We describe a “master equation” analysis for the bond amplitudes h(r) of an RVB wave function. Starting from any initial guess, h(r) evolves—in a manner dictated by the spin Hamiltonian under consideration—toward a steady-state distribution representing an approximation to the true ground state. Unknown transition coefficients in the master equation are treated as variational parameters. We illustrate the method by applying it to the J1–J2 antiferromagnetic Heisenberg model. Without frustration (J2 = 0), the amplitudes are radially symmetric and fall off as 1/r3 in the bond length. As the frustration increases, there are precursor signs of columnar or plaquette valence bond solid order: the bonds preferentially align along the axes of the square lattice and weight accrues in the nearest-neighbor bond amplitudes. The Marshall sign rule holds over a large range of couplings, J2/J1 ≲ 0.418. It fails when the r = (2,1) bond amplitude first goes negative, a point also marked by a cusp in the ground-state energy. A nonrigorous extrapolation of the staggered magnetic moment (through this point of nonanalyticity) shows it vanishing continuously at a critical value J2/J1 ≲ 0.447. This may be preempted by a first-order transition to a state of broken translational symmetry.

@article{
  title = {Master equation approach to computing RVB bond amplitudes},
  author = {Beach, K. S. D.},
  journal = {Physical Review B},
  volume = {79},
  issue = {22},
  pages = {224431},
  numpages = {8},
  year = {2009},
  month = {Jun},
  doi = {10.1103/PhysRevB.79.224431},
  url = {http://link.aps.org/doi/10.1103/PhysRevB.79.224431},
  publisher = {American Physical Society}
}