loos/DQMC
1
0
Fork 1
Code Issues Pull Requests Projects Releases Wiki Activity

OK with abstract and intro

Browse Source
This commit is contained in:
Pierre-Francois Loos 2022-09-14 15:57:39 +02:00
commit 4d1119a3a9
17 changed files with 6332 additions and 93 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

11
.gitignore vendored Normal file
View File

@ -0,0 +1,11 @@
g.aux
g.bbl
g.blg
g.fdb_latexmk
g.fls
g.log
g.out
g.pdf
gNotes.bib

BIN
fig1.pdf
View File

Binary file not shown.

BIN
fig2.pdf
View File

Binary file not shown.

161
g.tex
View File

@ -6,8 +6,6 @@
\usepackage[T1]{fontenc} \usepackage[T1]{fontenc}
\usepackage{txfonts,dsfont} \usepackage{txfonts,dsfont}
\usepackage{xspace} \usepackage{xspace}
\usepackage [french]{babel}
%\usepackage{lscape}
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
\newcommand{\roland}[1]{\textcolor{cyan}{\bf #1}} \newcommand{\roland}[1]{\textcolor{cyan}{\bf #1}}
@ -137,11 +135,9 @@
\begin{abstract} \begin{abstract}
\noindent \noindent
The sampling of the configuration space in diffusion Monte Carlo (DMC) The sampling of the configuration space in diffusion Monte Carlo (DMC) is done using walkers moving randomly.
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)}], 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 \titou{state} obeys a Poisson law and that it was shown that the probability for a walker to stay a certain amount of time in the same \titou{state} obeys a Poisson law and that the \titou{on-state} dynamics can be integrated out exactly, leading to an effective dynamics connecting only different states.
the on-\titou{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. 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 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. The larger the average (trapping) time spent by the walker within the domains, the greater the reduction in statistical fluctuations.
@ -154,86 +150,73 @@ Although this work presents the method for finite linear spaces, it can be gener
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} \section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Diffusion Monte Carlo (DMC) is a class of stochastic methods for evaluating Diffusion Monte Carlo (DMC) is a class of stochastic methods for evaluating the ground-state properties of quantum systems.
the ground-state properties of quantum systems. They have been extensively used They have been extensively used in virtually all domains of physics and chemistry where the many-body quantum problem plays a central role (condensed-matter physics,\cite{Foulkes_2001,Kolorenc_2011} quantum liquids,\cite{Holzmann_2006} nuclear physics,\cite{Carlson_2015,Carlson_2007} theoretical chemistry,\cite{Austin_2012} etc).
in virtually all domains of physics and chemistry where the many-body quantum problem plays a central role (condensed-matter physics,\cite{Foulkes_2001,Kolorenc_2011} DMC can be used either for systems defined in a continuous configuration space (typically, a set of particles moving in space) for which the Hamiltonian operator is defined in a Hilbert space of infinite dimension or systems defined in a discrete configuration space where the Hamiltonian reduces to a matrix.
quantum liquids,\cite{Holzmann_2006} Here, we shall consider only the discrete case, that is, the general problem of calculating the lowest eigenvalue and/or eigenstate of a (very large) matrix.
nuclear physics,\cite{Carlson_2015,Carlson_2007} theoretical chemistry,\cite{Austin_2012} etc).
DMC can be used either for systems defined in a continuous configuration space (typically, a set of particles moving in space) for which the Hamiltonian operator is defined in a (infinite-dimensional) Hilbert space or systems defined in a discrete configuration space where the Hamiltonian reduces to a matrix.
Here, we shall consider only the discrete case, that is, the general problem of calculating the lowest eigenvalue/eigenstate of a (very large) matrix.
The generalization to continuous configuration spaces presents no fundamental difficulty. The generalization to continuous configuration spaces presents no fundamental difficulty.
In essence, DMC is based on \textit{stochastic} power methods, an old and widely employed numerical approach to extract In essence, DMC is based on \textit{stochastic} power methods, a family of old and widely employed numerical approaches able to extract the largest or smallest eigenvalues of a matrix (see, \eg, Ref.~\onlinecite{Golub_2012}).
the largest or smallest eigenvalues of a matrix (see, \eg, Ref.~\onlinecite{Golub_2012}). These approaches are particularly simple as they merely consist in applying a given matrix (or some simple function of it) as many times as required on some arbitrary vector belonging to the linear space.
This approach is particularly simple as it merely consists in applying the matrix (or some simple function of it) as many times as Thus, the basic step of the corresponding algorithm essentially reduces to successive matrix-vector multiplications.
required on some arbitrary vector belonging to the linear space. In practice, power methods are employed under more sophisticated implementations, such as, \eg, the Lancz\`os algorithm (based on Krylov subspaces) \cite{Golub_2012} or Davidson's method where a diagonal preconditioning is performed. \cite{Davidson_1975}
Thus, the basic step of the corresponding algorithm essentially reduces to a matrix-vector multiplication. When the size of the matrix is too large, matrix-vector multiplications become unfeasible and probabilistic techniques to sample only the most important contributions of the matrix-vector product are required.
In practice, the power method is employed under some more sophisticated implementations, such as, \eg, This is the basic idea of DMC.
the Lancz\`os \cite{Golub_2012} or Davidson \cite{Davidson_1975} algorithms. There exist several variants of DMC known under various names: pure DMC, \cite{Caffarel_1988} DMC with branching, \cite{Reynolds_1982} reptation Monte Carlo, \cite{Baroni_1999} stochastic reconfiguration Monte Carlo, \cite{Sorella_1998,Assaraf_2000} etc.
When the size of the matrix is too large, the matrix-vector multiplication becomes unfeasible
and probabilistic techniques to sample only the most important contributions of the matrix-vector product are required.
This is the basic idea of DMC. There exist several variants of DMC known under various names:
pure DMC, \cite{Caffarel_1988} DMC with branching, \cite{Reynolds_1982} reptation Monte Carlo, \cite{Baroni_1999} stochastic reconfiguration Monte Carlo, \cite{Sorella_1998,Assaraf_2000} etc.
Here, we shall place ourselves within the framework of pure DMC whose mathematical simplicity is particularly appealing when developing new ideas, although it is usually not the most efficient variant of DMC. \titou{Why?} Here, we shall place ourselves within the framework of pure DMC whose mathematical simplicity is particularly appealing when developing new ideas, although it is usually not the most efficient variant of DMC. \titou{Why?}
However, all the ideas presented in this work can be adapted without too much difficulty to the other variants, so the denomination DMC must ultimately be understood here as a generic name for the broad class of DMC methods. However, all the ideas presented in this work can be adapted without too much difficulty to the other variants, so the denomination DMC must ultimately be understood here as a generic name for this broad class of methods.
Without entering into the mathematical details presented below, the main ingredient of DMC to perform the Without entering into the mathematical details (which are presented below), the main ingredient of DMC in order to perform the matrix-vector multiplications probabilistically is the stochastic matrix (or transition probability matrix) that generates stepwise a series of states over which statistical averages are evaluated.
matrix-vector multiplication probabilistically is the use of a stochastic matrix (or transition probability matrix) The critical aspect of any Monte Carlo scheme is the amount of computational effort required to reach a given statistical error.
to generate stepwise a series of states over which statistical averages are evaluated. Two important avenues to decrease the error are the use of variance reduction techniques (for example, by introducing improved estimators \cite{Assaraf_1999}) or to improve the quality of the sampling (minimization of the correlation time between states).
The critical aspect of any Monte Carlo scheme is the the amount of computational effort required to Another possibility, at the heart of the present work, is to integrate out exactly some parts of the dynamics, thus reducing the number of degrees of freedom and, hence, the amount of statistical fluctuations.
reach a given statistical error.
Two important avenues to decrease the error are the use of variance reduction techniques
(for example, by introducing improved estimators\cite{Assaraf_1999}) or to improve the quality of the sampling
(minimization of the correlation time between states).
Another possibility, at the heart of the present work, is to integrate out exactly some part of the dynamics, thus reducing the number of
degrees of freedom and, then, the amount of statistical fluctuations.
In a previous work,\cite{assaraf_99,caffarel_00} it has been shown that the probability for a walker to stay a certain amount of time on the same state obeys a Poisson law and that In a previous work,\cite{assaraf_99,caffarel_00} it has been 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 \titou{on-state} dynamics can be integrated out to generate an effective dynamics connecting only different states with some renormalized estimators for the properties.
the on-state dynamics can be integrated out to generate an effective dynamics connecting only different states with some
renormalized estimators for the properties.
Numerical applications have shown that the statistical errors can be very significantly decreased. Numerical applications have shown that the statistical errors can be very significantly decreased.
