AI 中文总结
研究四自旋环交换\(J_{ring}\)的提取方法，通过将四态方法推广到十六态方案，在\(La_2CuO_4\)和\(SrFeO_2\)上验证，实现了\(J_{ring}\)的有效提取，且提取过程依赖参考，完整工作流程在Mag4包中实现。
AI 中文摘要
四自旋环（循环）交换\(J_{ring}\)是铜酸盐和其他方形晶格磁体的海森堡自旋哈密顿量的重要组成部分，但它缺乏像四态方法为双线性交换\(J\)提供的那种直接的、局部的第一性原理提取。我们通过将该方法推广到十六态（\(2^4\)）方案来提供这种提取。对称性将十六种构型减少到六种或八种不等价能量，因此成本适中。推导还表明，传统的四态磁耦合\(J\)本身通过\(\pm 2 J_{ring} S^2\)进行环重整化，符号由参考态确定。\(T-La_2CuO_4\)定量地证实了这一点：三条独立路径在\(J_{ring}\)上的一致性达到\(0.2\%\)，得到\(J_{ring}/J_1 = 0.25\)，并且在此材料中未提及参考的四态\(J_1\)错误了\(12\%\)。直接的十六态提取本身证明是依赖参考的，奈尔态和铁磁态浴包围映射值：这是对成对加环模型之外相互作用的四阶指纹，额外的参考浴将其分解为裸\(J_{ring}\)以及六自旋和八自旋环耦合的收敛塔。\(SrFeO_2\)（\(S = 2\)）具有相同的格子但\(J_{ring}/J = 0.006\)，提供了负对照：格子对于环交换是必要的，但远远不够。完整的工作流程，包括认证任何此类提取的带隙和局部矩诊断，在公开可用的Mag4包中实现，因此获得\(J_{ring}\)并不比获得\(J\)花费更多精力。
英文摘要
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---Néel 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.
Comments42 pages, 6 figures, 6 Tables
