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%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-10-15 13:28:48 +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 = {2022-10-15 13:28:08 +0200},
doi = {https://doi.org/10.1016/0021-9991(83)90161-4},
issn = {0021-9991},
journal = {J. Comput. Phys.},
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}}
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},
year = {1983},
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},
volume = {128},
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}}
title = {Monte Carlo Calculations of the Ground State of Three- and Four-Body Nuclei},
author = {Kalos, M. H.},
journal = {Phys. Rev.},
volume = {128},
issue = {4},
pages = {1791--1795},
numpages = {0},
year = {1962},
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},
volume = {2},
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}}
title = {Energy of a Boson Fluid with Lennard-Jones Potentials},
author = {Kalos, M. H.},
journal = {Phys. Rev. A},
volume = {2},
issue = {1},
pages = {250--255},
numpages = {0},
year = {1970},
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 = {2022-10-15 13:28:48 +0200},
doi = {10.1063/1.451046},
journal = {J. Chem. Phys.},
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}}
author = {Moskowitz,Jules W. and Schmidt,K. E. },
title = {The domain Green’s function method},
journal = {The Journal of Chemical Physics},
volume = {85},
number = {5},
pages = {2868-2874},
year = {1986},
doi = {10.1063/1.451046},
URL = {
https://doi.org/10.1063/1.451046
},
eprint = {
https://doi.org/10.1063/1.451046
}
}
@article{Willow_2012,
author = {Willow,Soohaeng Yoo and Kim,Kwang S. and Hirata,So},
date-modified = {2022-09-30 16:10:46 +0200},
@ -337,12 +339,13 @@
year = {2012}}
@book{Ceperley_1979,
author = {D.M. Ceperley and M.H Kalos},
chapter = {4},
editor = {K.Binder},
publisher = {Springer, Berlin},
title = {Monte Carlo Methods in Statistical Physics},
year = {1979}}
author = {D.M. Ceperley and M.H Kalos},
editor={K.Binder},
chapter={4},
publisher = {Springer, Berlin},
title = {Monte Carlo Methods in Statistical Physics},
year = {1979}}
@article{Carlson_2007,
author = {J. Carlson},

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@ -122,33 +117,29 @@
\begin{document}
\title{Diffusion Monte Carlo using domains in configuration space}
\author{Roland Assaraf}
\email{assaraf@lct.jussieu.fr}
\affiliation{\LCT}
\author{Emmanuel Giner}
\email{giner@lct.jussieu.fr}
\affiliation{\LCT}
\author{Vijay Gopal Chilkuri}
\email{vijay.gopal.c@gmail.com}
\affiliation{\LCT}
\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Anthony Scemama}
\email{scemama@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Michel Caffarel}
\email{caffarel@irsamc.ups-tlse.fr}
\email{Corresponding author: caffarel@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\begin{abstract}
\noindent
The sampling of the configuration space in diffusion Monte Carlo (DMC) is done using walkers moving randomly.
In a previous work on the Hubbard model [\href{https://doi.org/10.1103/PhysRevB.60.2299}{Assaraf et al.~Phys.~Rev.~B \textbf{60}, 2299 (1999)}], it was shown that the probability for a walker to stay a certain amount of time in the same state obeys a Poisson law and that the on-state dynamics can be integrated out exactly, leading to an effective dynamics connecting only different states.
In a previous work on the Hubbard model [\href{https://doi.org/10.1103/PhysRevB.60.2299}{Assaraf et al.~Phys.~Rev.~B \textbf{60}, 2299 (1999)}],
it was shown that the probability for a walker to stay a certain amount of time in the same state obeys a Poisson law and that the on-state dynamics can be integrated out exactly, leading to an effective dynamics connecting only different states.
Here, we extend this idea to the general case of a walker trapped within domains of arbitrary shape and size.
The equations of the resulting effective stochastic dynamics are derived.
The larger the average (trapping) time spent by the walker within the domains, the greater the reduction in statistical fluctuations.
A numerical application to the Hubbard model is presented.
Although this work presents the method for (discrete) finite linear spaces, it can be generalized without fundamental difficulties to continuous configuration spaces.
Although this work presents the method for finite linear spaces, it can be generalized without fundamental difficulties to continuous configuration spaces.