Here, we extend this idea to the general case where a walker Here, we extend this idea to the general case where a walker remains a certain amount of time within a finite domain no longer restricted to a single state.
remains a certain amount of time within a finite domain no longer restricted to a single state. It is shown how to define an effective stochastic It is shown how to define an effective stochastic dynamics describing walkers moving from one domain to another.
dynamics describing walkers moving from one domain into another. The equations of the effective dynamics are derived. The equations of the effective dynamics are derived and a numerical application for a model (one-dimensional) problem is presented.
A numerical application for the 1D-problem is presented. In particular, it shows that the statistical convergence of the energy can be greatly In particular, it shows that the statistical convergence of the energy can be greatly enhanced when domains associated with large average trapping times are cnosidered.
enhanced when domains associated with large average trapping times are used.
It should be noted that the use of domains in quantum Monte Carlo is not new. In a pioneering work,\cite{Kalos_1974} It should be noted that the use of domains in quantum Monte Carlo is not new.
Kalos and collaborators introduced the so-called Domain's Green Function Monte Carlo approach in continuous space which they applied to In their pioneering work, \cite{Kalos_1974} Kalos and collaborators introduced the so-called domain Green's function Monte Carlo approach in continuous space that they applied to a system of bosons with hard-sphere interaction.
a system of bosons with hard-sphere interaction. The domain used was the Cartesian product of small spheres around each particle, the Hamiltonian The domain used was the Cartesian product of small spheres around each particle, the Hamiltonian being approximated by the kinetic part only within the domain.
being approximated by the kinetic part only within the domain. Some time later, Kalos proposed to extend these ideas to more general domains such as rectangular and/or cylindrical domains. \cite{Kalos_2000} In both works, the size of the domains is infinitely small in the limit of a vanishing time-step.
Some time later, Kalos proposed to extend these ideas to more general domains such as rectangular and/or Here, the domains are of arbitrary size, thus greatly increasing the efficiency of the approach.
cylindrical domains.\cite{Kalos_2000} In both works, the size of the domains is infinitely small in the limit of a vanishing time-step. Here, the Finally, note that some general equations for arbitrary domains in continuous space have also been proposed in Ref.~\onlinecite{Assaraf_1999B}.
domains are of arbitrary size, thus greatly increasing the efficiency of the approach.
Finally, note that some general equations for arbitary domains in continuous space have also been proposed in
[\onlinecite{Assaraf_1999B}].
The paper is organized as follows. Sec.\ref{Sec:DMC} presents the basic equations and notations of DMC. First, the path integral representation The paper is organized as follows.
of the Green's function is given in subsection \ref{sub:path}. The probabilistic framework allowing the Monte Carlo calculation of the Section \ref{sec:DMC} presents the basic equations and notations of DMC.
Green's function is presented in \ref{sub:proba}. First, the path integral representation of the Green's function is given in Subsec.~\ref{sec:path}.
Section \ref{sec:DMC_domains} is devoted to the use of domains in DMC. First, we recall in \ref{sub:single_domains} the case The probabilistic framework allowing the Monte Carlo calculation of the Green's function is presented in Subsec.~\ref{sec:proba}.
of a domain consisting of a single state,\cite{Assaraf_1999} then the general case, \ref{sub:general_domains}. In \ref{Green} both the time-dependent and Section \ref{sec:DMC_domains} is devoted to the use of domains in DMC.
energy dependent Green's function using domains are derived. Section \ref{numerical} presents the appplication of the approach to the one-dimensional First, we recall in Subsec.~\ref{sec:single_domains} the case of a domain consisting of a single state. \cite{Assaraf_1999}
Hubbard model. Finally in Sec.\ref{conclu} some conclusions and perspectives are given. The general case is then treated in Subsec.~\ref{sec:general_domains}.
In Subsec.~\ref{sec:Green}, both the time- and energy-dependent Green's function using domains is derived.
Section \ref{sec:numerical} presents the application of the approach to the one-dimensional Hubbard model.
Finally, in Sec.\ref{sec:conclu}, some conclusions and perspectives are given.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Diffusion Monte Carlo} \section{Diffusion Monte Carlo}
\label{Sec:DMC} \label{sec:DMC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%=======================================%
\subsection{Path-integral representation} \subsection{Path-integral representation}
\label{sub:path} \label{sec:path}
Diffusion Monte Carlo is a stochastic implementation of the power method. The operator used is %=======================================%
As previously mentioned, DMC is a stochastic implementation of the power method defined by the following operator:
\be \be
T = \mathds{1} -\tau (H-E\mathds{1}), T = \mathds{1} -\tau (H-E\mathds{1}),
\ee \ee
where $\mathds{1}$ is the identity matrix, $\tau$ a small positive parameter playing the role of a time-step, $E$ some arbitrary reference energy, and where $\mathds{1}$ is the identity operator, $\tau$ a small positive parameter playing the role of a time-step, $E$ some arbitrary reference energy, and $H$ the Hamiltonian operator. Starting from some initial vector, $\ket{\Psi_0}$, we have
$H$ the Hamiltonian matrix. Starting from some initial vector, $|\Psi_0\rangle$, we have
\be \be
\lim_{N \rightarrow \infty } T^N|\Psi_0 \rangle = |\Phi_0 \rangle \lim_{N \to \infty} T^N \ket{\Psi_0} = \ket{\Phi_0}
\ee \ee
where $|\Phi_0 \rangle$ is the ground-state. The equality is up to a global phase factor playing no role in physical quantum averages. where $\ket{\Phi_0}$ is the ground-state wave function.
This result is true for any $|\Psi_0 \rangle$ The equality is up to a global phase factor playing no role in physical quantum averages.
provided that $\langle \Phi_0 |\Psi_0 \rangle \ne 0$ and for $\tau$ sufficiently small. This result is true for any $\ket{\Psi_0}$ provided that $\braket{\Phi_0}{\Psi_0} \ne 0$ and for $\tau$ sufficiently small.
At large but finite $N$, the vector $T^N|\Psi_0\rangle$ differs from $|\Phi_0 \rangle$ only by an exponentially small correction, At large but finite $N$, the vector $T^N \ket{\Psi_0}$ differs from $\ket{\Phi_0}$ only by an exponentially small correction, making easy the extrapolation of the finite-N results to $N=\infty$.\\
making easy the extrapolation of the finite-N results to $N=\infty$.\\
Ground-state properties may be obtained at large $N$. For example, in the important case of the energy one can use the formula Ground-state properties may be obtained at large $N$. For example, in the important case of the energy one can use the formula
\be \be
@ -247,7 +230,7 @@ To proceed further we introduce the time-dependent Green's matrix $G^{(N)}$ defi
\be \be
G^{(N)}_{ij}=\langle j|T^N |i\rangle. G^{(N)}_{ij}=\langle j|T^N |i\rangle.
\ee \ee
The denomination \og time-dependent Green's matrix \fg is used here since $G$ may be viewed as a short-time approximation of the (time-imaginary) evolution operator, 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.\\ $e^{-N\tau H}$ which is usually referred to as the imaginary-time dependent Green's function.\\
Introducing the convenient notation, $i_k$, for the $N-1$ indices of the intermediate states in the $N$-th product of $T$, $G^{(N)}$ can be written in Introducing the convenient notation, $i_k$, for the $N-1$ indices of the intermediate states in the $N$-th product of $T$, $G^{(N)}$ can be written in
@ -273,15 +256,15 @@ the $i$-th component of the ground-state (obtained as $\lim_{N\rightarrow \infty
at vector $|i\rangle$. This result is independent of the initial at vector $|i\rangle$. This result is independent of the initial
vector $|i_0\rangle$, apart from some irrelevant global phase factor. vector $|i_0\rangle$, apart from some irrelevant global phase factor.
When the size of the linear space is small the explicit calculation of the full sums involving $M^N$ terms (here, $M$ is the size of the Hilbert space) When the size of the linear space is small the explicit calculation of the full sums involving $M^N$ terms (here, $M$ is the size of the Hilbert space)
can be performed. We are in the realm of what can be called the \og deterministic \fg power methods, such as can be performed. We are in the realm of what can be called the ``deterministic'' power methods, such as
the Lancz\`os or Davidson approaches. If not, probabilistic techniques for generating only the the Lancz\`os or Davidson approaches. If not, probabilistic techniques for generating only the
paths contributing significantly to the sums are to be used. paths contributing significantly to the sums are to be used.
\subsection{Probabilistic framework} \subsection{Probabilistic framework}
\label{sub:proba} \label{sec:proba}
In order to derive In order to derive
a probabilistic expression for the Green's matrix we introduce a so-called guiding vector, $|\Psi^+\rangle$, a probabilistic expression for the Green's matrix we introduce a so-called guiding vector, $|\Psi^+\rangle$,
having strictly positive components, $\Psi^+_i > 0$, and apply a similarity tranformation to the operators $G^{(N)}$ and $T$ having strictly positive components, $\Psi^+_i > 0$, and apply a similarity transformation to the operators $G^{(N)}$ and $T$
\be \be
{\bar T}_{ij}= \frac{\Psi^+_j}{\Psi^+_i} T_{ij} {\bar T}_{ij}= \frac{\Psi^+_j}{\Psi^+_i} T_{ij}
