We will review the fundamental challenge of fermion Monte Carlo for continuous systems, the "sign problem," and some of the proposals that have been made for its solution, including some approximate schemes and others whose computing requirements grow exponentially. The issue is to find not the fundamental eigenmode of the Schrodinger equation in many dimensions, but one that has a special inversion of sign--antisymmetry in the exchange of pairs of coordinates. Thus the function sought is not everywhere positive. We also describe a class of methods that depend upon the use of correlated dynamics for ensembles of correlated walkers that carry opposite signs. We discuss the algorithmic symmetry between such walkers that must be broken to create a method that is both exact and as effective as for symmetric functions. We explain the concept of marginally correct dynamics. Stable overlaps with an antisymmetric trial function given by such dynamics correspond to the lowest antisymmetric mode. As an elementary example, we will show how correlated walkers can solve an antisymmetric Schroedinger equation for a two-dimensional box. Many-body harmonic oscillator problems are particularly tractable: their stochastic dynamics permits the use of regular geometric structures for the ensembles, structures that are stable when appropriate correlations are introduced, and that avoid the decay of signal-to-noise that is a normal characteristic of the sign problem. Finally, we outline a generalization of the method for arbitrary potentials and describe the progress in treating some model problems and few-electron systems.
NATO ASI, Cornell Theory Center,Last modified: Thu Jun 11 17:20:05 EDT 1998