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Using high-accuracy variational Monte Carlo based on group-convolutionalneural networks (GCNNs), we obtain the symmetry-resolved low-energy spectrum ofthe spin-1/2 Heisenberg model on several highly symmetric fullerene geometries,including the famous C$_{60}$ buckminsterfullerene. We argue that as the degreeof frustration is lowered in large fullerenes, they display characteristicfeatures of incipient magnetic ordering: correlation functions showhigh-intensity Bragg peaks consistent with N\'eel-like ordering, while thelow-energy spectrum is organised into a tower of states. Competition withfrustration, however, turns the simple N\'eel order into a noncoplanar one.Remarkably, we find and predict chiral incipient ordering in a large number offullerene structures.

Noncoplanar and chiral spin states on the way towards N\'eel ordering in fullerene Heisenberg models