\label{defT} \label{defT}
@ -428,7 +411,7 @@ found, for example, in refs \onlinecite{Foulkes_2001,Kolorenc_2011}.
\section{DMC with domains} \section{DMC with domains}
\label{sec:DMC_domains} \label{sec:DMC_domains}
\subsection{Domains consisting of a single state} \subsection{Domains consisting of a single state}
\label{sub:single_domains} \label{sec:single_domains}
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 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
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 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
fluctuations. This idea was proposed some time ago\cite{assaraf_99,Assaraf_1999B,caffarel_00} and applied to the SU(N) one-dimensional Hubbard model. fluctuations. This idea was proposed some time ago\cite{assaraf_99,Assaraf_1999B,caffarel_00} and applied to the SU(N) one-dimensional Hubbard model.
@ -468,13 +451,13 @@ The time-averaged contribution of the on-state weight can be easily calculated t
\ee \ee
Details of the implementation of the effective dynamics can be in found in Refs. (\onlinecite{assaraf_99},\onlinecite{caffarel_00}). Details of the implementation of the effective dynamics can be in found in Refs. (\onlinecite{assaraf_99},\onlinecite{caffarel_00}).
\subsection{General domains} \subsection{General domains}
\label{sub:general_domains} \label{sec:general_domains}
Let us now extend the results of the preceding section to a general domain. For that, Let us now extend the results of the preceding section to a general domain. For that,
let us associate to each state $|i\rangle$ a set of states, called the domain of $|i\rangle$ and let us associate to each state $|i\rangle$ a set of states, called the domain of $|i\rangle$ and
denoted ${\cal D}_i$, consisting of the state $|i\rangle$ plus a certain number of states. No particular constraints on the type of domains denoted ${\cal D}_i$, consisting of the state $|i\rangle$ plus a certain number of states. No particular constraints on the type of domains
are imposed, for example domains associated with different states can be identical, or may have or not common states. The only important condition is are imposed, for example domains associated with different states can be identical, or may have or not common states. 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 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). In pratice, it is not difficult to impose such a condition. non-zero-probability to reach any other state in a finite number of steps). In practice, it is not difficult to impose such a condition.
Let us write an arbitrary path of length $N$ as Let us write an arbitrary path of length $N$ as
\be \be
@ -489,7 +472,7 @@ where $|I_0\rangle=|i_0\rangle$ is the initial state,
$n_0$ the number of times the walker remains within the domain of $|i_0\rangle$ ($n_0=1$ to $N+1$), $|I_1\rangle$ is the first exit state, $n_0$ the number of times the walker remains within the domain of $|i_0\rangle$ ($n_0=1$ to $N+1$), $|I_1\rangle$ is the first exit state,
that is not belonging to that is not belonging to
${\cal D}_{i_0}$, $n_1$ is the number of times the walker remains within ${\cal D}_{i_1}$ ($n_1=1$ to $N+1-n_0$), $|I_2\rangle$ the second exit state, and so on. ${\cal D}_{i_0}$, $n_1$ is the number of times the walker remains within ${\cal D}_{i_1}$ ($n_1=1$ to $N+1-n_0$), $|I_2\rangle$ the second exit state, and so on.
Here, the integer $p$ goes from 0 to $N$ and indicates the number of exit events occuring along the path. The two extreme cases, $p=0$ and $p=N$, Here, the integer $p$ goes from 0 to $N$ and indicates the number of exit events occurring along the path. The two extreme cases, $p=0$ and $p=N$,
correspond to the cases where the walker remains for ever within the initial domain, and to the case where the walker leaves the current domain at each step, correspond to the cases where the walker remains for ever within the initial domain, and to the case where the walker leaves the current domain at each step,
respectively. respectively.
In what follows, we shall systematically write the integers representing the exit states in capital letter. In what follows, we shall systematically write the integers representing the exit states in capital letter.
@ -556,7 +539,7 @@ t_{I}={\bar n}_{I} \tau= \frac{1}{\Psi^+_{I}} \langle I | P_{I} \frac{1}{H^+ -
In practice, the various quantities restricted to the domain are computed by diagonalizing the matrix $(H^+-E_L^+)$ in ${\cal D}_{I}$. Note that In practice, the various quantities restricted to the domain are computed by diagonalizing the matrix $(H^+-E_L^+)$ in ${\cal D}_{I}$. Note that
it is possible only if the dimension of the domains is not too large (say, less than a few thousands). it is possible only if the dimension of the domains is not too large (say, less than a few thousands).
\subsection{Expressing the Green's matrix using domains} \subsection{Expressing the Green's matrix using domains}
\label{Green} \label{sec:Green}
\subsubsection{Time-dependent Green's matrix} \subsubsection{Time-dependent Green's matrix}
\label{time} \label{time}
In this section we generalize the path-integral expression of the Green's matrix, Eqs.(\ref{G}) and (\ref{cn_stoch}), to the case where domains are used. In this section we generalize the path-integral expression of the Green's matrix, Eqs.(\ref{G}) and (\ref{cn_stoch}), to the case where domains are used.
@ -596,7 +579,7 @@ In that case, $p=N$ and $n_k=1$ for $k=0$ to $N$ and we are left only with the $
= \prod_{k=0}^{N-1} \langle I_k|F_{I_k}|I_{k+1} \rangle $ = \prod_{k=0}^{N-1} \langle I_k|F_{I_k}|I_{k+1} \rangle $
where $F=T$. where $F=T$.
To express the fundamental equation for $G$ under the form of a probabilitic average, we write the importance-sampled version of the equation To express the fundamental equation for $G$ under the form of a probabilistic average, we write the importance-sampled version of the equation
$$ $$
{\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N} + {\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N} +
$$ $$
@ -799,7 +782,7 @@ a tradoff has to be found between the possible bias in the extrapolation and the
required. required.
\section{Numerical application to the Hubbard model} \section{Numerical application to the Hubbard model}
\label{numerical} \label{sec:numerical}
Let us consider the one-dimensional Hubbard Hamiltonian for a chain of $N$ sites Let us consider the one-dimensional Hubbard Hamiltonian for a chain of $N$ sites
\be \be
\hat{H}= -t \sum_{\langle i j\rangle \sigma} \hat{a}^+_{i\sigma} \hat{a}_{j\sigma} \hat{H}= -t \sum_{\langle i j\rangle \sigma} \hat{a}^+_{i\sigma} \hat{a}_{j\sigma}
@ -953,9 +936,7 @@ so that the convergence at large $p$ is reached. The values of $E$ are $-0.780,-
the two-component expression. The estimate of the energy obtained from ${\cal E}(E)=E$ is $-0.76807(5)$ in full agreement with the exact value of $-0.768068...$. the two-component expression. The estimate of the energy obtained from ${\cal E}(E)=E$ is $-0.76807(5)$ in full agreement with the exact value of $-0.768068...$.
\begin{figure}[h!] \begin{figure}[h!]
\begin{center} \includegraphics[width=\columnwidth]{fig1}
\includegraphics[width=\linewidth]{fig1}
\end{center}
\caption{1D-Hubbard model, $N=4$, $U=12$. $H_p$ as a function of $p$ for $E=-1.6,-1.2,-1.,-0.9,-0.8$. $H_0$ is \caption{1D-Hubbard model, $N=4$, $U=12$. $H_p$ as a function of $p$ for $E=-1.6,-1.2,-1.,-0.9,-0.8$. $H_0$ is
computed analytically and $H_p$ (p > 0) by Monte Carlo. Error bars are smaller than the symbol size.} computed analytically and $H_p$ (p > 0) by Monte Carlo. Error bars are smaller than the symbol size.}
\label{fig1} \label{fig1}
@ -963,11 +944,9 @@ computed analytically and $H_p$ (p > 0) by Monte Carlo. Error bars are smaller t
\begin{figure}[h!] \begin{figure}[h!]
\begin{center} \includegraphics[width=\columnwidth]{fig2}
\includegraphics[width=\linewidth]{fig2} \caption{1D-Hubbard model, $N=4$ and $U=12$. $\mathcal{E}(E)$ as a function of $E$.
\end{center} The horizontal and vertical lines are at $\mathcal{E}(E_0)=E_0$ and $E=E_0$, respectively.
\caption{1D-Hubbard model, $N=4$ and $U=12$. ${\cal E}(E)$ as a function of $E$.
The horizontal and vertical lines are at ${\cal E}(E_0)=E_0$ and $E=E_0$, respectively.
$E_0$ is the exact energy of -0.768068.... The dotted line is the two-component extrapolation. $E_0$ is the exact energy of -0.768068.... The dotted line is the two-component extrapolation.
Error bars are smaller than the symbol size.} Error bars are smaller than the symbol size.}
\label{fig2} \label{fig2}
@ -1142,7 +1121,7 @@ $N$ & Size Hilbert space & Domain & Domain size & $\alpha,\beta$ &$\bar{t}_{I_0}
\end{table*} \end{table*}
\section{Summary and perspectives} \section{Summary and perspectives}
\label{conclu} \label{sec:conclu}
In this work it has been shown how to integrate out exactly within a DMC framework the contribution of all In this work it has been shown how to integrate out exactly within a DMC framework the contribution of all
stochastic trajectories trapped in some given domains of the Hilbert space and the corresponding general equations have been derived. stochastic trajectories trapped in some given domains of the Hilbert space and the corresponding general equations have been derived.
In this way a new effective stochastic dynamics connecting only the domains and not the individual states is defined. In this way a new effective stochastic dynamics connecting only the domains and not the individual states is defined.
@ -1180,7 +1159,7 @@ under grant agreement no.~952165.
For the simplest case of a two-state system the fundamental equation (\ref{eqfond}) writes For the simplest case of a two-state system the fundamental equation (\ref{eqfond}) writes
$$ $$
{\cal I}= {\cal I}=
\langle I_0|\frac{1}{H-E}|\Psi\rangle = \langle I_0|P_0\frac{1}{H-E} P_0|Psi\rangle \langle I_0|\frac{1}{H-E}|\Psi\rangle = \langle I_0|P_0\frac{1}{H-E} P_0|\Psi\rangle
+ \sum_{p=1}^{\infty} {\cal I}_p$$ + \sum_{p=1}^{\infty} {\cal I}_p$$
with with
$$ $$

