Mag4: An Automated First-Principles Workflow to Extract Magnetic Interactions, from Pair Couplings to Four-Spin Ring Exchange

Abstract

Chemists design magnetic materials through structure-property relationships built on pairwise exchange couplings $J$, now extracted routinely from first principles by energy-mapping. But wherever four magnetic centers close a loop---as in cuprate CuO$_2$ planes or infinite-layer square nets---a four-spin ring exchange $J_{\rm ring}$ can arise, reshaping ground states and corrupting the $J$ values used for design. Missing has been a direct, local way to obtain it, as the four-state method gives for $J$. We supply it by generalizing that method to a sixteen-state ($2^4$) scheme, automated in the open Mag4 package---the first automatic implementation of the four-state family. From one cif file, Mag4 proposes the supercell that isolates the couplings, generates DFT inputs in the user's chosen functional, extracts $J$ and $J_{\rm ring}$ with band-gap and local-moment diagnostics, and predicts the magnetic order, propagation vector, and critical temperature. Symmetry reduces the sixteen configurations to six or eight inequivalent energies, so the cost is modest. The derivation shows the conventional four-state $J$ is itself ring-renormalized by $\mp 2 J_{\rm ring} S^2$, the sign set by the reference state. T-La$_2$CuO$_4$ confirms this: three routes agree on $J_{\rm ring}$ to $0.2\%$, giving $J_{\rm ring}/J_1=0.25$, and a four-state $J_1$ quoted without its reference is wrong by $12\%$. The sixteen-state value is itself reference-dependent---N\'eel and ferromagnetic baths bracket it---a fourth-order fingerprint of interactions beyond the pair-plus-ring model, resolved by further baths into a bare $J_{\rm ring}$ and a converging tower of six- and eight-spin loop couplings. SrFeO$_2$ ($S=2$), same plaquette yet $J_{\rm ring}/J=0.006$, is the negative control: a plaquette is necessary for ring exchange but far from sufficient---a structure-property criterion for screening new multi-spin magnets.