Introduction to SUPERHEX

SUPERHEX is a Python code that generates all possible supercell structures within a specified size range, then ranks them based on the farthest Heisenberg exchange interactions that can be computed for each structure.

Using a supercell with various magnetic configurations allows us to map ab initio results, such as those from DFT, to the Heisenberg Hamiltonian. The following figure from Phys. Rev. B 108, 144413 (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.108.144413) illustrates this mapping:

Mapping of DFT to Heisenberg Hamiltonian

However, due to periodic boundary conditions, these results are limited to specific nearest neighbors. To push this limit and consider more interactions, larger supercells are needed, though this comes with increased computational costs.

Typically, researchers generate supercells by multiplying lattice vectors by integers. However, this method can lead to very time-consuming ab initio calculations. In contrast, SUPERHEX generates all possible supercell structures, analyzes the coefficient matrix obtained from the Heisenberg Hamiltonian, and then identifies the optimal supercell structure. This approach helps find the smallest supercell required for a given number of exchange interactions, potentially speeding up DFT calculations by 1 to 2 orders of magnitude.

When we map ab initio results of various magnetic configurations, we obtain the following matrix equation:

\[\mathbb{A} \mathbb{J} = \mathbb{E}\]
\[\begin{split}\begin{pmatrix} 1 & \alpha_{1,1} & \alpha_{1,2} & \cdots & \alpha_{1,m} \\ 1 & \alpha_{2,1} & \alpha_{2,2} & \cdots & \alpha_{2,m} \\ 1 & \alpha_{3,1} & \alpha_{3,2} & \cdots & \alpha_{3,m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_{n,1} & \alpha_{n,2} & \cdots & \alpha_{n,m} \end{pmatrix} \begin{pmatrix} c_0 \\ J_1 \\ J_2 \\ \vdots \\ J_m \end{pmatrix} = \begin{pmatrix} E_1 \\ E_2 \\ E_3 \\ \vdots \\ E_n \end{pmatrix}\end{split}\]

Using null space analysis of matrix \(\mathbb{A}\), we can identify the first dependent column (e.g., \(1 + q\)), indicating that exchange interactions can be calculated up to the \(q\)-th nearest neighbor. By generating all possible supercell structures within a range of volumes and applying null space analysis, we can identify the smallest supercell structures that capture the maximum number of exchange interactions. Using these optimal structures can significantly accelerate ab initio calculations for exchange interactions, potentially speeding them up by 1 to 2 orders of magnitude.

For more information on this technique, please refer to:

https://arxiv.org/abs/2410.14356

Please cite the above reference if you use this method.