102
plots/H_m0.8_fig1 Normal file
View File

@ -0,0 +1,102 @@
0 -0.33768685411477151 0.0000000000000000
0 -0.33768685411477151 0.0000000000000000
1 -0.21760366032739437 7.3348514269074486E-005
2 -0.30313060501711708 9.5121269826079597E-005
3 -0.19776983982162860 9.0592711207087962E-005
4 -0.29165151429925573 7.7303291787246290E-005
5 -0.18791755994996856 8.1027417133264367E-005
6 -0.28014425897798467 9.9147598045599497E-005
7 -0.17969979723492649 7.4391151679196194E-005
8 -0.26899542232328766 1.3775241236057995E-004
9 -0.17232325082938163 8.6485233426452439E-005
10 -0.25792763231321458 1.0402902952884553E-004
11 -0.16526643579833439 8.6174643195279473E-005
12 -0.24732332345389324 1.4608505216936497E-004
13 -0.15849641950502730 7.9870792795755916E-005
14 -0.23726527612173473 1.6578022951260120E-004
15 -0.15198265939914093 9.7424557864468332E-005
16 -0.22765721104766984 1.4237627139259898E-004
17 -0.14585713094432423 7.4127196358804247E-005
18 -0.21820725207805927 1.7516278000120542E-004
19 -0.13984639694944118 9.4535910190667169E-005
20 -0.20933618308662477 1.5816390390028823E-004
21 -0.13405980330910117 9.6945175289597590E-005
22 -0.20073091471291066 1.5062446664491962E-004
23 -0.12867030912207097 1.1342468999962055E-004
24 -0.19254385730389484 1.4685318046811410E-004
25 -0.12336844407386201 8.6495243909515704E-005
26 -0.18464417123281981 1.5015992286256227E-004
27 -0.11836494239493874 9.3768675023244794E-005
28 -0.17707140802899698 1.6151224299934780E-004
29 -0.11350112510342254 1.2079620942749168E-004
30 -0.16999126044241455 1.4592271724527244E-004
31 -0.10883888605954921 9.0016411562122029E-005
32 -0.16289429122281113 1.4026745892932623E-004
33 -0.10437684032995624 7.1953004752193980E-005
34 -0.15628071190700599 1.4259759092535019E-004
35 -0.10004146738625704 8.2424555802618280E-005
36 -0.14983639975434820 1.1901873444764650E-004
37 -9.5961845949245630E-002 6.8761956064047322E-005
38 -0.14375712860868062 1.1401897119065882E-004
39 -9.2078949826651074E-002 9.7115659804331540E-005
40 -0.13778630679809523 1.0892436067552820E-004
41 -8.8274449152963549E-002 8.5687392937495185E-005
42 -0.13217126718441691 1.4381581275061027E-004
43 -8.4664025472795140E-002 8.1029613197395502E-005
44 -0.12686719949298042 1.3284513012535166E-004
45 -8.1220979905887433E-002 7.6016400470732231E-005
46 -0.12150242499214618 1.3631390476605116E-004
47 -7.7799809209691984E-002 7.3885715938010592E-005
48 -0.11653021691261023 1.2012451314992711E-004
49 -7.4617462047346594E-002 8.7276657130191010E-005
50 -0.11181057642174722 1.1152597014946340E-004
51 -7.1642260100951288E-002 8.4557947019937564E-005
52 -0.10732031774292525 1.1919613921908822E-004
53 -6.8768445408819864E-002 8.1909856018789521E-005
54 -0.10293299098913881 1.1382558712177023E-004
55 -6.5947373922051120E-002 7.5279021164832753E-005
56 -9.8767132675845179E-002 1.0656154021874254E-004
57 -6.3226179213323283E-002 7.2422193579596518E-005
58 -9.4650953745398864E-002 1.0520300250630444E-004
59 -6.0592926566379615E-002 7.1117741219494518E-005
60 -9.0801355063662076E-002 1.0120333932630714E-004
61 -5.8166202171993933E-002 6.0044924430298955E-005
62 -8.7091291135095700E-002 1.1159701342765535E-004
63 -5.5749769475729682E-002 6.9922449919220369E-005
64 -8.3448792260772842E-002 1.0522820902709696E-004
65 -5.3416097428439280E-002 6.5867208024714789E-005
66 -8.0055501498096857E-002 1.2269714508167731E-004
67 -5.1217931495312820E-002 6.9078764823029146E-005
68 -7.6737506386577198E-002 9.1943788251627189E-005
69 -4.9119671270550391E-002 7.8720448762423264E-005
70 -7.3593584077912444E-002 9.7250729663106146E-005
71 -4.7141983429688498E-002 5.5377925433190131E-005
72 -7.0620629762652959E-002 1.0023771067150202E-004
73 -4.5321070049070114E-002 4.3601622096155715E-005
74 -6.7731838458925636E-002 8.8226315845087814E-005
75 -4.3395412894418145E-002 5.3937341099558384E-005
76 -6.5047019578746451E-002 9.5705191655180583E-005
77 -4.1595029138835907E-002 5.9608421622097094E-005
78 -6.2297910015713387E-002 1.1744779793088422E-004
79 -3.9927483438601051E-002 5.2561482512654839E-005
80 -5.9776287150985742E-002 9.2337118584136685E-005
81 -3.8332672424358540E-002 6.1966289224530592E-005
82 -5.7331120632202304E-002 1.1262181286669684E-004
83 -3.6724929819147045E-002 7.2037195723777004E-005
84 -5.4977549329548681E-002 9.6410417745686926E-005
85 -3.5254135673762024E-002 6.5717066108312639E-005
86 -5.2775171031882416E-002 1.0260738547698916E-004
87 -3.3830163866572172E-002 6.8679864106531321E-005
88 -5.0498837890432857E-002 9.0310866480494147E-005
89 -3.2362557409094196E-002 6.5656193887833939E-005
90 -4.8445373432756046E-002 9.8805777622051861E-005
91 -3.1014884904270134E-002 6.6276920065462864E-005
92 -4.6505033652946837E-002 9.8855622269780776E-005
93 -2.9780592295147977E-002 7.0891354648583866E-005
94 -4.4575812177607310E-002 1.0160494035635478E-004
95 -2.8512064740996990E-002 7.2491132000754654E-005
96 -4.2766612201251872E-002 8.4948361329025444E-005
97 -2.7361692554253504E-002 6.1356604436510595E-005
98 -4.0872009335771585E-002 8.1303128090534670E-005
99 -2.6238159524272588E-002 6.2097851108514461E-005
100 -3.9286130783933539E-002 8.0466763132744333E-005

