2.250},
url = {https://link.aps.org/doi/10.1103/PhysRevA.2.250}
}
@article{Moskowitz_1986,
author = {Moskowitz,Jules W. and Schmidt,K. E. },
date-modified = {2022-10-05 14:41:57 +0200},
doi = {10.1063/1.451046},
journal = {J. Chem. Phys.},
title = {The domain Green’s function method},
journal = {The Journal of Chemical Physics},
volume = {85},
number = {5},
pages = {2868-2874},
title = {The domain Green's function method},
volume = {85},
year = {1986},
bdsk-url-1 = {https://doi.org/10.1063/1.451046}}
doi = {10.1063/1.451046},
URL = {
https://doi.org/10.1063/1.451046
},
eprint = {
https://doi.org/10.1063/1.451046
}
}
@misc{note2,
note = {The property results from the fact that the series is a telescoping series and that the general term
$\mel{ I }{ \qty(T^+_{I})^{n} }{ \PsiG }$ goes to zero as $n$ goes to infinity.}}
@misc{note,
note = {As $\tau \rightarrow 0$ and $N \rightarrow \infty$ with $N\tau=t$, the operator $T^N$ converges to $e^{-t(H-E \Id)}$. We then have $G^E_{ij} \rightarrow \int_0^{\infty} dt \mel{i}{e^{-t(H-E \Id)}}{j}$, which is the Laplace transform of the time-dependent Green's function $\mel{i}{e^{-t(H-E \Id)}}{j}$.}}
@ -341,12 +348,13 @@
@book{Ceperley_1979,
author = {D.M. Ceperley and M.H Kalos},
chapter = {4},
editor={K.Binder},
chapter={4},
publisher = {Springer, Berlin},
title = {Monte Carlo Methods in Statistical Physics},
year = {1979}}
@article{Carlson_2007,
author = {J. Carlson},
date-modified = {2022-09-15 10:14:32 +0200},

90
g.tex
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@ -1,3 +1,4 @@
%\documentclass[aps,prb,reprint,showkeys]{revtex4-1}
\documentclass[aps,prb,reprint,showkeys,superscriptaddress]{revtex4-1}
%\usepackage{subcaption}
\usepackage{bm,graphicx,tabularx,array,booktabs,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx}
@ -305,7 +306,7 @@ To derive a probabilistic expression for the Green's matrix, we introduce a guid
\end{align}
Note that, thanks to the properties of similarity transformations, the path integral expression relating $G^{(N)}$ and $T$ [see Eq.~\eqref{eq:G}] remains unchanged for $\bar{G}^{(N)}$ and $\bar{T}$.
Next, the matrix elements of $\bar{T}$ are expressed as those of a stochastic matrix multiplied by some residual weights, namely,
Now, the key idea to take advantage of probabilistic techniques is to rewrite the matrix elements of $\bar{T}$ as those of a stochastic matrix multiplied by some residual weights (here, not necessarily positive), namely
\be
\label{eq:defTij}
\bar{T}_{ij}= p_{i \to j} w_{ij}.
@ -315,7 +316,42 @@ Here, we recall that a stochastic matrix is defined as a matrix with positive en
\label{eq:sumup}
\sum_j p_{i \to j}=1.
\ee
To build the transition probability density, the following operator is introduced
Using this representation for $\bar{T}_{ij}$ the similarity-transformed Green's matrix components can be rewritten as
\be
\label{eq:GN_simple}
\bar{G}^{(N)}_{i_0 i_N} =
\sum_{i_1,\ldots,i_{N-1}} \qty( \prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}} ) \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}},
\ee
which is amenable to Monte Carlo calculations by generating paths using the transition probability matrix $p_{i \to j}$.
