Rigorous Anderson-type lower bounds on the ground-state energy of the pyrochlore Heisenberg antiferromagnet

Abstract

We construct rigorous Anderson-type lower bounds on the ground-state energy of the spin-$S$ Heisenberg antiferromagnet on the pyrochlore lattice. By formulating and optimizing a hierarchy of local cluster motifs ordered by size, we generate a sequence of increasingly tight bounds. A seven-site "hourglass" cluster composed of two corner-sharing tetrahedra furnishes an optimal lower bound that admits a closed-form expression for arbitrary spin $S$. We also derive exact lower bounds for generalized models with further-neighbor exchange, ring exchange, and scalar spin-chirality interactions. For $S=1/2$ and $S=1$, numerical optimization of an 18-site "crown" cluster containing a hexagonal loop yields rigorous lower bounds on the ground-state energy per site of the nearest-neighbor Heisenberg model with unit exchange, $e_\mathrm{GS} \geq -0.549832$ and $e_\mathrm{GS} \geq -1.632985$, respectively. We compare the resulting bounds with numerical ground-state energy estimates from the literature.