Variational ground states of two-dimensional antiferromagnets in the valence bond basis

We study a variational wave function for the ground state of the two-dimensional S=1/2 Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes h(x,y) for valence bonds connecting spins separated by (x,y) lattice spacings. In contrast to previous studies, in which a functional form for h(x,y) was assumed, we here optimize all the amplitudes for lattices with up to 32x32 spins. We use two different schemes for optimizing the amplitudes; a Newton conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only approximate to 0.06% from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling approximate to 2% below the exact ones at long distances (corresponding to an approximate to 1% underestimation of the sublattice magnetization). The amplitudes h(r) for valence bonds of long length r decay as r(-3). We also discuss some results for small frustrated lattices.