2026/07/21 by Xavier Rocquefelte, Peter Blaha
#cond-mat.mtrl-sci #cond-mat.str-el #quant-ph
Four-spin ring (cyclic) exchange Jring is an essential ingredient of the Heisenberg spin Hamiltonian of cuprates and other square-lattice magnets, yet it has lacked the kind of direct, local first-principles extraction that the four-state method provides for bilinear exchange, J. We supply it by generalizing that method to a sixteen-state (24) scheme. Symmetry reduces the sixteen configurations to six or eight inequivalent energies, so the cost is modest. The derivation also shows that the conventional four-state magnetic coupling, J, is itself ring-renormalized, by ± 2 Jring S2 with the sign set by the reference state. T-La2CuO4 confirms this quantitatively: three independent routes agree on Jring to 0.2%, giving Jring/J1 = 0.25, and a four-state J1 quoted without naming its reference is wrong by 12% in this material. The direct sixteen-state extraction itself proves reference-dependent, the Néel and ferromagnetic baths bracketing the mapping value: a fourth-order fingerprint of interactions beyond the pair-plus-ring model, which additional reference baths resolve into a bare Jring and a converging tower of six- and eight-spin loop couplings. SrFeO2 (S = 2), with the same plaquette yet Jring/J = 0.006, provides the negative control: a plaquette is necessary for ring exchange, far from sufficient. The complete workflow, including the band-gap and local-moment diagnostics that certify any such extraction, is implemented in the openly available Mag4 package, so that Jring costs no more effort to obtain than J.