Unlocking the Potential of Machine Learning in the Derivation of Low-Energy Models for Metallic Magnets

Condensed matter physics often grapples with complex many-particle problems lacking definitive closed-form solutions, necessitating approximation strategies to investigate low-energy sectors of the Hilbert space. Perturbation theory, though widely used for this purpose, is limited when expansion terms diverge. This work introduces a machine learning (ML) assisted protocol to extract effective low-energy models for lattice models of fermions interacting with classical fields, specifically focusing on the Kondo Lattice Model (KLM).

Skyrmions, featuring whirling spin texture and topological protection, are promising candidates for future spintronic devices. Materials featuring conduction electrons coupled to localized $f$-electrons' net moment are ideal for realizing skyrmions and can be modeled using the KLM. The RKKY model approximates the KLM in the weak-coupling regime. Away from this regime, significant four-spin interactions that are non-analytic functions of the coupling constant emerge, rendering perturbative treatment inapplicable. We employ our supervised ML-assisted protocol to extract effective two- and four-spin interactions by integrating out the KLM’s conduction electrons. Utilizing the resultant low-energy effective Hamiltonian, both in real and momentum space, we explore multiple-${\bm Q}$ magnetic orderings on a triangular KLM. We demonstrate the existence of the skyrmion phase even in the absence of spin anisotropy, unattainable within the pure RKKY model. The effective spin models reproduce the original KLM’s phase diagram, revealing the effective four-spin interactions stabilizing field-induced skyrmion crystal phases. The ML-derived minimal spin models enable efficient computation of static and dynamical properties at a significantly reduced numerical cost. A comparison of the dynamical spin structure factor in the fully polarized phase computed with the effective models and the original KLM reveals good agreement for the magnon dispersion, even though this information was not incorporated during training.

Our ML-assisted protocol identifies patterns from limited data sets, generating effective models that save substantial computational time while reproducing the original model's low-energy physics. Free from human bias, this approach unveils unconventional skyrmion stabilization mechanisms and debunks prior incorrect assumptions. It efficiently aids the search for new magnetic skyrmion hosts, accelerating simulations and paving the way for efficient exploration a more expansive class of materials.