\end{abstract}
\maketitle
@ -185,15 +176,22 @@ It is shown how to define an effective stochastic dynamics describing walkers mo
The equations of the effective dynamics are derived and a numerical application for a model (one-dimensional) problem is presented.
In particular, it shows that the statistical convergence of the energy can be greatly enhanced when domains associated with large average trapping times are considered.
It should be noted that the use of domains in quantum Monte Carlo is not new.
Domains have been introduced within the context of Green's function Monte Carlo (GFMC) pioneered by Kalos \cite{Kalos_1962,Kalos_1970} and later developed and applied by Kalos and others. \cite{Kalos_1974,Ceperley_1979,Ceperley_1983,Moskowitz_1986}
In GFMC, an approximate Green's function that can be sampled is required for the stochastic propagation of the wave function.
In the so-called domain GFMC version of GFMC introduced in Ref.~\onlinecite{Kalos_1970} and \onlinecite{Kalos_1974}, the sampling is realized by using the restriction of the Green's function to a small domain consisting of the cartesian product of small spheres around each particle, the potential being considered as constant within the domain.
Fundamentally, the method presented in this work is closely related to domain GFMC, although the way we present the formalism in terms of walkers trapped within domains and derive the equations may appear different.
However, we show here to use domains of arbitrary size, a new feature that greatly enhances the simulation efficiency when domains are suitably chosen, as illustrated in our numerical application.
It should be noted that the use of domains in quantum Monte Carlo is not new. Domains have been introduced within the context of the
Green's function Monte Carlo (GFMC) method pioneered by Kalos\cite{Kalos_1962,Kalos_1970}
and later developped and applied by Kalos and others.\cite{Kalos_1974,Ceperley_1979,Ceperley_1983,Moskowitz_1986}
In GFMC, an approximate Green's function that can be sampled is needed to propagate stochastically the wavefunction.
In the so-called Domain's GFMC version of GFMC introduced in Ref.\onlinecite{Kalos_1970} and \onlinecite{Kalos_1974}
the sampling is realized by using the restriction of the Green's function
to a small domain consisting of the Cartesian product of small spheres around each particle, the potential being considered as constant within the domain.
Fundamentally, the method presented in this work is closely related to the Domain's GFMC, although the way we present the formalism in terms of walkers
trapped within domains
and derive the equations may appear as different.
However, a key difference here is that we show how to use
domains of arbitrary size, a point which can greatly enhance the efficiency of the simulations when domains are suitably chosen, as illustrated in our numerical
application.
Finally, from a general perspective, it is interesting to mention that the method proposed here is an illustration of how valuable and efficient can be the combination of stochastic and deterministic techniques.
Finally, from a general perspective, it is interesting to emphasize that the present method illustrates how suitable combinations of stochastic and deterministic techniques lead to a more efficient and valuable method.
In recent years, a number of works have exploited this idea and proposed hybrid stochastic/deterministic schemes.
Let us mention, for example, the semi-stochastic approach of Petruzielo {\it et al.}, \cite{Petruzielo_2012} two different hybrid algorithms for evaluating the second-order perturbation energy in selected configuration interaction methods, \cite{Garniron_2017b,Sharma_2017} the approach of Willow \textit{et al.} for computing stochastically second-order many-body perturbation energies, \cite{Willow_2012} or the zero variance Monte Carlo scheme for evaluating two-electron integrals in quantum chemistry. \cite{Caffarel_2019}
@ -233,7 +231,7 @@ The equality in Eq.~\eqref{eq:limTN} holds up to a global phase factor playing n
At large but finite $N$, the vector $T^N \ket{\Psi_0}$ differs from $\ket{\Phi_0}$ only by an exponentially small correction, making it straightforward to extrapolate the finite-$N$ results to $N \to \infty$.
Likewise, ground-state properties may be obtained at large $N$.
For example, in the important case of the energy, one can project out the vector $T^N \ket{\Psi_0}$ on some approximate vector, $\ket{\PsiT}$, as follows:
For example, in the important case of the energy, one can project out the vector $T^N \ket{\Psi_0}$ on some approximate vector, $\ket{\PsiT}$, as follows
\be
\label{eq:E0}
E_0 = \lim_{N \to \infty } \frac{\mel{\Psi_0}{T^N}{H\Psi_T}}{\mel{\Psi_0}{T^N}{\Psi_T}}.
