Theory

We utilize the maximally localized Wannier functions (MLWF) of Marzari and Vanderbilt [N. Marzari and D. Vanderbilt, PRB (1997); N. Marzari et al., Rev. Mod. Phys. (2012)] within the wannier90 package [A. A. Mostofi et al., Comput. Phys. Commun. (2008); A. A. Mostofi et al., Comput. Phys. Commun. (2014)] to construct effective tight-binding representation of the ab initio-calculated ground state.

Accessing the spin expectation values of the interpolated band structure with Wannier tight-binding models requires deriving the Wannier real-space spin operator.

Just like any other operator that can be obtained from the ab initio calculation (for all the ab initio k-points \(\textbf{q}\)), the spin operator in the basis of ab initio eigenstates \(\mathcal{S}_{mn}^\mathrm{H}(\textbf{q})\) can be converted into its real-space representation by first applying the semi-unitary matrix \(V_{m^{\prime} m}(\textbf{q})\) to convert \(\mathcal{S}\) from the Hamiltonian gauge to the Wannier gauge

\[\begin{equation} \mathcal{S}_{m n}^{\mathrm{W}}(\textbf{q})=\sum_{m^{\prime} n^{\prime}} V_{m^{\prime} m}^\dagger(\textbf{q}) \cdot \mathcal{S}_{m^{\prime} n^{\prime}}^\mathrm{H}(\textbf{q}) \cdot V_{n^{\prime} n}(\textbf{q}), \end{equation}\]

where the spin operator \(\hat{\mathcal{S}}\) projected onto the basis of the ab initio eigenstates \(\psi\) in its matrix form \(\mathcal{S}_{m^{\prime} n^{\prime}}^\mathrm{H}(\textbf{q}) = \left\langle\psi_{m^{\prime} \textbf{q}}|\hat{\mathcal{S}}| \psi_{n^{\prime} \textbf{q}}\right\rangle\) is obtained with the help of the vaspspn module of WannierBerri (S. Tsirkin, npj Comp. Mater. (2021)). Primed indices run over the (larger) space of disentanglement bands.

Follows the direct Fourier sum over the ab initio grid

\[\begin{equation} \mathcal{S}_{m n}(\textbf{R}) \equiv \frac{1}{N_{\textbf{q}}} \sum_{\textbf{q}} \mathcal{S}_{m n}^{\mathrm{W}}(\textbf{q}) \cdot e^{-i \textbf{q} \cdot \textbf{R}} \,, \end{equation}\]

with \(N_{\textbf{q}}\) the number of ab initio grid points \(\textbf{q}\).

Having \(\mathcal{S}_{m n}(\textbf{R})\) at hand, its interpolation \(\overline{\mathcal{S}}_{mn}^\mathrm{H} (\textbf{k})\) to an arbitrary k-vector \(\textbf{k}\) involves an inverse Fourier sum \(\overline{\mathcal{S}}_{mn}^\mathrm{W} (\textbf{k}) = \sum_\textbf{R} \mathcal{S}_{mn} (\textbf{R}) \cdot e^{i \textbf{k} \cdot \textbf{R}}\) over the real-space lattice vectors \(\textbf{R}\) followed by a rotation back to the Hamiltonian gauge of the original eigenstates \(\overline{\mathcal{S}}_{mn}^\mathrm{H} (\textbf{k}) = (U^\dagger \cdot \mathcal{S}^\mathrm{W} \cdot U)_{mn}\), where \(U_{mn}\) is a unitary matrix which diagonalizes the interpolated Hamiltonian \(\overline{\mathcal{H}}_{mn}^\mathrm{W} (\textbf{k}) = \sum_\textbf{R} \mathcal{H}_{mn} (\textbf{R}) \cdot e^{i \textbf{k} \cdot \textbf{R}}\).

See the Supplementary of L. Vojáček*, J. M. Dueñas* et al., Nano Letters (2024).