Let us illustrate this in the case of the energy as given by Eq.~\eqref{eq:E0}. Taking $\ket{\Psi_0}=\ket{i_0}$ as initial state, we have
\be
E_0 = \lim_{N \to \infty }
\frac{ \sum_{i_N} G^{(N)}_{i_0 i_N} {(H\PsiT)}_{i_N} }
{ \sum_{i_N} {G}^{(N)}_{i_0 i_N} {\PsiT}_{i_N} }.
\ee
which can be rewritten probabilistically as
\be
E_0 = \lim_{N \to \infty }
\frac{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \frac{ {(H\PsiT)}_{i_N} }{ \PsiG_{i_N} }}}
{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \frac{ {\PsiT}_{i_N} } {\PsiG_{i_N}} }},
\ee
where $\expval{...}$ is the probabilistic average defined over the set of paths $\ket{i_1},\ldots,\ket{i_N}$ occurring with probability
\be
\text{Prob}_{i_0}(i_1,\ldots,i_{N}) = \prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}}.
\ee
Using Eq.~\eqref{eq:sumup} and the fact that $p_{i \to j} \ge 0$, one can easily verify that $\text{Prob}_{i_0}$ is positive and obeys
\be
\sum_{i_1,\ldots,i_{N}} \text{Prob}_{i_0}(i_1,\ldots,i_{N}) = 1,
\ee
as it should.
The rewriting of $\bar{T}_{ij}$ as a product of a stochastic matrix times some general real weight
does not introduce any constraint on the choice of the stochastic matrix, so that, in theory, any stochastic matrix could be used. However,
in practice, it is highly desirable that the magnitude of the fluctuations of the weight during the Monte Carlo simulation be as small as
possible. A natural solution is to choose a stochastic matrix equal to a good approximation of $\bar{T}_{ij}$. This is done as follows.
Let us introduce the following operator
\be
\label{eq:T+}
T^+= \Id - \tau \qty( H^+ - \EL^+ \Id ),
@ -384,46 +420,6 @@ This is done thanks to Eq.~\eqref{eq:defTij} that connects $p_{i \to j}$ and $T_
w_{ij}=\frac{T_{ij}}{T^+_{ij}},
\ee
derived from Eqs.~\eqref{eq:defT} and \eqref{eq:pij}.
Using these notations the similarity-transformed Green's matrix components can be rewritten as
\be
\label{eq:GN_simple}
\bar{G}^{(N)}_{i_0 i_N} =
\sum_{i_1,\ldots,i_{N-1}} \qty( \prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}} ) \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}},
\ee
which is amenable to Monte Carlo calculations by generating paths using the transition probability matrix $p_{i \to j}$.
Let us illustrate this in the case of the energy as given by Eq.~\eqref{eq:E0}. Taking $\ket{\Psi_0}=\ket{i_0}$ as initial state, we have
\be
E_0 = \lim_{N \to \infty }
\frac{ \sum_{i_N} G^{(N)}_{i_0 i_N} {(H\PsiT)}_{i_N} }
{ \sum_{i_N} {G}^{(N)}_{i_0 i_N} {\PsiT}_{i_N} }.
\ee
which can be rewritten probabilistically as
\be
E_0 = \lim_{N \to \infty }
\frac{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \frac{ {(H\PsiT)}_{i_N} }{ \PsiG_{i_N} }}}
{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \frac{ {\PsiT}_{i_N} } {\PsiG_{i_N}} }},
\ee
where $\expval{...}$ is the probabilistic average defined over the set of paths $\ket{i_1},\ldots,\ket{i_N}$ occurring with probability
\be
\text{Prob}_{i_0}(i_1,\ldots,i_{N}) = \prod_{k=0}^{N-1} p_{i_{k} \to i_{k+1}}.
\ee
Using Eq.~\eqref{eq:sumup} and the fact that $p_{i \to j} \ge 0$, one can easily verify that $\text{Prob}_{i_0}$ is positive and obeys
\be
\sum_{i_1,\ldots,i_{N}} \text{Prob}_{i_0}(i_1,\ldots,i_{N}) = 1,
\ee
as it should.
