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\usepackage[T1]{fontenc}
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\usepackage{txfonts,dsfont}
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\usepackage{xspace}
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\usepackage [french]{babel}
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%\usepackage{lscape}
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\usepackage[normalem]{ulem}
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\newcommand{\roland}[1]{\textcolor{cyan}{\bf #1}}
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@ -137,11 +135,9 @@
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\begin{abstract}
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\noindent
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The sampling of the configuration space in diffusion Monte Carlo (DMC)
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is done using walkers moving randomly.
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The sampling of the configuration space in diffusion Monte Carlo (DMC) is done using walkers moving randomly.
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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)}],
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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
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the on-\titou{state} dynamics can be integrated out exactly, leading to an effective dynamics connecting only different states.
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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.
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Here, we extend this idea to the general case of a walker trapped within domains of arbitrary shape and size.
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The equations of the resulting effective stochastic dynamics are derived.
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The larger the average (trapping) time spent by the walker within the domains, the greater the reduction in statistical fluctuations.
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@ -154,86 +150,73 @@ Although this work presents the method for finite linear spaces, it can be gener
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Diffusion Monte Carlo (DMC) is a class of stochastic methods for evaluating
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the ground-state properties of quantum systems. They have been extensively used
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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}
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quantum liquids,\cite{Holzmann_2006}
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nuclear physics,\cite{Carlson_2015,Carlson_2007} theoretical chemistry,\cite{Austin_2012} etc).
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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.
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Here, we shall consider only the discrete case, that is, the general problem of calculating the lowest eigenvalue/eigenstate of a (very large) matrix.
|
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Diffusion Monte Carlo (DMC) is a class of stochastic methods for evaluating the ground-state properties of quantum systems.
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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).
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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.
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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.
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The generalization to continuous configuration spaces presents no fundamental difficulty.
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In essence, DMC is based on \textit{stochastic} power methods, an old and widely employed numerical approach to extract
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the largest or smallest eigenvalues of a matrix (see, \eg, Ref.~\onlinecite{Golub_2012}).
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This approach is particularly simple as it merely consists in applying the matrix (or some simple function of it) as many times as
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required on some arbitrary vector belonging to the linear space.
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Thus, the basic step of the corresponding algorithm essentially reduces to a matrix-vector multiplication.
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In practice, the power method is employed under some more sophisticated implementations, such as, \eg,
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the Lancz\`os \cite{Golub_2012} or Davidson \cite{Davidson_1975} algorithms.
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When the size of the matrix is too large, the matrix-vector multiplication becomes unfeasible
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and probabilistic techniques to sample only the most important contributions of the matrix-vector product are required.
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This is the basic idea of DMC. There exist several variants of DMC known under various names:
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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.
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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}).
|
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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.
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Thus, the basic step of the corresponding algorithm essentially reduces to successive matrix-vector multiplications.
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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}
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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.
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This is the basic idea of DMC.
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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.
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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?}
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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.
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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.
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Without entering into the mathematical details presented below, the main ingredient of DMC to perform the
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matrix-vector multiplication probabilistically is the use of a stochastic matrix (or transition probability matrix)
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to generate stepwise a series of states over which statistical averages are evaluated.
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The critical aspect of any Monte Carlo scheme is the the amount of computational effort required to
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reach a given statistical error.
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Two important avenues to decrease the error are the use of variance reduction techniques
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(for example, by introducing improved estimators\cite{Assaraf_1999}) or to improve the quality of the sampling
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(minimization of the correlation time between states).
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Another possibility, at the heart of the present work, is to integrate out exactly some part of the dynamics, thus reducing the number of
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degrees of freedom and, then, the amount of statistical fluctuations.
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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.
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The critical aspect of any Monte Carlo scheme is the amount of computational effort required to reach a given statistical error.
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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).
|
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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.
|
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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
|
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the on-state dynamics can be integrated out to generate an effective dynamics connecting only different states with some
|
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renormalized estimators for the properties.
|
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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.
|
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Numerical applications have shown that the statistical errors can be very significantly decreased.
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Here, we extend this idea to the general case where a walker
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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
|
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dynamics describing walkers moving from one domain into another. The equations of the effective dynamics are derived.
