Calculation of the biquadratic spin interactions based on the spin cluster expansion for \textit{ab initio} tight-binding models

TL;DR

The paper develops the SCE-DLM method to extract bilinear and biquadratic spin interactions from ab initio tight-binding models by merging spin cluster expansion with disordered local moments. It provides a general, reference-state-robust route to map electronic structure to a classical spin Hamiltonian H = -2 \sum_{⟨i,j⟩} [ J_{ij}(\bm{e}_i·\bm{e}_j) + B_{ij}(\bm{e}_i·\bm{e}_j)^2 ], enabling systematic access to higher-order spin couplings without heavy supercell fittings. The methodology is validated against a 1D two-sublattice Hubbard model and benchmarked on Fe, Co, and Co-based compounds, showing good agreement with LKAG/KKR results and experimental trends, including noncollinear states in Co$_{1/3}$TaS$_{2}$ and interstitial-moment systems like K$_4$Al$_3$(SiO$_4$)$_3$. The work demonstrates broad applicability to complex materials, offering an ab initio path to understand electronic mechanisms driving biquadratic interactions and related nontrivial magnetic orders. It also outlines limitations related to longitudinal fluctuations and nonlocal correlations, suggesting future extensions to cluster-CPA and relativistic terms for enhanced realism.

Abstract

We develop a calculation scheme using \textit{ab initio} tight-binding Hamiltonians to evaluate biquadratic magnetic interactions. This approach relies on the spin cluster expansion combined with the disordered local moment (DLM) method, originally developed within the multiple scattering Korringa-Kohn-Rostoker method. Applying it to a single-orbital Hubbard model with two sublattices, we show that the evaluated DLM biquadratic interactions are in good agreement with those obtained from the strongly correlated limit, demonstrating the wide applicability of the method to various magnetic systems with large local moments. We then apply it to the \textit{ab initio} tight-binding models for elemental magnetic metals; the resulting magnetic interactions align well with previous literature. Finally, we explore its performance in more complex compounds, such as transition metal dichalcogenides with intercalation of 3\textit{d} transition metals and potassium electrosodalite. The obtained results for both compounds show good agreement with experiments. The present approach offers a convenient \textit{ab initio} path for evaluating biquadratic interactions and understanding the electronic mechanisms controlling them.

Figures (30)

• Figure 1: One-dimensional single-orbital Hubbard model with two sublattices. The Hamiltonian has three types of transfer integrals, $t$, $t'$, and $t"$, defined for the pairs of sublattices shown with the solid, dashed, and double lines, respectively.
• Figure 2: BL ($J^{\mathrm{C}}_{ij}/J^{\mathrm{Q}}_{ij}$) and BQ ($B^{\mathrm{C}}_{ij}/B^{\mathrm{Q}}_{ij}$) interactions in the classical/quantum spin Hamiltonian for $t=t'=t"$ in the limit of strong correlation $t\ll U$. We set $B=U/2=10$ in the calculations. $J^{\mathrm{C}}_{ij}$ (solid red line) and $B^{\mathrm{C}}_{ij}$ (solid blue line) are evaluated by SCE-DLM, and $J^{\mathrm{Q}}_{ij}$ (dashed red line) and $B^{\mathrm{Q}}_{ij}$ (dashed blue line) are evaluated perturbatively (see Eq. (\ref{['eq:jij-pert']})).
• Figure 3: Band structure of (a) the FM state of bcc Fe. (b)-(d) are those of the FM, AFM, and AFM-$ab$ states of hcp Co. (e) and (f) are those of the FM and AFM states of fcc Co. Blue and red lines are those for the up and down spin components obtained by SDFT calculation, and green and orange lines are those obtained by Wannier interpolation. The labels of the Brillouin zone are defined based on the primitive cell of each lattice.
• Figure 4: Density of states (DOS) for the spin-up and spin-down components (a) and the integrated DOS (b) for bcc Fe. In panel (a), the blue (red) line represents the DOS for the FM (DLM) state. The spin-up and spin-down components of both the FM and DLM states are plotted on the positive and negative sides, respectively. In panel (b), the vertical black line indicates the chemical potential of the FM state ($\mu=0$), and the vertical red line indicates the chemical potential $\mu_c$ for the DLM state. The horizontal line indicates the number of electrons at $\mu=0$ in the ab initio tight-binding Hamiltonian.
• Figure 5: Density of states of the DLM states obtained by the ab initio tight-binding method (blue) and the KKR method (red). The energy axis is calibrated by subtracting $\mu_c$ from each result.