102
plots/H_m0.9_fig1 Normal file
View File

@ -0,0 +1,102 @@
0 -0.30016609254646365 1.2735131497718755E-017
0 -0.30016609254646359 0.0000000000000000
1 -0.19192605355332870 6.4693263231019947E-005
2 -0.23657652372389060 7.8444224843353052E-005
3 -0.15412179110014998 7.1720616275217501E-005
4 -0.20162534507771138 5.6118769807352067E-005
5 -0.12971625620853650 5.7506511678928538E-005
6 -0.17158814975943504 6.4044298847299201E-005
7 -0.10988067015690270 4.6650408292995979E-005
8 -0.14596500486248737 7.9521041214521040E-005
9 -9.3348825460471005E-002 4.8277219103398136E-005
10 -0.12398201434375071 5.2132525967110484E-005
11 -7.9307943570408726E-002 4.1957305771303458E-005
12 -0.10531287776902858 6.6027045304442608E-005
13 -6.7376684993415786E-002 3.4775048814059726E-005
14 -8.9503174519872192E-002 6.6848942213262904E-005
15 -5.7232812542546141E-002 3.8491986125725953E-005
16 -7.6079098466638823E-002 5.0428895119299008E-005
17 -4.8659564829665991E-002 2.6513375269140156E-005
18 -6.4592545679425878E-002 5.4987252166287370E-005
19 -4.1327027520479072E-002 2.8972830500472514E-005
20 -5.4898239230251653E-002 4.4138799792023181E-005
21 -3.5094881692752947E-002 2.6595347130366256E-005
22 -4.6633985343815132E-002 3.6514453387242215E-005
23 -2.9842715027168489E-002 2.7346071567557133E-005
24 -3.9626693326399308E-002 3.2004357276842206E-005
25 -2.5345694204830422E-002 1.8376882565322658E-005
26 -3.3663568160899653E-002 2.8390728917164872E-005
27 -2.1543542048227979E-002 1.8297928618608534E-005
28 -2.8598013378335195E-002 2.8097162661918815E-005
29 -1.8300252112767090E-002 2.0273936737379241E-005
30 -2.4322303460703795E-002 2.1857917359310102E-005
31 -1.5545061635248103E-002 1.3560306698855363E-005
32 -2.0645859801000775E-002 1.8685459779283190E-005
33 -1.3206005778632254E-002 9.6720323910849635E-006
34 -1.7547673613696628E-002 1.7215814118890300E-005
35 -1.1212241646457529E-002 9.7697379053409134E-006
36 -1.4902086402121231E-002 1.2800094543805626E-005
37 -9.5276071236610766E-003 7.0464653615846114E-006
38 -1.2667136458420774E-002 1.0810771058306354E-005
39 -8.0988348437147448E-003 9.1778969016704379E-006
40 -1.0753574029418716E-002 9.2518340881499613E-006
41 -6.8788910275917742E-003 7.3804330603091786E-006
42 -9.1392946277924318E-003 1.0765454451599525E-005
43 -5.8434604908128244E-003 6.1029165541253229E-006
44 -7.7728814210678053E-003 8.7862903346228444E-006
45 -4.9670754589932805E-003 5.1121306700138458E-006
46 -6.5923880797842411E-003 8.0473507873383072E-006
47 -4.2138257162904445E-003 4.2236660533459980E-006
48 -5.6015152788397625E-003 6.1643987502942225E-006
49 -3.5801394564210173E-003 4.2967704746595603E-006
50 -4.7609478628567233E-003 4.8809379739655495E-006
51 -3.0451536753422050E-003 3.6188439211370041E-006
52 -4.0480899458627765E-003 4.6593746978566373E-006
53 -2.5896164948364057E-003 3.1563626306825435E-006
54 -3.4400313282858174E-003 3.8725086684585543E-006
55 -2.1998376322817587E-003 2.4451495658803682E-006
56 -2.9233328033801834E-003 3.1868847133210753E-006
57 -1.8681004840119340E-003 2.1692783294432064E-006
58 -2.4815653007123329E-003 2.7362575527607768E-006
59 -1.5857862817333511E-003 1.9073251670612535E-006
60 -2.1088008695809768E-003 2.3518539999133073E-006
61 -1.3488874019370258E-003 1.4162258958626199E-006
62 -1.7916966627105977E-003 2.3768754435158329E-006
63 -1.1451903943283926E-003 1.4694775322062126E-006
64 -1.5213055159474687E-003 1.9203014463293078E-006
65 -9.7204583339410258E-004 1.2182433147155669E-006
66 -1.2926068686298373E-003 1.9921749185140804E-006
67 -8.2552393279161271E-004 1.1550872267785537E-006
68 -1.0975463344166643E-003 1.2645151166945428E-006
69 -7.0118339532185704E-004 1.1392261208878685E-006
70 -9.3235269298005442E-004 1.3017225371100087E-006
71 -5.9626219852185040E-004 7.1268902987004004E-007
72 -7.9283049519827020E-004 1.1487144933529473E-006
73 -5.0779283875790023E-004 5.0728059755584994E-007
74 -6.7342672861163595E-004 9.4024918670002046E-007
75 -4.3075854693075372E-004 5.6585702063145195E-007
76 -5.7305169446623223E-004 9.0436532308608414E-007
77 -3.6571466781761393E-004 5.6070064471920889E-007
78 -4.8603921557061243E-004 9.3352574644413807E-007
79 -3.1095521178021464E-004 4.1513008018843970E-007
80 -4.1318999801435530E-004 6.4837662945382356E-007
81 -2.6451550081604282E-004 4.4707835914605085E-007
82 -3.5107724345373334E-004 7.0746618774484791E-007
83 -2.2451186390893681E-004 4.7510289930723102E-007
84 -2.9820726752705469E-004 5.4504380021006431E-007
85 -1.9089654193492993E-004 3.8605066838279815E-007
86 -2.5358220255730093E-004 5.1525580555595669E-007
87 -1.6223364141255216E-004 3.3579586220178861E-007
88 -2.1487928333189824E-004 3.9631274062396933E-007
89 -1.3750659661597683E-004 2.8165294517751828E-007
90 -1.8267365130366644E-004 3.9778783077861972E-007
91 -1.1673989753043384E-004 2.6769889236457950E-007
92 -1.5535580372428386E-004 3.7945632185824568E-007
93 -9.9299770521424035E-005 2.6398348747898608E-007
94 -1.3190149400144391E-004 3.3219861672295356E-007
95 -8.4215069671523521E-005 2.1472946787620809E-007
96 -1.1211271199570519E-004 2.4658118951492859E-007
97 -7.1595379295494923E-005 1.7129285826354936E-007
98 -9.4865235491646413E-005 1.9248391555934189E-007
99 -6.0839681576678517E-005 1.4901609628594746E-007
100 -8.0816334625495627E-005 1.7142954827081591E-007

