The spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice (PAFH) is arguably the most well-known strongly frustrated quantum magnet in three spatial dimension. However, because of the rapid scaling of Hilbert space with the linear size of such a three-dimensional system, most previous studies of this model are conducted on rather small clusters and the nature of its ground state in the thermodynamic limit remains elusive after about 30 years of intense debate. Here, we apply a recently developed powerful algorithm to perform large-scale unrestricted variational optimization of the ground state of the spin-$\frac{1}{2}$ PAFH from the perspective of the resonating valence bond (RVB) theory. The largest cluster that we have attempted contains as many as 2048 sites and the corresponding variational wave function contains as many as 16 777 216 variational parameters. Through such a large-scale variational optimization, we find a highly competitive candidate ground state for the spin-$\frac{1}{2}$ PAFH. This state features a maximally resonating valence bond crystal (VBC) pattern with $2{\vec{a}}_1\times 2{\vec{a}}_2\times 2{\vec{a}}_3$ 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. As a result, there is strong disparity in the extent of ice rule violation between up and down tetrahedrons in this state. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state obtained from dressing hard hexagon covering of the pyrochlore lattice, although the extent of ice rule violation is totally uniform in the latter case. We find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization. In particular, strong disparity in the extent of ice rule violation for up and down tetrahedrons will also emerge in this 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 VBC state.

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• Exchange interaction
• Quantum spin liquid
• Antiferromagnets
• Pyrochlores
• Heisenberg model
• Monte Carlo methods
• Resonating valence bond theory
• Trial wave functions