We report on a valence bond projector Monte Carlo simulation of the cubic lattice quantum Heisenberg model with additional higher-order exchange interactions in each unit cell. The model supports two different valence bond solid (VBS) ground states. In one of these states, the dimer pattern is a three-dimensional analogue of the columnar pattern familiar from two dimensions. In the other, the dimers are regularly arranged along the four main diagonals in 1/8 of the unit cells. The phases are separated from one another and from a Néel phase by strongly first-order boundaries. Our results strengthen the case for exotic transitions in two dimensions, where no discontinuities have been detected at the Heisenberg Néel-VBS transition driven by four-spin plaquette interactions.
The large-N limit of a nearest-neighbor SU(N) antiferromagnet on a bipartite lattice exhibits in dimensions d\ensuremath{\ge}2 a zero-temperature phase transition between a N\'eel ordered state and a resonating-valence-bond state. Here it is shown in d=1,2 that topological effects produce spin-Peierls or valence-bond-solid order in the non-N\'eel phase with a ground-state degeneracy which varies periodically with ``spin'' for fixed N with periodicity given by the coordination number of the lattice. Thus a non-N\'eel phase of the spin-(1/2 Heisenberg model on a square lattice would be a spin-Peierls state with a fourfold degeneracy due to broken lattice rotational symmetry.
Closely competing valence bond crystal orders in the ground state of the spin-21 antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study
The spin-21 antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin-21 PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with 2a1×2a2×2a3 periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in ⟨S^u2⟩ for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.