Variational ground states of 2D antiferromagnets in the valence-bond basis

We use a variational method to study two-dimensional \(S=\frac {1}{2} \) Heisenberg antiferromagnets in the valence bond basis. The wave function is of the form \[ \mid \psi \rangle =\prod h(x_{ij},y_{ij})(i,j),\] where \((i,j)\) represents a singlet formed by the spins at sites $i$ and $j$; \[ (i,j)=\frac{1}{\sqrt{2}}(\uparrow_i \downarrow_j - \downarrow_i\uparrow_j), \] and \(h(x_{ij},y_{ij}) \) is the amplitude corresponding to a bond coneenting two spins with seperation \( (x_{ij},y_{ij}) \). The form \( h \sim \frac{1}{r^p} \), where $r$ is the distance, was studied prevously. The best variational energy was obtained for $p = 4$. Now we optimize all \(h(x,y)\)by combining a standard Newton method and a conjugate gradient method. For systems with up to $16\times 16$ spins, the energy of the optimized wave function deviates by less than 0.1\% from the exact ground state energy. The spin-spin correlations are also very well reproduced. The exponent $p=3$ in agreement with recent Monte Carlo simulations. We also investigates this class of wave functions for a quantum-critical bilayer model.