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

TL;DR

This paper develops an optimized variational wave function in the valence bond basis for 2D S=1/2 Heisenberg antiferromagnets, achieving high accuracy in energy and correlations for large lattices.

Contribution

It introduces a method to optimize all valence bond amplitudes without assuming a functional form, improving accuracy over previous approaches.

Findings

Energy deviation from exact results is only 0.06% for large systems.

Long-range valence bond amplitudes decay as 1/r^3.

Spin correlations are accurately reproduced within 2% at long distances.

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*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. The amplitudes h(r) for valence bonds of long length r decay as 1/r^3. We also discuss some results for small frustrated lattices.