102
plots/H_m1.0_fig1 Normal file
View File

@ -0,0 +1,102 @@
0 -0.27014948329181721 0.0000000000000000
0 -0.27014948329181721 0.0000000000000000
1 -0.17140472936581536 5.7776060470249295E-005
2 -0.18930138340253552 6.6157194278073620E-005
3 -0.12314045978186812 5.8297969997645534E-005
4 -0.14469412752862287 4.2401881606299597E-005
5 -9.2949427798302259E-002 4.2470500013762028E-005
6 -0.11046234642264763 4.3714672527601725E-005
7 -7.0618305071547122E-002 3.0987907332799199E-005
8 -8.4288960893518242E-002 4.8818078611948899E-005
9 -5.3814091928964206E-002 2.8831795694083941E-005
10 -6.4215180327410831E-002 2.8245451959173582E-005
11 -4.1007902373304081E-002 2.2202925947586440E-005
12 -4.8923142926584756E-002 3.2685191564443551E-005
13 -3.1247668980299098E-002 1.6619662215822144E-005
14 -3.7295576004409650E-002 2.9813321289070389E-005
15 -2.3807446096318814E-002 1.6870014853033970E-005
16 -2.8435634680536854E-002 2.0062683318949115E-005
17 -1.8156175325433087E-002 1.0600737814105856E-005
18 -2.1652360961388957E-002 1.9492555983787091E-005
19 -1.3830140569412069E-002 1.0106006058615165E-005
20 -1.6507506351294648E-002 1.4164641582161474E-005
21 -1.0534057576183364E-002 8.4553598876133995E-006
22 -1.2577590658268207E-002 1.0359952132031018E-005
23 -8.0352816391457728E-003 7.7611846920052290E-006
24 -9.5862455660019087E-003 8.3060741016008639E-006
25 -6.1207576812953753E-003 4.6647824755308998E-006
26 -7.3043790835895447E-003 6.4723657796259664E-006
27 -4.6667079275016302E-003 4.2787209983628100E-006
28 -5.5656557869080796E-003 5.9473802754667555E-006
29 -3.5555722346460516E-003 4.1321380431317922E-006
30 -4.2459812719340189E-003 4.0424299807685888E-006
31 -2.7088950440397283E-003 2.5225324953455609E-006
32 -3.2326311105026970E-003 3.1013439562794595E-006
33 -2.0640828656008398E-003 1.6161311368021520E-006
34 -2.4645107528239113E-003 2.6303043999456632E-006
35 -1.5718041721516913E-003 1.4651742746765543E-006
36 -1.8769924781585484E-003 1.7727419367817223E-006
37 -1.1979955840386689E-003 9.2936364645443233E-007
38 -1.4312606791622548E-003 1.3323680554902328E-006
39 -9.1341776508336767E-004 1.1161275633869637E-006
40 -1.0896529506785966E-003 1.0225452876490816E-006
41 -6.9598972133875733E-004 8.2369408017606095E-007
42 -8.3076495152398894E-004 1.0621159190381908E-006
43 -5.3019606318554136E-004 6.0231631652066696E-007
44 -6.3388913966123419E-004 7.7060217062121080E-007
45 -4.0432115421180274E-004 4.5770657671282763E-007
46 -4.8203721718967943E-004 6.4094045931501207E-007
47 -3.0758170758367635E-004 3.2577074081648928E-007
48 -3.6743099799689988E-004 4.3532062940944923E-007
49 -2.3439494631291580E-004 2.8777554808162057E-007
50 -2.8008902146100503E-004 2.9855723911858470E-007
51 -1.7882881520616881E-004 2.1368987671421249E-007
52 -2.1360752010762666E-004 2.5580098798584368E-007
53 -1.3642218788703777E-004 1.7184529925359646E-007
54 -1.6284521484645278E-004 1.8735174171063471E-007
55 -1.0394032390493785E-004 1.1100905862250214E-007
56 -1.2408872869917111E-004 1.3880202559351285E-007
57 -7.9156475545743679E-005 9.4787887257574767E-008
58 -9.4472761759906157E-005 1.0548020542145087E-007
59 -6.0261169752965541E-005 7.4514451758036912E-008
60 -7.2001717027039567E-005 8.2141125760520427E-008
61 -4.5990438530043311E-005 4.9776194583716203E-008
62 -5.4868407216719035E-005 7.6923329058789945E-008
63 -3.5019297041177359E-005 4.6673299696985345E-008
64 -4.1803705219721920E-005 5.3533157309257480E-008
65 -2.6661742224995719E-005 3.4691637845413219E-008
66 -3.1851385416680935E-005 5.0669839436925807E-008
67 -2.0306223152589325E-005 3.0262330091119592E-008
68 -2.4256927797145250E-005 2.7686100720725078E-008
69 -1.5466062192056640E-005 2.6261232980301952E-008
70 -1.8479407817812662E-005 2.8331917902446073E-008
71 -1.1799321435157268E-005 1.4971272451717717E-008
72 -1.4100405234229030E-005 2.1845593364053628E-008
73 -9.0118064605158558E-006 1.0256236412963806E-008
74 -1.0738877587682335E-005 1.7094340485083609E-008
75 -6.8581068234648139E-006 1.0093748522424016E-008
76 -8.1984475518816380E-006 1.4477361275561628E-008
77 -5.2213565454666016E-006 8.9094105308453739E-009
78 -6.2345555196654330E-006 1.2718747045868444E-008
79 -3.9813694920953924E-006 5.7661431562963816E-009
80 -4.7545936260137411E-006 8.1496959680697307E-009
81 -3.0384132418627167E-006 5.5629857814240963E-009
82 -3.6236334818501281E-006 7.7799133982528983E-009
83 -2.3133356071047361E-006 5.4613205105411841E-009
84 -2.7602866798492989E-006 5.4429445169010287E-009
85 -1.7640023236981123E-006 4.0276717710473842E-009
86 -2.1053577511635629E-006 4.6807664175670148E-009
87 -1.3440063615780071E-006 2.9701093523532143E-009
88 -1.5995523046784525E-006 3.1845094615621520E-009
89 -1.0221400477255824E-006 2.1769132031273092E-009
90 -1.2202318339394027E-006 2.9386939057106659E-009
91 -7.7839882081539054E-007 1.9809695401495311E-009
92 -9.3086720825568093E-007 2.6625803564672424E-009
93 -5.9383311722569277E-007 1.8106454367192415E-009
94 -7.0878595051901198E-007 2.0299458675965006E-009
95 -4.5171344965796612E-007 1.1735886917570485E-009
96 -5.4042195064170321E-007 1.3734945909744059E-009
97 -3.4447751625848848E-007 9.1344413085181797E-010
98 -4.0994142873779791E-007 8.8526331121507855E-010
99 -2.6267056981626724E-007 6.8844325383762986E-010
100 -3.1340710493498088E-007 7.2953305302812118E-010

102
plots/H_m1.2_fig1 Normal file
View File

@ -0,0 +1,102 @@
0 -0.22512456940984768 0.0000000000000000
0 -0.22512456940984768 0.0000000000000000
1 -0.14067307334281831 4.7417221345793050E-005
2 -0.12833139406402125 4.9471529731521528E-005
3 -8.3227583940345559E-002 4.0920779217007719E-005
4 -8.1167547791933789E-002 2.6444482021628751E-005
5 -5.1984259782251796E-002 2.5344369647954739E-005
6 -5.1301037066764622E-002 2.3017141939963990E-005
7 -3.2686591843926902E-002 1.5575453302662377E-005
8 -3.2404947168099989E-002 2.1105956651888442E-005
9 -2.0619514345668902E-002 1.1989040991642458E-005
10 -2.0433428976157429E-002 9.8930101992867294E-006
11 -1.3005152078838580E-002 7.5261283144631261E-006
12 -1.2884543992030354E-002 9.8254839322239292E-006
13 -8.2021206102574733E-003 4.7031667230812608E-006
14 -8.1306879686248550E-003 7.4352202132655380E-006
15 -5.1723341020327631E-003 4.0938643548858033E-006
16 -5.1314155541250335E-003 4.1261631096251640E-006
17 -3.2652839376218308E-003 2.1776800026390915E-006
18 -3.2334506940823052E-003 3.2251179221161492E-006
19 -2.0583909092800499E-003 1.6516495952150413E-006
20 -2.0407555180908406E-003 2.0153060002105776E-006
21 -1.2976482141769340E-003 1.2061877150946047E-006
22 -1.2870423234407462E-003 1.2022944924263911E-006
23 -8.1944292721275480E-004 9.1540834215394012E-007
24 -8.1191716623930617E-004 8.3262546532125898E-007
25 -5.1658796503039283E-004 4.5157285162551359E-007
26 -5.1204582430876863E-004 5.1809110774921027E-007
27 -3.2604476591963289E-004 3.5179386343255615E-007
28 -3.2291438470220628E-004 4.1445892894766046E-007
29 -2.0560489614741107E-004 2.6813590477215892E-007
30 -2.0393743088631955E-004 2.2362448852035152E-007
31 -1.2964381674734847E-004 1.3984506249507430E-007
32 -1.2850784490177269E-004 1.4219777862103543E-007
33 -8.1758922516707622E-005 7.3658075870851571E-008
34 -8.1102993894084083E-005 1.0492409168539331E-007
35 -5.1532405585205565E-005 5.6055473883228700E-008
36 -5.1108318502822870E-005 5.9750022916369399E-008
37 -3.2507982646598835E-005 2.8768324212802841E-008
38 -3.2266960346566409E-005 3.6562586929900379E-008
39 -2.0516517373226253E-005 2.9233963885058851E-008
40 -2.0326434719285795E-005 2.2728092970884407E-008
41 -1.2945134861935793E-005 1.8397293433210792E-008
42 -1.2831113880463161E-005 1.9452294841646733E-008
43 -8.1589284849549368E-006 1.0796344978142572E-008
44 -8.1079423122332365E-006 1.1228897815418221E-008
45 -5.1522038987283331E-006 6.9491251666401699E-009
46 -5.0990563126083610E-006 8.0461176547616321E-009
47 -3.2425601912733119E-006 3.8407783902585975E-009
48 -3.2188372410363160E-006 4.5884472172027045E-009
49 -2.0453781091460065E-006 2.6639329028648218E-009
50 -2.0304062935141502E-006 2.4786424694605638E-009
51 -1.2916946278912489E-006 1.5898526435080612E-009
52 -1.2817708939428670E-006 1.7119349950860918E-009
53 -8.1596427695404953E-007 1.2049423276626244E-009
54 -8.0917269740462677E-007 1.0461539527350636E-009
55 -5.1449954121455772E-007 4.9442609339621063E-010
56 -5.1001984829308440E-007 6.5596619370333128E-010
57 -3.2416422477760185E-007 4.4402479564349884E-010
58 -3.2130207626398323E-007 4.1133754966328134E-010
59 -2.0418493058507266E-007 2.7278485052998559E-010
60 -2.0262957380905720E-007 2.6107630201899273E-010
61 -1.2908051177894808E-007 1.5379386690812940E-010
62 -1.2780592353471635E-007 2.1762140110475156E-010
63 -8.1353594316775577E-008 1.2400453750570241E-010
64 -8.0679932361223695E-008 1.0973334002492716E-010
65 -5.1262670883436863E-008 7.7488288858348612E-011
66 -5.0854439277693701E-008 9.6261593538386032E-011
67 -3.2306895644468852E-008 5.9106021140240744E-011
68 -3.2056333368696008E-008 4.1913702652936463E-011
69 -2.0352617715441651E-008 4.0544812418381067E-011
70 -2.0202221913081458E-008 4.0008095438956431E-011
71 -1.2860870244367662E-008 2.0866643873736579E-011
72 -1.2772871731173527E-008 2.5400519432973823E-011
73 -8.1243673625679052E-009 1.3359562594074142E-011
74 -8.0444742432727441E-009 1.8254562151622072E-011
75 -5.1211726159462146E-009 1.0305508865311268E-011
76 -5.0856634841383041E-009 1.2034683614106763E-011
77 -3.2247758416148209E-009 7.3813331667399839E-012
78 -3.1978034507004199E-009 8.2434737162888721E-012
79 -2.0340599156811150E-009 3.9158956732037275E-012
80 -2.0191818565756725E-009 4.7549444935146577E-012
81 -1.2855015851218641E-009 2.9748981738388690E-012
82 -1.2737125403394305E-009 3.4341261571493877E-012
83 -8.1032088147188636E-010 2.4993459535001295E-012
84 -8.0265364508145725E-010 1.9478609347149748E-012
85 -5.1128607656560815E-010 1.5372712242327781E-012
86 -5.0682844336277092E-010 1.4708994189271958E-012
87 -3.2185591714855973E-010 9.2779034497571892E-013
88 -3.1838720170554392E-010 8.1124407584844357E-013
89 -2.0291429148255577E-010 4.9892641520473018E-013
90 -2.0130933747159448E-010 6.3434150963542191E-013
91 -1.2794484699788354E-010 4.2274428392241788E-013
92 -1.2712404568479556E-010 5.0103260441620614E-013
93 -8.0762491877278106E-011 3.3730642362543232E-013
94 -8.0082455454848161E-011 3.0494149681538096E-013
95 -5.0848758193274283E-011 1.4538252269730179E-013
96 -5.0559198366823712E-011 1.7897196862873705E-013
97 -3.2108850988129618E-011 1.1165701868958913E-013
98 -3.1713247227790270E-011 8.8480901162655900E-014
99 -2.0288035423398922E-011 6.8158353003770159E-014
100 -2.0090036316644218E-011 6.6347490335420047E-014

