Anatomy of linear and non-linear intermolecular exchange in S = 1 nanographenes

Nanographene triangulenes with a S = 1 ground state have been used as building blocks of antiferromagnetic Haldane spin chains realizing a symmetry protected topological phase. By means of inelastic electron spectroscopy, it was found that the intermolecular exchange contains both linear and non-linear interactions, realizing the bilinear-biquadratic Hamiltonian. Starting from a Hubbard model, and mapping it to an interacting Creutz ladder, we analytically derive these effective spin-interactions using perturbation theory, up to fourth order. We find that for chains with more than two units other interactions arise, with same order-of-magnitude strength, that entail second neighbor linear, and three-site non-linear exchange. Our analytical expressions compare well with experimental and numerical results. We discuss the extension to general S = 1 molecules, and give numerical results for the strength of the non-linear exchange for several nanographenes. Our results pave the way towards rational design of spin Hamiltonians for nanographene based spin chains.