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

Figure 1

(Color online) The deviation $\Delta E=E_{\mathrm{var}}\left(L\right)-E_{\text{exact}}\left(L\right)$ of the variational ground state energy from the exact energy as a function of lattice size. Results obtained with both the Newton method and the stochastic opitimization scheme are shown. The error bars were obtained by carrying out several independent optimization runs for each $L$.

Figure 2

(Color online) The staggered structure factor $S(\pi ,\pi )$ versus lattice size, compared with unbiased QMC results. The inset shows the long-distance spin-spin correlation. Statistical errors are smaller than the symbols.

Figure 3

(Color online) The amplitude $h(L∕2,L∕2-1)$ versus the system size. Statistical errors are of the order of the size of the squares. The line shows the power law $h\sim L^{-3}$.

Figure 4

(Color online) Overlap between the exact $4\times 4$ wave function and the VB wave function versus the single independent amplitude $h(2,1)$. The value of $h(2,1)$ obtained in the variational calculation is indicated.

Figure 5

(Color online) Overlap between the exact $4\times 4$ ground state and the VB wave function for different values of the frustration $J_2∕J_1$. The best amplitudes obtained in variational Monte Carlo calculations are indicated with the dashed lines.