Systematic improvement of variational Monte Carlo using Lanczos iterations

Abstract

We present a new numerical technique which combines the variational Monte Carlo and the Lanczos methods without suffering from the fermion sign problem. Lanczos iterations allow systematic improvement of trial wavefunctions while Monte Carlo sampling permits treatment of large lattices. As in the usual Lanczos method we find it useful to symmetrize the starting wavefunction in order to accelerate convergence. We apply our method to the 2D AFM Heisenberg model in the fermionic electron representation, which allows us to compare with results from the equivalent bosonic spin representation. Using d-wave RVB states as starting wavefunctions shows that after only one iteration between 70 and 80% of the difference between the variational energy and the ground state energy (as determined by GFMC) is recovered, and a similar improvement is observed in the second iteration. Leaving the spin-singlet sector by introducing antiferromagnetic correlations reduces the symmetry and the relative improvement in energy drops below 50% for one iteration. Our method allows us also to see trends in observables. Relative to the d-wave RVB states we find an enhancement in the spinspin correlations, consistent with the expectation that the true ground state has long-range order.