%For a given path $i_1,\ldots,i_{N-1}$, the probabilistic average associated with this probability is then defined as
%\be
%\label{eq:average}
% \expval{F} = \sum_{i_1,\ldots,i_{N-1}} F(i_0,\ldots,i_N) \text{Prob}_{i_0 \to i_N}(i_1,\ldots,i_{N-1}),
%\ee
%where $F$ is an arbitrary function.
%Finally, the path-integral expressed as a probabilistic average reads
%\be
%\label{eq:cn_stoch}
% \bar{G}^{(N)}_{i_0 i_N}= \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}}}.
%\ee
To calculate the probabilistic averages, an artificial (mathematical) ``particle'' called walker (or psi-particle) is introduced.
During the Monte Carlo simulation, the walker moves in configuration space by drawing new states with
@ -509,7 +505,7 @@ Let us write an arbitrary path of length $N$ as
\ket{i_0} \to \ket{i_1} \to \cdots \to \ket{i_N},
\ee
where the successive states are drawn using the transition probability matrix $p_{i \to j}$.
This series can be recast
This path belongs to the set of paths that can be represented as follows
\be
\label{eq:eff_series}
(\ket*{I_0},n_0) \to (\ket*{I_1},n_1) \to \cdots \to (\ket*{I_p},n_p),
@ -566,7 +562,7 @@ where the operator $F^+_I = P_I T^+ (1-P_I)$, corresponding to the last move co
\be
(F^+_I)_{ij} =
\begin{cases}
T^+_{ij}, & \qif* \ket{i} \in \cD_{I} \land \ket{j} \notin \cD_{I},
T^+_{ij}, & \qif* \ket{i} \in \cD_{I} \;\;{\rm and}\;\; \ket{j} \notin \cD_{I},
\\
0, & \text{otherwise}.
\end{cases}
@ -584,7 +580,7 @@ The normalization of this probability can be verified using the fact that
\label{eq:relation}
\qty(T^+_{I})^{n-1} F^+_I = \qty(T^+_{I})^{n-1} T^+ - \qty(T^+_I)^n,
\ee
leading to
leading to\cite{note2}
\be
\sum_{n=0}^{\infty} P_{I}(n)
= \frac{1}{\PsiG_{I}} \sum_{n=1}^{\infty} \qty[ \mel{ I }{ \qty(T^+_{I})^{n-1} }{ \PsiG }
@ -594,7 +590,7 @@ The average trapping time defined as ${\bar t}_{I}={\bar n}_{I} \tau$ where $ {\
\be
{\bar t}_{I}=\frac{1}{\PsiG_I} \mel{I}{ { \qty[ P_I \qty( H^+ - \EL^+ \Id ) P_I ] }^{-1} }{ \PsiG }.
\ee
In practice, the various quantities restricted to the domain will be computed by diagonalizing the matrix $(H^+-\EL^+ \Id)$ in $\cD_{I}$.
In practice, the various quantities restricted to the domain will be computed by inverting the matrix $(H^+-\EL^+ \Id)$ in $\cD_{I}$.
Note that it is possible only if the dimension of the domains is not too large (say, less than a few thousands).
%=======================================%
@ -928,7 +924,7 @@ The contribution of the other states vanishes as $U$ increases with a rate incre
In addition, for a given number of double occupations, configurations with large values of $n_A(n)$ are favored due to their high electronic mobility.
Therefore, we build domains associated with small $n_D$ and large $n_A$ in a hierarchical way as described below.
For simplicity and reducing the number of diagonalizations to perform, we shall consider only one non-trivial domain called here the main domain and denoted as $\cD$.
For simplicity and reducing the number of matrix inversions to perform, we shall consider only one non-trivial domain called here the main domain and denoted as $\cD$.
This domain will be chosen common to all states belonging to it, that is,
\be
\cD_i= \cD \qq{for} \ket{i} \in \cD.