101
plots/H_m1.6_fig1 Normal file
View File

@ -0,0 +1,101 @@
0 -0.16884342705738575 0.0000000000000000
1 -0.10240172250695223 3.4516947891265065E-005
2 -6.8882071392691369E-002 3.1533614090075040E-005
3 -4.4394991185261842E-002 2.3736816348283004E-005
4 -3.2209690450591119E-002 1.2881931981059281E-005
5 -2.0505105099309712E-002 1.1440875556603347E-005
6 -1.5070609371341881E-002 8.7125882148917241E-006
7 -9.5381426229961454E-003 5.5263201493275689E-006
8 -7.0456000398059996E-003 5.6603594583081482E-006
9 -4.4535746090049290E-003 3.1134189617294586E-006
10 -3.2874304340441639E-003 1.9526000629760129E-006
11 -2.0783290701546072E-003 1.4386774927678127E-006
12 -1.5337334483624258E-003 1.5173918688880039E-006
13 -9.6990068127071844E-004 6.7172626617113152E-007
14 -7.1631309954608752E-004 8.4005236872561313E-007
15 -4.5257893729886681E-004 4.4890101035835080E-007
16 -3.3458398127653941E-004 3.4950701266441743E-007
17 -2.1146956421661548E-004 1.8105199144674772E-007
18 -1.5593276656595111E-004 1.8509290397804983E-007
19 -9.8604468924524787E-005 9.9500778414446935E-008
20 -7.2853935447657691E-005 1.0031614207591647E-007
21 -4.5994520598857135E-005 6.3579617421676855E-008
22 -3.4000518192572172E-005 4.4396093356846513E-008
23 -2.1497437042167478E-005 3.6093260321460479E-008
24 -1.5869874783670137E-005 2.3874073638792534E-008
25 -1.0026299766665607E-005 1.2260205630014703E-008
26 -7.4052822045126645E-006 1.0543126421296120E-008
27 -4.6835563173883834E-006 6.9886948865377357E-009
28 -3.4547349818755444E-006 6.4290662683611469E-009
29 -2.1853854058259924E-006 3.8887038056153495E-009
30 -1.6155559071271601E-006 2.5379298099067258E-009
31 -1.0195761435313238E-006 1.4956682710006772E-009
32 -7.5339933788357934E-007 1.2889242887748707E-009
33 -4.7567096004115120E-007 6.0516205723232169E-010
34 -3.5200146385133285E-007 7.3193759172561133E-010
35 -2.2194513665001587E-007 3.5313087813548784E-010
36 -1.6398174504155940E-007 3.0642696121253454E-010
37 -1.0356348854110525E-007 1.4067120995580961E-010
38 -7.6659900744082016E-008 1.4128978864901443E-010
39 -4.8374309620872453E-008 9.8328313777568827E-011
40 -3.5710345162333792E-008 6.1561007206134596E-011
41 -2.2615597780721738E-008 4.5854328237409120E-011
42 -1.6693909305179762E-008 3.8387710075834696E-011
43 -1.0535632762520894E-008 2.1532635475094437E-011
44 -7.8169764749281033E-009 1.4560365044969765E-011
45 -4.9286774930814858E-009 9.0296153397746511E-012
46 -3.6308246041814847E-009 8.4419900313993387E-012
47 -2.2942689379433804E-009 4.7516631556758728E-012
48 -1.7007634097834043E-009 5.5728104802745101E-012
49 -1.0714799721986820E-009 2.7177542600409106E-012
50 -7.9280393320348484E-010 2.2472362597934371E-012
51 -5.0059618374067312E-010 1.1544850022622903E-012
52 -3.7061475717249471E-010 9.4150065123891415E-013
53 -2.3457218156500420E-010 9.7074693885070933E-013
54 -1.7341609697972443E-010 6.0407802780072769E-013
55 -1.0935027631016161E-010 2.2045291061156055E-013
56 -8.0752246106058936E-011 2.6426226079611277E-013
57 -5.0908788203988633E-011 1.4308401342483215E-013
58 -3.7588054217059001E-011 1.1627068133611037E-013
59 -2.3690393747755766E-011 5.2545973410313896E-014
60 -1.7523406899260708E-011 4.7989283078098191E-014
61 -1.1107120817133203E-011 2.3450500578389463E-014
62 -8.1813131403768924E-012 3.1345705650281625E-014
63 -5.1853006069294439E-012 1.5959707598717458E-014
64 -3.8321562084391296E-012 1.1814349944728047E-014
65 -2.4154581505906950E-012 7.5899444425231823E-015
66 -1.7857979579368521E-012 7.5424972178017179E-015
67 -1.1257362419764545E-012 4.0718694242091225E-015
68 -8.3299509562516693E-013 2.6907559933748514E-015
69 -5.2360409109681268E-013 1.7060816536983771E-015
70 -3.8757906108015929E-013 1.4185067721048245E-015
71 -2.4546366845933088E-013 9.3141601319567295E-016
72 -1.8189167206986861E-013 7.1758043164047327E-016
73 -1.1429490116278992E-013 3.7786271140822010E-016
74 -8.4529106052488852E-014 4.0501058636475411E-016
75 -5.3483372313257633E-014 2.1408396690760941E-016
76 -3.9577038631407705E-014 1.8532249368215504E-016
77 -2.4846695967402978E-014 1.1322786243839362E-016
78 -1.8357773454263101E-014 9.2729541569632598E-017
79 -1.1565744970522853E-014 4.4309124164619045E-017
80 -8.5817390730824005E-015 4.2093242380516575E-017
81 -5.4171792787636975E-015 2.3764663948707981E-017
82 -4.0046118636100735E-015 2.0762752060828060E-017
83 -2.5295656485257986E-015 1.3449214789896888E-017
84 -1.8634340358468111E-015 7.5821606605432950E-018
85 -1.1804315419559570E-015 5.7664688395998897E-018
86 -8.7191067665999900E-016 4.8392429747616713E-018
87 -5.4501211093362523E-016 3.7628087703131418E-018
88 -4.0373761344715064E-016 1.9365349501222389E-018
89 -2.5624949417971153E-016 1.0685229418555911E-018
90 -1.8983829934561817E-016 1.1610560478768992E-018
91 -1.1972287787490124E-016 6.9530600496709315E-019
92 -8.8732022285578295E-017 6.4834388493039168E-019
93 -5.5844345378711899E-017 4.6510511429467828E-019
94 -4.1278511360039205E-017 2.8837690333070259E-019
95 -2.6006954655976249E-017 1.3475675702975903E-019
96 -1.9321022886536565E-017 1.3530183670855149E-019
97 -1.2176596839798274E-017 8.1596934158184900E-020
98 -8.9549811386537832E-018 5.9980902651350810E-020
99 -5.7122138705206385E-018 4.3233903717636202E-020
100 -4.2047751363061788E-018 3.5464871667060900E-020