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A numerical application for the 1D-problem is presented. In particular, it shows that the statistical convergence of the energy can be greatly
|
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enhanced when domains associated with large average trapping times are used.
|
||||
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.
|
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It is shown how to define an effective stochastic dynamics describing walkers moving from one domain to another.
|
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The equations of the effective dynamics are derived and a numerical application for a model (one-dimensional) problem is presented.
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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.
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It should be noted that the use of domains in quantum Monte Carlo is not new. In a pioneering work,\cite{Kalos_1974}
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Kalos and collaborators introduced the so-called Domain's Green Function Monte Carlo approach in continuous space which they applied to
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a system of bosons with hard-sphere interaction. The domain used was the Cartesian product of small spheres around each particle, the Hamiltonian
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being approximated by the kinetic part only within the domain.
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Some time later, Kalos proposed to extend these ideas to more general domains such as rectangular and/or
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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
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domains are of arbitrary size, thus greatly increasing the efficiency of the approach.
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Finally, note that some general equations for arbitary domains in continuous space have also been proposed in
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[\onlinecite{Assaraf_1999B}].
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It should be noted that the use of domains in quantum Monte Carlo is not new.
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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.
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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.
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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.
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Here, the domains are of arbitrary size, thus greatly increasing the efficiency of the approach.
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Finally, note that some general equations for arbitrary domains in continuous space have also been proposed in Ref.~\onlinecite{Assaraf_1999B}.
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The paper is organized as follows. Sec.\ref{Sec:DMC} presents the basic equations and notations of DMC. First, the path integral representation
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of the Green's function is given in subsection \ref{sub:path}. The probabilistic framework allowing the Monte Carlo calculation of the
|
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Green's function is presented in \ref{sub:proba}.
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Section \ref{sec:DMC_domains} is devoted to the use of domains in DMC. First, we recall in \ref{sub:single_domains} the case
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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
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energy dependent Green's function using domains are derived. Section \ref{numerical} presents the appplication of the approach to the one-dimensional
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Hubbard model. Finally in Sec.\ref{conclu} some conclusions and perspectives are given.
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The paper is organized as follows.
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Section \ref{sec:DMC} presents the basic equations and notations of DMC.
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First, the path integral representation of the Green's function is given in Subsec.~\ref{sec:path}.
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The probabilistic framework allowing the Monte Carlo calculation of the Green's function is presented in Subsec.~\ref{sec:proba}.
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Section \ref{sec:DMC_domains} is devoted to the use of domains in DMC.
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First, we recall in Subsec.~\ref{sec:single_domains} the case of a domain consisting of a single state. \cite{Assaraf_1999}
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The general case is then treated in Subsec.~\ref{sec:general_domains}.
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In Subsec.~\ref{sec:Green}, both the time- and energy-dependent Green's function using domains is derived.
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Section \ref{sec:numerical} presents the application of the approach to the one-dimensional Hubbard model.
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Finally, in Sec.\ref{sec:conclu}, some conclusions and perspectives are given.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Diffusion Monte Carlo}
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\label{Sec:DMC}
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\label{sec:DMC}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%=======================================%
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\subsection{Path-integral representation}
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\label{sub:path}
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Diffusion Monte Carlo is a stochastic implementation of the power method. The operator used is
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\label{sec:path}
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%=======================================%
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As previously mentioned, DMC is a stochastic implementation of the power method defined by the following operator:
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\be
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T= \mathds{1} -\tau (H-E\mathds{1}),
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T = \mathds{1} -\tau (H-E\mathds{1}),
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\ee
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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
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$H$ the Hamiltonian matrix. Starting from some initial vector, $|\Psi_0\rangle$, we have
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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
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\be
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\lim_{N \rightarrow \infty } T^N|\Psi_0 \rangle = |\Phi_0 \rangle
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\lim_{N \to \infty} T^N \ket{\Psi_0} = \ket{\Phi_0}
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\ee
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where $|\Phi_0 \rangle$ is the ground-state. The equality is up to a global phase factor playing no role in physical quantum averages.
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This result is true for any $|\Psi_0 \rangle$
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provided that $\langle \Phi_0 |\Psi_0 \rangle \ne 0$ and for $\tau$ sufficiently small.
