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

TL;DR

This work develops a systematic, motif-based extension of Anderson’s rigorous lower bounds to the frustrated pyrochlore Heisenberg antiferromagnet for spin-$S$ spins. By organizing local sub-Hamiltonians into a hierarchy of clusters—the star, tetrahedron, seven-site hourglass, and the 18-site crown—the authors derive increasingly tight lower bounds, with a closed-form bound for arbitrary $S$ in the hourglass motif and exact results for $S=\tfrac{1}{2}$ and $S=1$ on the crown cluster. The approach generalizes to extended models including $J_1$-$J_2$-$J_{3b}$ exchanges, ring exchange, and scalar spin chirality, often producing piecewise-analytic bounds governed by level crossings; notably, the Klein point saturates the hourglass bound in the quadrilinear case. The results provide tight, rigorous benchmarks that complement numerical estimates and variational bounds, and they point to a scalable path toward even larger motifs and broader lattice geometries in frustrated quantum magnets.

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.

Figures (8)

• Figure 1: Pyrochlore lattice consisting of corner-sharing tetrahedra, with an embedded hourglass cluster (highlighted in red) serving as a representative motif for defining the sub-Hamiltonian $\mathcal{H}_i$. In this case, there is an hourglass for each site ($M=N$ in Eq. (\ref{['eq:lower_bound_sum']})).
• Figure 2: Clusters used to derive rigorous lower bounds on the ground‑state energy. (a) The star cluster of Anderson’s original argument, comprising a central spin (labelled '1') coupled to its neighbouring spins (sites '2'--'7'). (b) The tetrahedron cluster, which incorporates the intrinsic frustration of the pyrochlore lattice and therefore gives a tighter bound. (c) The seven‑site hourglass cluster, which merges the star and tetrahedron motifs and further improves the lower bound.
• Figure 3: The energy spectrum of the 7-site hourglass subHamiltonian, Eq. (\ref{['eq:motifham7_nn']}), as a function of the free parameter $J'$ (we set $J=1$). The optimal lower bound $-0.5625$ is achieved for $J'=J^*=3/8$, where the energies of the different irreducible representations cross. For $J'=0.5$, we recover the lower bound (red circle) from the star-cluster (Sec. \ref{['sec:star']}), while for $J' = 1/4$, the blue circle indicates the case of equal exchange couplings on all bonds.
• Figure 4: (a) Ground-state symmetry sectors of the hourglass Hamiltonian with second-neighbor coupling $J_2$, as a function of $J'$. Grey lines denote transitions between distinct symmetry sectors. The black curve marks the loci of maximal ground-state energies across symmetry sectors, which define a lower bound on the true ground-state energy. (b) Lower bound $E_0$ for different values of $J_2$. The lower bound closely follows the boundaries between regions where the ground state becomes degenerate.
• Figure 5: (a) Ground-state symmetry sectors of the hourglass Hamiltonian with bilinear and quadrilinear couplings, as a function of $\tilde{J}$, which is defined in Sec. \ref{['sec:foursite_appendix']}. Grey lines denote transitions between distinct symmetry sectors. The black curve marks the loci of maximal ground-state energies across symmetry sectors, which define a lower bound on the true ground-state energy. The burgundy dot marks the Klein point for $K/J=4/5$. (b) Lower bound $E_0$ for different values of $K$. The lower bound closely follows the boundaries between regions where the ground state becomes degenerate. The burgundy dot marks the Klein point.