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

Abstract

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 $32\ifmmode\times\else\texttimes\fi{}32$ 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 $\ensuremath{\approx}0.06%$ from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling $\ensuremath{\approx}2%$ below the exact ones at long distances (corresponding to an $\ensuremath{\approx}1%$ underestimation of the sublattice magnetization). The amplitudes $h(r)$ for valence bonds of long length $r$ decay as ${r}^{\ensuremath{-}3}$. We also discuss some results for small frustrated lattices.