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At large but finite $N$, the vector $T^N|\Psi_0\rangle$ differs from $|\Phi_0 \rangle$ only by an exponentially small correction,
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making easy the extrapolation of the finite-N results to $N=\infty$.\\
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where $\ket{\Phi_0}$ is the ground-state wave function.
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The equality is up to a global phase factor playing no role in physical quantum averages.
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This result is true for any $\ket{\Psi_0}$ provided that $\braket{\Phi_0}{\Psi_0} \ne 0$ and for $\tau$ sufficiently small.
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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 $N=\infty$.\\
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Ground-state properties may be obtained at large $N$. For example, in the important case of the energy one can use the formula
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\be
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@ -247,7 +230,7 @@ To proceed further we introduce the time-dependent Green's matrix $G^{(N)}$ defi
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\be
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G^{(N)}_{ij}=\langle j|T^N |i\rangle.
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\ee
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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,
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The denomination ``time-dependent Green's matrix'' is used here since $G$ may be viewed as a short-time approximation of the (time-imaginary) evolution operator,
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$e^{-N\tau H}$ which is usually referred to as the imaginary-time dependent Green's function.\\
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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
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@ -273,15 +256,15 @@ the $i$-th component of the ground-state (obtained as $\lim_{N\rightarrow \infty
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at vector $|i\rangle$. This result is independent of the initial
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vector $|i_0\rangle$, apart from some irrelevant global phase factor.
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When the size of the linear space is small the explicit calculation of the full sums involving $M^N$ terms (here, $M$ is the size of the Hilbert space)
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can be performed. We are in the realm of what can be called the \og deterministic \fg power methods, such as
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can be performed. We are in the realm of what can be called the ``deterministic'' power methods, such as
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the Lancz\`os or Davidson approaches. If not, probabilistic techniques for generating only the
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paths contributing significantly to the sums are to be used.
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\subsection{Probabilistic framework}
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\label{sub:proba}
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\label{sec:proba}
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In order to derive
|
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a probabilistic expression for the Green's matrix we introduce a so-called guiding vector, $|\Psi^+\rangle$,
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having strictly positive components, $\Psi^+_i > 0$, and apply a similarity tranformation to the operators $G^{(N)}$ and $T$
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having strictly positive components, $\Psi^+_i > 0$, and apply a similarity transformation to the operators $G^{(N)}$ and $T$
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\be
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{\bar T}_{ij}= \frac{\Psi^+_j}{\Psi^+_i} T_{ij}
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\label{defT}
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@ -428,7 +411,7 @@ found, for example, in refs \onlinecite{Foulkes_2001,Kolorenc_2011}.
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\section{DMC with domains}
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\label{sec:DMC_domains}
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\subsection{Domains consisting of a single state}
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\label{sub:single_domains}
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\label{sec:single_domains}
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During the simulation, walkers move from state to state with the possibility of being trapped a certain number of times on the same state before
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exiting to a different state. This fact can be exploited in order to integrate out some part of the dynamics, thus leading to a reduction of the statistical
|
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fluctuations. This idea was proposed some time ago\cite{assaraf_99,Assaraf_1999B,caffarel_00} and applied to the SU(N) one-dimensional Hubbard model.
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@ -468,13 +451,13 @@ The time-averaged contribution of the on-state weight can be easily calculated t
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\ee
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Details of the implementation of the effective dynamics can be in found in Refs. (\onlinecite{assaraf_99},\onlinecite{caffarel_00}).
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\subsection{General domains}
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\label{sub:general_domains}
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\label{sec:general_domains}
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Let us now extend the results of the preceding section to a general domain. For that,
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let us associate to each state $|i\rangle$ a set of states, called the domain of $|i\rangle$ and
|
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denoted ${\cal D}_i$, consisting of the state $|i\rangle$ plus a certain number of states. No particular constraints on the type of domains
|
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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
|
||||
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.
|
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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.
|
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|
||||
Let us write an arbitrary path of length $N$ as
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\be
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@ -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,
|
||||
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.
|
||||
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,
|
||||
respectively.
|
||||
In what follows, we shall systematically write the integers representing the exit states in capital letter.
|
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@ -556,7 +539,7 @@ t_{I}={\bar n}_{I} \tau= \frac{1}{\Psi^+_{I}} \langle I | P_{I} \frac{1}{H^+ -
|
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In practice, the various quantities restricted to the domain are computed by diagonalizing the matrix $(H^+-E_L^+)$ in ${\cal D}_{I}$. Note that
|
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it is possible only if the dimension of the domains is not too large (say, less than a few thousands).
