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Model Fermion Monte Carlo with Antithetical Pairs

Zhiping Liu, Shiwei Zhang, Malvin H. Kalos

Published Feb 1, 1994• 0 citations
OpenAlex

Abstract

We consider certain model systems within which we seek solutions of the Schrodinger equation that are antisymmetric on inversion in the origin. We find geometrical constructions that define sequences of correlated random walks for which the overlap with the antisymmetrical test functions are asymptotically constant. The emphasis here is on problems in two dimensions, but the methods generalize to many dimensions and to certain other model problems.

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We consider certain model systems within which we seek solutions of the Schrodinger equation that are antisymmetric on inversion in the origin. We find geometrical constructions that define sequences of correlated random walks for which the overlap with the antisymmetrical test functions are asymptotically constant. The emphasis here is on problems in two dimensions, but the methods generalize to many dimensions and to certain other model problems.
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.
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.
We suggest an exact approach to help remedy the fermion sign problem in diffusion quantum Monte Carlo simulations. The approach is based on an explicit suppression of symmetric modes in the Schrödinger equation by means of a modified stochastic diffusion process (antisymmetric diffusion process). We introduce this algorithm and illustrate it on potential models in one dimension (1D) and show that there it solves the fermion sign problem exactly and converges to the lowest antisymmetric state of the system. Then, we discuss extensions of this approach to many-dimensional systems on examples of quantum oscillator in 2D-20D and a toy model of three and four fermions on harmonic strings in 2D and 3D. We show that in all these cases our method shows a performance comparable to that of a fixed-node approximation with an exact node.
We present, in this dissertation, a numerical simulation method to study interacting fermion systems. The general simulation procedures are discussed in connection with a description of a Quantum Monte Carlo simulation algorithm for interacting electrons on lattices. The algorithm presented here has been used to simulate interacting electrons on lattices, and it makes possible the study of substantially larger systems than can be studied by other numerical methods. As long as certain limits of applicability are respected, model Hamiltonians of interacting electrons can be studied without resort to uncontrolled approximations. The method then provides nearly exact solutions to model Hamiltonians of many-body systems, in the sense that the degree of error can be controlled. We also discuss some results obtained from simulations of the extended Hubbard model and and the periodic Anderson model. For the extended Hubbard model in two dimensions, different regions are identified where electron correlation produces antiferromagnetism, charge-density-waves, and singlet pairing superconducting behaviors. We also find regions where transitions from one type of correlation to another occur. For the symmetric periodic Anderson model, we observe the formation of local spin moments at high temperatures and their quenching at low temperatures, as in the single impurity Anderson model. In addition, we find antiferromagnetic interaction between f local moments at low temperatures which are not present in a single impurity problem.
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.