Green function Monte Carlo with stochastic reconfiguration: An effective remedy for the sign problem

A recent technique, proposed to alleviate the “sign problem disease,” is discussed in detail. As is well known, the ground state of a given Hamiltonian H can be obtained by applying the propagator $e^{-H\tau }$ to a trial wave function ${\psi }_T$ and sampling statistically the state ${\psi }_{\tau }{=e}^{-H\tau }{\psi }_T$ for large imaginary time $\tau .$ However, the sign problem may appear in the simulation and such statistical propagation would be practically impossible without employing some approximation such as the “fixed node” (FN) one. The present method allows the improvement of the FN dynamics with a systematic correction scheme. This is possible by the simple requirement that, after a short imaginary time propagation via the FN Hamiltonian, a number p of correlation functions can be further constrained to be exact by small perturbations of the FN state, which is free from the sign problem. By iterating this procedure, the Monte Carlo average sign, which is almost zero when there is a sign problem, remains stable and finite even for large $\tau .$ The proposed algorithm is tested against exact diagonalization data available on finite lattices. It is also shown, in some test cases, that the dependence of the results upon the few parameters entering the stochastic technique can be very easily controlled, unless for exceptional cases.