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\subsection{Expressing the Green's matrix using domains}
|
||||
\label{Green}
|
||||
\label{sec:Green}
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||||
\subsubsection{Time-dependent Green's matrix}
|
||||
\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.
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@ -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 $
|
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= \prod_{k=0}^{N-1} \langle I_k|F_{I_k}|I_{k+1} \rangle $
|
||||
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
|
||||
$$
|
||||
{\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N} +
|
||||
$$
|
||||
@ -799,7 +782,7 @@ a tradoff has to be found between the possible bias in the extrapolation and the
|
||||
required.
|
||||
|
||||
\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
|
||||
\be
|
||||
\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 $-0.768068...$.
|
||||
|
||||
\begin{figure}[h!]
|
||||
\begin{center}
|
||||
\includegraphics[width=\linewidth]{fig1}
|
||||
\end{center}
|
||||
\includegraphics[width=\columnwidth]{fig1}
|
||||
\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.}
|
||||
\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{center}
|
||||
\includegraphics[width=\linewidth]{fig2}
|
||||
\end{center}
|
||||
\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.
|
||||
\includegraphics[width=\columnwidth]{fig2}
|
||||
\caption{1D-Hubbard model, $N=4$ and $U=12$. $\mathcal{E}(E)$ as a function of $E$.
|
||||
The horizontal and vertical lines are at $\mathcal{E}(E_0)=E_0$ and $E=E_0$, respectively.
|
||||
$E_0$ is the exact energy of -0.768068.... The dotted line is the two-component extrapolation.
|
||||
Error bars are smaller than the symbol size.}
|
||||
\label{fig2}
|
||||
@ -1058,7 +1037,7 @@ The exact QMC 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 for the reference energy. The extrapolated QMC
|
||||
energies are in full agreement
|
||||
with the exact value within error bars. However, an increase of the statistical error is observed when the size increases. To get lower error bars
|
||||
we need to use better trial wavefunctions, better domains, and also larger simulation times.
|
||||
we need to use better trial wave functions, better domains, and also larger simulation times.
|
||||
laptop. 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.
|
||||
|
||||
@ -1142,7 +1121,7 @@ $N$ & Size Hilbert space & Domain & Domain size & $\alpha,\beta$ &$\bar{t}_{I_0}
|
||||
\end{table*}
|
||||
|
||||
\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
|
||||
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.
|
||||
@ -1180,7 +1159,7 @@ under grant agreement no.~952165.
|
||||
For the simplest case of a two-state system the fundamental equation (\ref{eqfond}) writes
|
||||
$$
|
||||
{\cal I}=
|
||||
\langle I_0|\frac{1}{H-E}|\Psi\rangle = \langle I_0|P_0\frac{1}{H-E} P_0|Psi\rangle
|
||||
\langle I_0|\frac{1}{H-E}|\Psi\rangle = \langle I_0|P_0\frac{1}{H-E} P_0|\Psi\rangle
|
||||
+ \sum_{p=1}^{\infty} {\cal I}_p$$
|
||||
with
|
||||
$$
|
||||
|
||||
102
plots/H_m0.8_fig1
Normal file
102
plots/H_m0.8_fig1
Normal file
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Normal file
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plots/H_m0.9_fig1
Normal file
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||||
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||||
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|
||||
102
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Normal file
@ -0,0 +1,102 @@
|
||||
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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
101
plots/H_m1.8_fig1
Normal 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
5
plots/data_fig2_accurate
Normal 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
19
plots/fig1.plt
Normal 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
3158
plots/fig1.ps
Normal file
File diff suppressed because it is too large
Load Diff
23
plots/fig2.plt
Normal file
23
plots/fig2.plt
Normal 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
1429
plots/fig2.ps
Normal file
File diff suppressed because it is too large
Load Diff
BIN
plots/figures.tar
Normal file
BIN
plots/figures.tar
Normal file
Binary file not shown.
1005
plots/two_compo_fig2
Normal file
1005
plots/two_compo_fig2
Normal file
File diff suppressed because it is too large
Load Diff
Loading…
Reference in New Issue
Block a user