Roland Assaraf

Zero-variance zero-bias principle for observables in quantum Monte Carlo: Application to forces

A simple and stable method for computing accurate expectation values of observables with variational Monte Carlo (VMC) or diffusion Monte Carlo (DMC) algorithms is presented. The basic idea consists in replacing the usual “bare” estimator associated with the observable by an improved or “renormalized” estimator. Using this estimator more accurate averages are obtained: Not only the statistical fluctuations are reduced but also the systematic error (bias) associated with the approximate VMC or (fixed-node) DMC probability densities. It is shown that improved estimators obey a zero-variance zero-bias property similar to the usual zero-variance zero-bias property of the energy with the local energy as improved estimator. Using this property improved estimators can be optimized and the resulting accuracy on expectation values may reach the remarkable accuracy obtained for total energies. As an important example, we present the application of our formalism to the computation of forces in molecular systems. Cal...

Zero-Variance Principle for Monte Carlo Algorithms

We present a general approach to greatly increase at little cost the efficiency of Monte Carlo algorithms. To each observable to be computed we associate a renormalized observable (improved estimator) having the same average but a different variance. By writing down the zero-variance condition a fundamental equation determining the optimal choice for the renormalized observable is derived (zero-variance principle for each observable separately). We show, with several examples including classical and quantum Monte Carlo calculations, that the method can be very powerful.

Metal-insulator transition in the one-dimensional SU ( N ) Hubbard model

We investigate the metal-insulator transition of the one-dimensional SU(N) Hubbard model for repulsive interaction. Using the bosonization approach a Mott transition in the charge sector at half-filling (k_F=\pi/Na_0) is conjectured for N > 2. Expressions for the charge and spin velocities as well as for the Luttinger liquid parameters and some correlation functions are given. The theoretical predictions are compared with numerical results obtained with an improved zero-temperature quantum Monte Carlo approach. The method used is a generalized Greens function Monte Carlo scheme in which the stochastic time evolution is partially integrated out. Very accurate results for the gaps, velocities, and Luttinger liquid parameters as a function of the Coulomb interaction U are given for the cases N=3 and N=4. Our results strongly support the existence of a Mott-Hubbard transition at a {\it non-zero} value of the Coulomb interaction. We find

A pedagogical introduction to Quantum Monte-Carlo

U_c \sim 2.2

Dynamical Symmetry Enlargement Versus Spin-Charge Decoupling in the One-Dimensional SU(4) Hubbard Model

for N=3 and

Time-dependent linear-response variational Monte Carlo

U_c \sim 2.8

Ab initio lifetime correction to scattering states for time-dependent electronic-structure calculations with incomplete basis sets

for N=4.

Hubbard model on hypercubes

Quantum Monte Carlo (QMC) methods are powerful stochastic approaches to calculate ground-state properties of quantum systems. They have been applied with success to a great variety of problems described by a Schrodinger-like Hamiltonian (quantum liquids and solids, spin systems, nuclear matter, ab initio quantum chemistry, etc ... ). In this paper we give a pedagogical presentation of the main ideas of QMC. We develop and exemplify the various concepts on the simplest system treatable by QMC, namely a 2 x 2 matrix. First, we discuss the Pure Diffusion Monte Carlo (PDMC) method which we consider to be the most natural implementation of QMC concepts. Then, we discuss the Diffusion Monte Carlo (DMC) algorithms based on a branching (birth-death) process. Next, we present the Stochastic Reconfiguration Monte Carlo (SRMC) method which combines the advantages of both PDMC and DMC approaches. A very recently introduced optimal version of SRMC is also discussed. Finally, two methods for accelerating QMC calculations are sketched: (a) the use of integrated Poisson processes to speed up the dynamics (b) the introduction of “renormalized” or “improved” estimators to decrease the statistical fluctuations.

Diffusion monte carlo methods with a fixed number of walkers

We investigate dynamical symmetry enlargement in the half-filled SU(4) Hubbard chain using non-perturbative renormalization group and Quantum Monte Carlo techniques. A spectral gap is shown to open for arbitrary Coulombic repulsion

Improved Monte Carlo estimators for the one-body density.

U