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 32×32 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 ≈0.06% from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling ≈2% below the exact ones at long distances (corresponding to an ≈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.