101
plots/H_m1.8_fig1 Normal file
View File

@ -0,0 +1,101 @@
0 -0.15008304627323180 0.0000000000000000
1 -8.9704568508453042E-002 3.0237068684151705E-005
2 -5.3198376733895901E-002 2.6271374048689459E-005
3 -3.4177100552428029E-002 1.9088598411711318E-005
4 -2.1951830160013189E-002 9.6510612226051610E-006
5 -1.3932752472089343E-002 8.3058456947683928E-006
6 -9.0706811942826392E-003 5.9109053917784065E-006
7 -5.7216500718837108E-003 3.6699925442023071E-006
8 -3.7446023992357426E-003 3.3033188772138762E-006
9 -2.3592579608772766E-003 1.8123956195975867E-006
10 -1.5427452619322106E-003 1.0202174398446820E-006
11 -9.7204757261821410E-004 7.4312946638715333E-007
12 -6.3548806075220928E-004 7.1056336238671755E-007
13 -4.0054451063465730E-004 3.0782894721957425E-007
14 -2.6208781830648869E-004 3.4407043517170059E-007
15 -1.6503149950569828E-004 1.8319514339412023E-007
16 -1.0810485181302386E-004 1.2823060437843481E-007
17 -6.8095831847503807E-005 6.6026454108134176E-008
18 -4.4473058601454302E-005 5.7095650843167961E-008
19 -2.8029911891977868E-005 3.2630018566387077E-008
20 -1.8351347835455677E-005 3.0539988261250971E-008
21 -1.1544119721718054E-005 2.0037971678379027E-008
22 -7.5622386253917485E-006 1.1972826111152724E-008
23 -4.7642555958773756E-006 1.0081778826324582E-008
24 -3.1163400853506684E-006 5.6828823960017059E-009
25 -1.9618729867081245E-006 2.8346215699166620E-009
26 -1.2839030199900646E-006 2.1911746784487507E-009
27 -8.0924416509199289E-007 1.4070437869245217E-009
28 -5.2875453634494936E-007 1.1731836062423818E-009
29 -3.3342901812866973E-007 7.2923576910497314E-010
30 -2.1845647519393818E-007 4.3261528533375821E-010
31 -1.3736513026669937E-007 2.4373537334048250E-010
32 -8.9977429599297707E-008 2.0977962001291938E-010
33 -5.6572252371113106E-008 9.4074804425301140E-011
34 -3.7132124344778598E-008 1.0409735907697210E-010
35 -2.3318563027606245E-008 4.7917893572569747E-011
36 -1.5264583768869108E-008 3.7682371374687095E-011
37 -9.6039150086823248E-009 1.7817984138255096E-011
38 -6.3026906606442388E-009 1.5937970233973316E-011
39 -3.9614625943771158E-009 1.0242680243011378E-011
40 -2.5913357310920749E-009 6.1167077744664104E-012
41 -1.6366970818293399E-009 4.1073626210711258E-012
42 -1.0702314905860721E-009 3.2226386745997202E-012
43 -6.7266026987729147E-010 2.0465438953133099E-012
44 -4.4286277288100007E-010 1.0706745332799704E-012
45 -2.7807366462356560E-010 6.1783653475965821E-013
46 -1.8146731763371916E-010 5.5573596826886513E-013
47 -1.1435296069345295E-010 4.1904283120608149E-013
48 -7.5267546852551154E-011 4.5024857470471981E-013
49 -4.7184657400907662E-011 2.1578054517813879E-013
50 -3.0936408304304696E-011 1.6038832973741819E-013
51 -1.9455126438575129E-011 7.8846222350128401E-014
52 -1.2779735213142957E-011 5.5462635465781107E-014
53 -8.0718485243300040E-012 6.2163718086494127E-014
54 -5.2883997030195741E-012 3.4667805399459097E-014
55 -3.3185691047810696E-012 1.3619997365625801E-014
56 -2.1723290098877423E-012 1.3103770208909739E-014
57 -1.3628179042276291E-012 6.4390844167210875E-015
58 -8.9146494134291890E-013 4.8942261680363475E-015
59 -5.5913747324783894E-013 2.0568900156945681E-015
60 -3.6666943026215800E-013 1.7998950621470222E-015
61 -2.3181934242245154E-013 8.2986512096322382E-016
62 -1.5122033854271536E-013 9.7066557833234613E-016
63 -9.5683396210251686E-014 4.8929261225029992E-016
64 -6.2640824621835110E-014 3.7313018578341281E-016
65 -3.9312972397843879E-014 2.0312372621026180E-016
66 -2.5768666963108078E-014 1.7601418142719491E-016
67 -1.6169188661292848E-014 8.7915949180846463E-017
68 -1.0610799544896232E-014 5.7910818232216244E-017
69 -6.6291935906038100E-015 2.9935445814261517E-017
70 -4.3513540147834491E-015 2.2814767101254601E-017
71 -2.7497664667072149E-015 1.6810147168671253E-017
72 -1.8064196518811890E-015 1.0306307393475669E-017
73 -1.1264997239998418E-015 5.1938293708687849E-018
74 -7.3973725154285087E-016 5.1005832094486157E-018
75 -4.6667943048438547E-016 2.6682234686897492E-018
76 -3.0595421227137120E-016 2.0584416146233118E-018
77 -1.9097977087434586E-016 1.2237866150340573E-018
78 -1.2504618550956113E-016 8.9380889860746889E-019
79 -7.8311627882144597E-017 4.2424491922434885E-019
80 -5.1612987325893524E-017 3.5788159072854734E-019
81 -3.2427112423144782E-017 2.0371387958791390E-019
82 -2.1261559859350078E-017 1.5474491148651913E-019
83 -1.3381168532291091E-017 9.1800103349985530E-020
84 -8.7203767833349034E-018 4.9089569177223874E-020
85 -5.5120237509454559E-018 3.3824093691963943E-020
86 -3.6075669933017596E-018 2.8366606086893595E-020
87 -2.2288066587847980E-018 2.4990346882529037E-020
88 -1.4701377448526320E-018 9.8815881912803488E-021
89 -9.3188718250933159E-019 5.6064184958451948E-021
90 -6.1267815631169933E-019 5.2973590658965679E-021
91 -3.8483170866222993E-019 2.9454429659530535E-021
92 -2.5292805251474194E-019 2.5033491972120768E-021
93 -1.5833844597960494E-019 1.8481533643205036E-021
94 -1.0369676145917045E-019 1.0105940930442871E-021
95 -6.5061055794869659E-020 5.0692177242901795E-022
96 -4.2928357822687635E-020 4.1708039122778156E-022
97 -2.6943508451132377E-020 2.5123301329106747E-022
98 -1.7562019101404436E-020 1.8124814870122045E-022
99 -1.1183137396887359E-020 1.2667111037046020E-022
100 -7.2819979260256388E-021 9.2997458617163063E-023

5
plots/data_fig2_accurate Normal file
View File

@ -0,0 +1,5 @@
-0.8 -0.765468678667415 0.000000197272900
-0.795 -0.765862264591136 0.000000233475946
-0.79 -0.766260752174222 0.000000282577334
-0.785 -0.766664278559794 0.000000352807987
-0.78 -0.767072993403423 0.000000463059077

19
plots/fig1.plt Normal file
View File

@ -0,0 +1,19 @@
set term pdf enhanced font "Times,18" rounded linewidth 2.0
set key bottom right
set format y "%.2f"
set grid
set xlabel "p"
set ylabel "H_p"
set output "fig1.pdf"
set xrange[0.:60]
set yrange[-0.35:0.05]
plot \
"H_m1.6_fig1" u 1:2:3 title "E=-1.6" w yerrorlines ls 1,\
"H_m1.2_fig1" u 1:2:3 title "E=-1.2" w yerrorlines ls 2,\
"H_m1.0_fig1" u 1:2:3 title "E=-1.0" w yerrorlines ls 3,\
"H_m0.9_fig1" u 1:2:3 title "E=-0.9" w yerrorlines ls 4,\
"H_m0.8_fig1" u 1:2:3 title "E=-0.8" w yerrorlines ls 6,\

3158
plots/fig1.ps Normal file
View File

File diff suppressed because it is too large Load Diff

23
plots/fig2.plt Normal file
View File

@ -0,0 +1,23 @@
set term pdf enhanced font "Times,18" rounded linewidth 2.2
set key bottom right
set format y "%.4f"
set format x "%.3f"
set grid
# set term X11 enhanced
set xlabel "E"
set ylabel "ℰ(E)"
set label 1 "E_0" at -0.793, -0.7678, 0 left norotate
set label 2 "E_0" at -0.7675, -0.7657, 0 left norotate
#set term x11
set output "fig2.pdf"
set xrange[-0.8:-0.76]
set yrange[-0.7685:-0.765]
set arrow from -0.768068,-0.7685 to -0.768068,-0.765 nohead ls 3
plot \
"two_compo_fig2" u 1:2:3 notitle w l ls 2, \
"data_fig2_accurate" u 1:2:3 notitle w p ls 1, \
-0.768068 notitle, \

1429
plots/fig2.ps Normal file
View File

File diff suppressed because it is too large Load Diff

BIN
plots/figures.tar Normal file
View File

Binary file not shown.

1005
plots/two_compo_fig2 Normal file
View File

File diff suppressed because it is too large Load Diff