Bilinear Quantum Monte Carlo: Expectations and Energy Differences

We propose a bilinear sampling algorithm in Green's function Monte Carlo for expectation values of operators that do not commute with the Hamiltonian and for differences between eigenvalues of different Hamiltonians. The integral representations of the Schroedinger equations are transformed into two equations whose solution has the form $\psi_a(x) t(x,y) \psi_b(y)$, where $\psi_a$ and $\psi_b$ are the wavefunctions for the two related systems and $t(x,y)$ is a kernel chosen to couple $x$ and $y$. The Monte Carlo process, with random walkers on the enlarged configuration space $x \otimes y$, solves these equations by generating densities whose asymptotic form is the above bilinear distribution. With such a distribution, exact Monte Carlo estimators can be obtained for the expectation values of quantum operators and for energy differences. We present results of these methods applied to several test problems, including a model integral equation, and the hydrogen atom.Comment: 27 page

An Exact Monte Carlo Method for Continuum Fermion Systems

We offer a new proposal for the Monte Carlo treatment of many-fermion systems in continuous space. It is based upon Diffusion Monte Carlo with significant modifications: correlated pairs of random walkers that carry opposite signs; different functions ``guide'' walkers of different signs; the Gaussians used for members of a pair are correlated; walkers can cancel so as to conserve their expected future contributions. We report results for free-fermion systems and a fermion fluid with 14 $^3$He atoms, where it proves stable and correct. Its computational complexity grows with particle number, but slowly enough to make interesting physics within reach of contemporary computers.Comment: latex source, 3 separated figures (2 in jpg format, 1 in eps format

The Fermion Monte Carlo revisited

In this work we present a detailed study of the Fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies [M.H. Kalos and F. Pederiva, Phys. Rev. Lett. vol. 85, 3547 (2000)]. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact Diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose-Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full sign problem. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.Comment: 49 pages with 4 postscript figure

Fermionic Shadow Wavefunction Variational calculations of the vacancy formation energy in 3^3He

We present a novel technique well suited to study the ground state of inhomogeneous fermionic matter in a wide range of different systems. The system is described using a Fermionic Shadow wavefunction (FSWF) and the energy is computed by means of the Variational Monte Carlo technique. The general form of FSWF is useful to describe many--body systems with the coexistence of different phases as well in the presence of defects or impurities, but it requires overcoming a significant sign problem. As an application, we studied the energy to activate vacancies in solid $^3$He.Comment: 4 pages, 2 figure

Entropic effects in large-scale Monte Carlo simulations

The efficiency of Monte Carlo samplers is dictated not only by energetic effects, such as large barriers, but also by entropic effects that are due to the sheer volume that is sampled. The latter effects appear in the form of an entropic mismatch or divergence between the direct and reverse trial moves. We provide lower and upper bounds for the average acceptance probability in terms of the Renyi divergence of order 1/2. We show that the asymptotic finitude of the entropic divergence is the necessary and sufficient condition for non-vanishing acceptance probabilities in the limit of large dimensions. Furthermore, we demonstrate that the upper bound is reasonably tight by showing that the exponent is asymptotically exact for systems made up of a large number of independent and identically distributed subsystems. For the last statement, we provide an alternative proof that relies on the reformulation of the acceptance probability as a large deviation problem. The reformulation also leads to a class of low-variance estimators for strongly asymmetric distributions. We show that the entropy divergence causes a decay in the average displacements with the number of dimensions n that are simultaneously updated. For systems that have a well-defined thermodynamic limit, the decay is demonstrated to be n^{-1/2} for random-walk Monte Carlo and n^{-1/6} for Smart Monte Carlo (SMC). Numerical simulations of the LJ_38 cluster show that SMC is virtually as efficient as the Markov chain implementation of the Gibbs sampler, which is normally utilized for Lennard-Jones clusters. An application of the entropic inequalities to the parallel tempering method demonstrates that the number of replicas increases as the square root of the heat capacity of the system.Comment: minor corrections; the best compromise for the value of the epsilon parameter in Eq. A9 is now shown to be log(2); 13 pages, 4 figures, to appear in PR

