Loop updates for variational and projector quantum Monte Carlo simulations in the valence-bond basis

We show how efficient loop updates, originally developed for Monte Carlo simulations of quantum spin systems at finite temperature, can be combined with a ground-state projector scheme and variational calculations in the valence-bond basis. The methods are formulated in a combined space of spin z components and valence bonds. Compared to schemes formulated purely in the valence-bond basis, the computational effort is reduced from up to O(N²) to O(N) for variational calculations, where N is the system size, and from O(m²) to O(m) for projector simulations, where m≫N is the projection power. These improvements enable access to ground states of significantly larger lattices than previously. We demonstrate the efficiency of the approach by calculating the sublattice magnetization M_s of the two-dimensional Heisenberg model to high precision, using systems with up to 256×256 spins. Extrapolating the results to the thermodynamic limit gives M_s=0.30743(1) . We also discuss optimized variational amplitude-product states, which were used as trial states in the projector simulations, and compare results of projecting different types of trial states.