Energy landscape of the kagome antiferromagnet: characterization of multiple energy scales

We investigate the energy landscape of the kagome Heisenberg antiferromagnet within its coplanar ground-state manifold. Although coplanar states are degenerate at harmonic order, transitions between them require collective weathervane-loop rotations whose barriers grow strongly with loop size. To characterize this structure, we construct disconnectivity graphs using two complementary approaches: exact enumeration and minimax-barrier calculations for small lattices, and a statistical construction for large lattices based on random walks through configuration space, with loop length used as a proxy for barrier height. The exact landscape reveals a dominant low-barrier scale associated with elementary six-spin loops and a broader higher-barrier sector from longer rearrangements. For large systems, the statistical analysis exposes a hierarchy of barrier scales, including a pronounced six-spin-loop peak and an intermediate scale-free regime of loop lengths. This hierarchy provides a natural basis for multiple dynamical time scales: six-spin loops govern the fastest local relaxation, while slower collective dynamics arise from activation of longer loops. These results show that the coplanar manifold is dynamically rugged, with its low-energy dynamics governed by a hierarchy of loop-mediated barriers.