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Molecular Physics

Cancellation time for correlated random walkers

F. Arias de Saavedra, Malvin H. Kalos, Francesco Pederiva

Published Aug 4, 2011• 1 citation
Publisher

Abstract

We have studied the dynamics of the correlated diffusion of pairs of random walkers of opposite signs. The use of populations of such pairs has been proposed for the Monte Carlo treatment of many-fermion systems, where the possibility of their cancellation might prevent the characteristic decay of the signal-to-noise ratio. For four model systems – free fermions, the harmonic oscillator, an N-body system of attractive and repulsive harmonic forces, and an extensive system interacting by Pöschl–Teller potentials – we have explored analytically and by computation the behavior of the time to cancellation as a function of initial conditions and, equally important, as a function of system size. We find that for these systems the computational efficiency does not decay either with large imaginary time or with large N.

DOI
10.1080/00268976.2011.604647

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We have studied the dynamics of the correlated diffusion of pairs of random walkers of opposite signs. The use of populations of such pairs has been proposed for the Monte Carlo treatment of many-fermion systems, where the possibility of their cancellation might prevent the characteristic decay of the signal-to-noise ratio. For four model systems – free fermions, the harmonic oscillator, an N-body system of attractive and repulsive harmonic forces, and an extensive system interacting by Pöschl–Teller potentials – we have explored analytically and by computation the behavior of the time to cancellation as a function of initial conditions and, equally important, as a function of system size. We find that for these systems the computational efficiency does not decay either with large imaginary time or with large N.
We review the fundamental challenge of fermion Monte Carlo for continuous systems, the “sign problem”. We seek that eigenfunction of the many-body Schrödinger equation that is antisymmetric under interchange of the coordinates of pairs of particles. We describe methods that depend upon the use of correlated dynamics for pairs of correlated walkers that carry opposite signs. There is an algorithmic symmetry between such walkers that must be broken to create a method that is both exact and as effective as for symmetric functions. In our new method, it is broken by using different “guiding” functions for walkers of opposite signs, and a geometric correlation between steps of their walks. With a specific process of cancellation of the walkers, overlaps with antisymmetric test functions are preserved. Finally, we describe the progress in treating free-fermion systems and a fermion fluid with 14 3He atoms.
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, and walkers can cancel so as to conserve their expected future contributions. We report results for free-fermion systems and a fermion fluid with 14 3He atoms, where it proves stable and correct. Its computational complexity grows with particle number, but slowly enough to make interesting physics within the reach of contemporary computers.
The issues that prevent the development of efficient and stable algorithms for fermion Monte Carlo calculations in continuum systems are reexamined with special reference to the implications of the ``plus-minus'' symmetry. This is a property of many algorithms that use signed walkers, namely, that the dynamics are unchanged when the signs of the walkers are interchanged. Algorithms that obey this symmetry cannot exhibit the necessary stability. Specifically, estimates of the overlap with any antisymmetric test function cannot be bounded away from zero in the limit of many iterations. Within the framework of a diffusion Monte Carlo treatment of the Schr\"odinger equation, it is shown that this symmetry is easily broken for pairs of walkers while at the same time preserving the correct marginal dynamics for each member of the pair. The key is to create different classes of correlations between members of pairs and to use (at least) two distinct correlations for a pair and for the same pair with signs exchanged. The ideas are applied successfully for a class of simple model problems in two dimensions. \textcopyright{} 1996 The American Physical Society.
Abstract A method for treating the nodes for small systems was described in this paper. The general procedure involves the use of both positive and negative walkers in Green’s function QMC, along with cancellation of walkers of opposite sign in close proximity. The method is illustrated for the first excited state of a particle in a one-dimensional box and for three model problems with three particles in three dimensions, interacting with each other through square-well potentials with repulsive and attractive parts. The calculations were executed with the Green’s function procedure, made convenient by availability of known Green’s functions satisfying the conditions of the sample problems. A partial cancellation of positive and negative walkers was made on the basis of overlapping Green’s functions, and the distributions of steps for the walkers involved were altered accordingly. Without cancellations, the distributions of positive and negative walkers proceed to the boson ground-state distribution. With cancellation and with control of their numbers, the positive and negative walkers tend to accumulate in separate regions and to commingle in the vicinity of a node. The arithmetic sum of the walkers yields the excited or fermion wavefunction with a node separating positive and negative regions. For the model problems the method successfully produced accuracies within a few percent of the exact energies, where known. It was noted that for larger problems, the total number of walkers required to maintain an adequate density for sufficient cancellation might be prohibitive.
Our approach to the solution of the Schrödinger Equation for many-fermion systems has been extensively revised. We have devised a generalization of "acceptance/rejection" that applies to signed random walkers. We have introduced a new class of importance functions for two walkers that better reflects the structure of the enlarged Euclidean space of the pair. For greater flexibility, we no longer rely on the "local energy" of the importance function to determine the dynamics of the walk. We sketch these technical changes and give new results for the two-dimensional electron gas.
A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015E_{h} of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10^{-5}E_{h} without an importance sampling or Jastrow factor.