@ -572,13 +570,15 @@ 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\footnote{The property results from the fact that Eq.~\eqref{eq:relation} is a telescoping series and that the general term $\mel{ I }{ \qty(T^+_{I})^{n} }{ \PsiG }$ goes to zero as $n\to\infty$.}
leading to\footnote{
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.}
\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 }
- \mel{ I }{ \qty(T^+_{I})^{n} }{ \PsiG } ] = 1.
\ee
The average trapping time defined as ${\bar t}_{I}={\bar n}_{I} \tau$, where $ {\bar n}_{I}=\sum_n n P_{I}(n)$ is calculated to be
The average trapping time defined as ${\bar t}_{I}={\bar n}_{I} \tau$ where $ {\bar n}_{I}=\sum_n n P_{I}(n)$ is calculated to be
\be
{\bar t}_{I}=\frac{1}{\PsiG_I} \mel{I}{ { \qty[ P_I \qty( H^+ - \EL^+ \Id ) P_I ] }^{-1} }{ \PsiG }.
\ee
@ -719,7 +719,7 @@ Let us define the energy-dependent Green's matrix
\ee
The denomination ``energy-dependent'' is chosen here since
this quantity is the discrete version of the Laplace transform of the time-dependent Green's function in a continuous space,
usually known under this name.\footnote{As $\tau \to 0$ and $N \to \infty$ with $N\tau=t$, the operator $T^N$ converges to $e^{-t(H-E \Id)}$. We then have $G^E_{ij} \to \int_0^{\infty} dt \mel{i}{e^{-t(H-E \Id)}}{j}$, which is the Laplace transform of the time-dependent Green's function $\mel{i}{e^{-t(H-E \Id)}}{j}$.}
usually known under this name.\footnote{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 $\mel{i}{e^{-t(H-E \Id)}}{j}$.}
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 sums, as follows
\begin{multline}
\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)
@ -880,7 +880,7 @@ Let us consider the one-dimensional Hubbard Hamiltonian for a chain of $N$ sites
H= -t \sum_{\expval{ i j } \sigma} \hat{a}^+_{i\sigma} \hat{a}_{j\sigma}
+ U \sum_i \hat{n}_{i\uparrow} \hat{n}_{i\downarrow},
\ee
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.
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.
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 can be written as
\be
@ -1109,7 +1109,7 @@ We also report the variational and exact energies together with the values of th
As $U$ increases the configurations with zero or one double occupation become more and more predominant and the average trapping time increases.
For very large values of $U$ (say, $U \ge 12$) the increase of $\bar{t}_{I_0}$ becomes particularly steep.
Finally, in Table \ref{tab4}, we report the results obtained for larger systems at $U=12$ for a chain size ranging from $N=4$ (36 states)
Finally, in Table \ref{tab5}, we report the results obtained for larger systems at $U=12$ for a chain size ranging from $N=4$ (36 states)
to $N=12$ ($\sim 10^6$ states).
No careful construction of domains maximizing the average trapping time has been performed, we have merely chosen domains of reasonable size (no more than 2682) by taking not too large number of double occupations (only, $n_D=0,1$) and not too small number of nearest-neighbor antiparallel pairs.
As seen, as the number of sites increases, the average trapping time for the chosen domains decreases.
@ -1119,7 +1119,7 @@ The exact DMC energies extrapolated using the two-component function are also re
Similarly to what has been done for $N=4$, the extrapolation is performed using about five values of the reference energy.
The extrapolated DMC energies are in full agreement with the exact value within error bars.
However, an increase of the statistical error is observed when the system size increases.
To get lower error bars, a more accurate trial wave functions may be considered, better domains, and also larger simulation times.
To get lower error bars, more accurate trial wave functions may be considered, better domains, and also larger simulation times.
Of course, it will also be particularly interesting to take advantage of the fully parallelizable character of the algorithm to get much lower error bars.
All these aspects will be considered in a forthcoming work.
@ -1173,7 +1173,8 @@ $E_0$ exact & $-0.768068\ldots$\\
%%% TABLE IV %%%
\begin{table}
\caption{One-dimensional Hubbard model with $N=4$.
\caption{One-dimensional Hubbard model with $N=4$. Average trapping time as a
function of $U$.
The main domain is $\cD(0,1) \cup \cD(1,0)$.}
\label{tab4}
\begin{ruledtabular}