Complex spin Hamiltonian represented by an artificial neural network

The effective spin Hamiltonian method is very useful for simulating and understanding the behavior of magnetism. However, it is not easy to construct an appropriate spin Hamiltonian for a magnetic system, especially for complex magnets such as itinerant topological magnets. Here, we put forward a machine learning (ML) approach, applying an artificial neural network (ANN) and a local spin descriptor to construct an effective spin Hamiltonian for any magnetic system. The obtained Hamiltonians include an explicit Heisenberg part and an implicit nonlinear ANN part. Such a method successfully reproduces artificially constructed models and also accurately describes the itinerant magnetism of bulk ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$. Our work paves a new way for investigating complex magnetic phenomena (e.g., skyrmions) using ML techniques.

Figure 1

Workflow of the present ANN method to obtain spin Hamiltonians. (a) Illustration of the local structure centered on spin ${\mathit{S}}_0$ within a cutoff radius of $R_c$. The $\mathrm{Cu}M{\mathrm{O}}_2$ structure is shown here as an example, where the red balls indicate magnetic sites of $M$. (b) The local spin descriptor centered on spin ${\mathit{S}}_0$, involving all neighboring spins within cutoff radius $R_c$. (c) Demonstration that one set of NN is enough for two different spin classes, as the descriptor of $A$ spin class includes the spin interaction information of the $B$ site. (d) The whole Hamiltonian that contains an explicit Heisenberg part and an implicit part, which includes all other forms of interactions. The total energy consists of energies from both parts.

Figure 2

Schematization of the local structure that is involved in a descriptor. (a) The input spin structure under $e$ point group operation. (b) The spin structure under $s_6^1$ point group operation with site 0 as the center. The descriptor with the lower value of ${\mathbf{S}}_0·{\mathbf{S}}_1$ is selected as the final descriptor. Blue balls represent the magnetic sites with the red vectors denoting the spin vectors. The order of the spins that appear in the descriptor is denoted by the number imposed on the magnetic sites.

Figure 3

(a) Schematization of the $\mathrm{Cu}M{\mathrm{O}}_2$-type structure, where only the magnetic $M$ sites are shown. The atoms connected by red lines form an example of the four-body cluster, for which the fourth-order interaction is considered in Eq. (2). (b) Comparison of the energies among three different fitting methods—only Heisenberg terms, only NN and both Heisenberg and NN. The mean absolute error (MAE) of each fitting method is shown in the legend. (c) The $\left(\uparrow \uparrow \downarrow \downarrow \right)$ AFM magnetic ground state of the exact model Hamiltonian is reproduced by the HB + NN method. (d) The frustration induced a short period spiral, which is the ground state predicted by the pure Heisenberg method.

Figure 4

(a) Crystal structure of bulk ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$. The unit cell consists of two van der Waals layers with six Fe atoms, that are categorized into two types, two ${\mathrm{Fe}}^{\mathrm{mid}}$ and four ${\mathrm{Fe}}^{\mathrm{top}}/\mathrm{F}{\mathrm{e}}^{\mathrm{bot}}$. (b) Comparison of the CPU time between the single-NN method and two-NN method, as a function of magnetic atom number. Both sets of simulations are performed on Intel Xeon Gold 6244. The inset indicates similar accuracy of the two methods. (c) Comparison among energies from DFT and three different fitting strategies. The MAE of each method is shown in the legend.