Self-Healing Diffusion Monte Carlo applied to a simple fermionic model: A critical assessment of the method

Abstract

We investigate the Self-Healing Diffusion Monte Carlo (SHDMC) method using a one-dimensional model with periodic boundary conditions. An inversion symmetry is introduced to mimic the antisymmetry property of fermionic wavefunctions, with the bosonic and fermionic sectors being modeled by the even and odd eigenstates, respectively. As in realistic fermionic systems, the nodal structure is only partially constrained by symmetry, making this model a non-trivial testbed for nodal optimization algorithms such as SHDMC. We show that the nodal evolution under SHDMC iterations can be cast into a dynamical system exhibiting both attractive and repulsive fixed points. In the standard formulation of SHDMC applied to this model, the fixed-node energy is found to increase upon iteration, and the node converges to a wrong value, indicating that SHDMC does not always converge to the correct solution. We further show that this problem is partially cured by modifying the nodal update criterion to give more importance to the nodal region. Achieving convergence in the general case very likely requires the use of a localized basis set, as is the case for this model.