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

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 optimized lower bound that admits a closed-form expression for arbitrary spin S . We also derive rigorous lower bounds for generalized models with further-neighbor exchange, ring exchange, and scalar spin-chirality interactions. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>S</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:math> 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 GS ≥ −0.549832 and e GS ≥ −1.632985, respectively. We compare the resulting bounds with numerical estimates of the ground-state energy from the literature.