Visualization of the local contribution to the nodal surface of a many-fermion wave function

An essential part of variational Monte Carlo or Green’s function Monte Carlo (GFMC) algorithms is the trial wave function. In the case of particles obeying Fermi statistics, this wave function is antisymmetric and cannot be interpreted directly as a probability distribution, thereby making calculations difficult. Some progress can be made in GFMC algorithms, however, by requiring the trial wave function to have the same “fixed” nodes as the variational function. The sensitivity of the energy to changes in the nodal surface has remained a significant unresolved issue for two reasons: (a) the many-dimensional nodal surface is hard to quantify or visualize and (b) it has been difficult to vary this surface in a controlled fashion. As a first step toward gaining further information, we have developed a method of quantifying the many-dimensional nodal surface by creating and visually characterizing a related surface. The related surface is the surface of the local volume in which any given particle can range without changing the overall sign of the wave function, and this surface is mapped for different configurations and wave functions. A comparison of these surfaces allows us to characterize the local contributions to the nodal surface of a particular wave function and thereby visually differentiate between different types of wave functions. We show that introducing backflow correlations into the trial wave function results in a constrained volume with the undesired effect of significantly slowing down GFMC calculations. This study demonstrates that advanced visualization methods can serve a useful role in the process of algorithm development rather than just in the presentation of results.