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 h͑x , y͒ for valence bonds connecting spins separated by ͑x , y͒ lattice spacings. In contrast to previous studies, in which a functional form for h͑x , y͒ was assumed, we here optimize all the amplitudes for lattices with up to 32ϫ 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 Ϸ0.06% from its exact value ͑calculated using unbiased quantum Monte Carlo simulations͒. The spin correlations are also well reproduced, falling Ϸ2% below the exact ones at long distances ͑corresponding to an Ϸ1% underestimation of the sublattice magnetization͒. The amplitudes h͑r͒ for valence bonds of long length r decay as r −3 . We also discuss some results for small frustrated lattices.

Resonating valence bond trial wave functions with both static and dynamically determined Marshall sign structure

“…(Still, the total number of parameters grows only linearly with the number of spins, which is radically slower than the number of states in the total spin singlet sector.) Previous calculations of this kind 34,35 considered only the checkerboard AB pattern and imposed on h(r) = h(x, y) the full symmetry of the lattice, such that h(x, y) = h(|x|, |y|) = h(|y|, |x|). In this calculation, we impose a less restrictive condition, h(x, y) = h(|x|, |y|), that respects reflection symmetry across the lines x = 0 and y = 0 but not across the lines y = ±x.…”

“…Our approach is inspired by Ref. 34, but there are several important differences. The first is simply the scale of the calculation: we have simulated a large number of lattice sizes up to L = 32 on a dense grid of relative coupling values (g = J 2 /J 1 ranging from 0 to 1 in steps of δ g = 0.01).…”

“…As in Ref. 34, we make use of an unbiased, stochastic optimization scheme. Changes to the h(r) values are made in the downhill direction of the local energy gradient.…”

Plaquette renormalization scheme for tensor network states

“…We then update the parameters based on the derivatives, normally using a stochastic optimization scheme of the kind discussed in Ref. 18. The plaquette renormalizations and the derivative calculations are highly parallelizable, which we take advantage of by using a massively parallel computer [19].…”

“…We update the tensors, using the steepest decent method or a stochastic scheme were only the sign of the derivatives is used [18] (which to a large extent avoids trapping in local minimas). In this process we normalize the elements of the T 0 (σ) and S n tensors such that the largest (in magnitude) elements in each are of order 1 [in the case of T 0 (σ) taken as the largest among all elements for σ = ±1, since the σ =↑ and ↓ tensors cannot be rescaled independently of each other).…”