The scaling properties of exchange and correlation holes of the valence shell of second row atoms

We study the exchange and correlation hole of the valence shell of second row atoms using variational Monte Carlo techniques, especially correlated estimates, and norm-conserving pseudopotentials. The well-known scaling of the valence shell provides a tool to probe the behavior of exchange and correlation as a functional of the density and thus test models of density functional theory. The exchange hole shows an interesting competition between two scaling forms -- one caused by self-interaction and another that is approximately invariant under particle number, related to the known invariance of exchange under uniform scaling to high density and constant particle number. The correlation hole shows a scaling trend that is marked by the finite size of the atom relative to the radius of the hole. Both trends are well captured in the main by the Perdew-Burke-Ernzerhof generalized-gradient approximation model for the exchange-correlation hole and energy.Comment: 18 pages, 8 figure

Pressure-induced diamond to beta-tin transition in bulk silicon: a near-exact quantum Monte Carlo study

The pressure-induced structural phase transition from diamond to beta-tin in silicon is an excellent test for theoretical total energy methods. The transition pressure provides a sensitive measure of small relative energy changes between the two phases (one a semiconductor and the other a semimetal). Experimentally, the transition pressure is well characterized. Density-functional results have been unsatisfactory. Even the generally much more accurate diffusion Monte Carlo method has shown a noticeable fixed-node error. We use the recently developed phaseless auxiliary-field quantum Monte Carlo (AFQMC) method to calculate the relative energy differences in the two phases. In this method, all but the error due to the phaseless constraint can be controlled systematically and driven to zero. In both structural phases we were able to benchmark the error of the phaseless constraint by carrying out exact unconstrained AFQMC calculations for small supercells. Comparison between the two shows that the systematic error in the absolute total energies due to the phaseless constraint is well within 0.5 mHa/atom. Consistent with these internal benchmarks, the transition pressure obtained by the phaseless AFQMC from large supercells is in very good agreement with experiment.Comment: 9 pages, 5 figure

On the efficient Monte Carlo implementation of path integrals

We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter products enjoys several properties that make it extremely suitable for path-integral Monte Carlo simulations: fast computation of paths, fast Monte Carlo sampling, and the ability to use different numbers of time slices for the different degrees of freedom, commensurate with the quantum effects. It is demonstrated that a Monte Carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all-particle or all-variable update sampler. The sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent. In the Levy-Ciesielski representation, the path variables are grouped in a small number of layers, with the variables from the same layer being statistically independent. The superior performance of the fast sampling algorithm is shown to be a consequence of these observations. Both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler, the Levy-Ciesielski implementation of path integrals, and the fast sampling algorithm.Comment: 14 pages, 3 figures; submitted to Phys. Rev.

A Constrained Path Quantum Monte Carlo Method for Fermion Ground States

We propose a new quantum Monte Carlo algorithm to compute fermion ground-state properties. The ground state is projected from an initial wavefunction by a branching random walk in an over-complete basis space of Slater determinants. By constraining the determinants according to a trial wavefunction $|\Psi_T \rangle$, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if $|\Psi_T\rangle$ is exact. We report results on the two-dimensional Hubbard model up to size $16\times 16$, for various electron fillings and interaction strengths.Comment: uuencoded compressed postscript file. 5 pages with 1 figure. accepted by PRL

Monte Carlo Methods in the Physical Sciences

I will review the role that Monte Carlo methods play in the physical sciences. They are very widely used for a number of reasons: they permit the rapid and faithful transformation of a natural or model stochastic process into a computer code. They are powerful numerical methods for treating the many-dimensional problems that derive from important physical systems. Finally, many of the methods naturally permit the use of modern parallel computers in efficient ways. In the presentation, I will emphasize four aspects of the computations: whether or not the computation derives from a natural or model stochastic process; whether the system under study is highly idealized or realistic; whether the Monte Carlo methodology is straightforward or mathematically sophisticated; and finally, the scientific role of the computation