Systematic reduction of sign errors in many-body problems: generalization of self-healing diffusion Monte Carlo to excited states

A recently developed self-healing diffusion Monte Carlo algorithm [PRB 79, 195117] is extended to the calculation of excited states. The formalism is based on an excited-state fixed-node approximation and the mixed estimator of the excited-state probability density. The fixed-node ground state wave-functions of inequivalent nodal pockets are found simultaneously using a recursive approach. The decay of the wave-function into lower energy states is prevented using two methods: i) The projection of the improved trial-wave function into previously calculated eigenstates is removed. ii) The reference energy for each nodal pocket is adjusted in order to create a kink in the global fixed-node wave-function which, when locally smoothed out, increases the volume of the higher energy pockets at the expense of the lower energy ones until the energies of every pocket become equal. This reference energy method is designed to find nodal structures that are local minima for arbitrary fluctuations of the nodes within a given nodal topology. We demonstrate in a model system that the algorithm converges to many-body eigenstates in bosonic-like and fermionic cases.