Nonperturbative calculation of exchange coupling parameters

TL;DR

This work demonstrates that a nonperturbative, self-consistent supercell approach ((SC)$^2$) to extract exchange parameters $J_{ij}$ yields consistent results for finite-angle spin fluctuations across SrMnO$_3$, Nd$_2$Fe$_{14}$B, Nd$_2$Co$_{14}$B, and elemental 3d metals, addressing failures of perturbative magnetic force theorem mappings. By explicitly accounting for finite-angle-induced changes in charge and spin densities, the method reveals density-change contributions, $igO( abla n^2, abla m^2)$, as the primary source of discrepancies with MFT and explains material-specific trends in Curie temperatures. The study contrasts SC$^2$ with the spin-spiral method and situates it within spin-cluster expansion, showing that higher-order cluster effects renormalize into $J_{ij}$ and depend on the sampled magnetic configurations. The results yield practically reliable spin models and Tc predictions, suggesting a complementary path to conventional MFT-based approaches for designing magnetic materials. This nonperturbative framework provides a practical route to more quantitative spin models, with potential extensions to include SOC, lattice vibrations, and higher-order interactions, and the authors plan to release the code to the community.

Abstract

Exchange coupling parameters $J_{ij}$ within the Heisenberg model and its extensions are crucial for understanding magnetic behavior at the atomic level. Perturbative approaches based on the magnetic force theorem (MFT, often called the Liechtenstein method) are well established in the infinitesimal-rotation limit, yet the accuracy of such mappings for finite-angle spin fluctuations -- including slightly to moderately disordered spin configurations -- remains to be clarified. Here we evaluate $J_{ij}$ nonperturbatively for systems of both fundamental and practical interest, including perovskite SrMnO$_3$, neodymium-magnet compounds, and elemental 3$d$ transition metals, and compare the results with MFT-based calculations. The nonperturbative approach provides results that remain consistent for both small and large finite spin rotations, thereby complementing MFT-based methods by extending validated applicability beyond the infinitesimal-rotation limit. Further analysis shows that this consistency arises from explicitly incorporating finite-angle-induced changes in charge and spin densities -- effects that are neglected in perturbative mappings -- into the extracted $J_{ij}$. This provides a practical route to more quantitative spin models for the analysis and design of magnetic materials.

Figures (10)

• Figure 1: Exchange coupling parameters at the first-nearest neighbor Mn-Mn pair $J_{01}$ in cubic SrMnO$_3$ with the type--G antiferromagnetic state. The black and gray bars represent the results obtained from the total energy method in Eq. (\ref{['ediff']}) and the MFT-based method, respectively. The other colored bars show the results from the (SC)$^2$ method with three different upper bounds for the polar angle $\theta$: 10$^\circ$ (blue), 15$^\circ$ (green), and 20$^\circ$ (orange). The lattice constants were set to $a/a_{\rm eq} =$ 1.0 and 1.05, where $a_{\rm eq} =$ 3.79 Å.
• Figure 2: Total energy variation (per formula unit) for the lattice constant $a/a_{\rm eq} = 1.05$ as a function of the rotation angle $\alpha$ of Mn magnetic moments, illustrated in the lower-right. The type--G antiferromagnetic state corresponds to $\alpha = 180^\circ$. The black line represents DFT total energy. The blue, dashed green, and red lines depict the energy variations using Eq. (\ref{['energy_variation']}) with $J_{ij}$ obtained from the (SC)$^2$ method for $\theta_{\rm max} = 10^\circ$, $\theta_{\rm max} = 20^\circ$, and the MFT-based method, respectively. The inset shows a magnified view of the energy variations in the range between $170^\circ$ and $180^\circ$.
• Figure 3: Element-decomposed band dispersions of SrMnO$_3$ with and without rotation of Mn magnetic moments, as illustrated in the inset. Panel (a) corresponds to the equilibrium lattice constant $a/a_{\rm eq} = 1.0$, whereas panel (b) corresponds to $a/a_{\rm eq} = 1.05$. The blue, red, and green circles denote the contributions from Mn, O, and Sr, respectively. The band dispersions and the element contributions were plotted using VASPKIT Wang2021-vv.
• Figure 4: Charge density differences $\Delta n(\bm{r})$ in SrMnO$_3$ as defined in Eq. (\ref{['diffden']}) between rotation angles $\alpha = 170^\circ$ and $\alpha = 180^\circ$, drawn using VESTA software Momma2011-yj. Panel (a) corresponds to the equilibrium lattice constant $a/a_{\rm eq} = 1.0$, whereas panel (b) corresponds to $a/a_{\rm eq} = 1.05$. The isosurface levels represented in yellow and blue are set to $+2\times10^{-5}$ and $-2\times10^{-5}$ bohr$^{-3}$, respectively.
• Figure 5: Differences in $T_{\rm C}$ of Nd$_2$Co$_{14}$B and Nd$_2$Fe$_{14}$B. White and gray bars show MFT-based results obtained from FM and LMD reference states, respectively. Black bars denote experimental values. Blue, green, and orange bars indicate (SC)$^2$ results with the maximum rotation angle set to $\theta_{\rm max} = 10^\circ$, $20^\circ$, and $30^\circ$, respectively. All theoretical values are obtained using the mean-field approximation. Experimental values are taken from Ref. Sagawa1984-rpFuerst1988-qm.