From Complexity to Principles: Machine Learning and Minimal Models in Physics

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Magnetic skyrmions are emergent mesoscale particles  that can play a crucial role in future  spintronic memory devices. These diminutive ‘hedgehog balls’ can be pushed around,  while maintaining their shape and exhibiting particle-like behavior. Owing to their minuscule size and remarkable stability, they are considered as prospective carriers for future information storage and logic technologies. While magnetic skyrmions and skyrmion crystals continue to captivate scientific interest, finding them in real materials remains  challenging. The ongoing pursuit to unravel the underlying principles behind their stabilization is hindered by the intricate nature of the microscopic models describing the electronic structure of the hosting material. Even modern supercomputers struggle with direct numerical simulations of these electronic models.  Physicists have traditionally tackled these challenges by deriving minimal  effective models designed to capture the essence of the original models, while retaining only the relevant  degrees of freedom. Unfortunately, the usual mathematical tools employed to derive efficient low-energy models  cannot be applied to metallic magnets, which are the predominant hosts for magnetic skyrmions. In parallel, other traditional phenomenological approaches grounded in intuition and simpler modeling have also shown severe limitations in revealing the stabilization mechanisms of skyrmion crystals. 

In our work, we address the challenge of deriving a minimal effective low-energy spin model from a "Kondo Lattice Model" (KLM). This model describes the physics of specific classes of materials, such as Lanthanide-based metallic compounds, where the spin degrees of freedom of itinerant (conduction) and localized electrons interact via the the so-called Kondo exchange J. In this scenario, the minimal low-energy model retains only the spin degrees of freedom of the localized electrons, while the effective spin-spin interactions are derived by integrating out the degrees of freedom associated with the conduction electrons. The non-analytic dependence of these spin-spin  interactions  on  J excludes the possibility of using perturbation theory beyond the second order  that leads to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. To surmount this challenge, we propose a machine learning (ML) assisted protocol to extract effective two- and four-spin interactions. The resulting spin model reproduces the phase diagram obtained with the original KLM as a function of magnetic field and easy-axis anisotropy. It also illuminates the effective four-spin interactions  that stabilize the field induced skyrmion crystal phase. Furthermore, the effective spin model enables the efficient calculation of both static and dynamical properties at a significantly reduced numerical cost compared to the original KLM. A comparison of the dynamical spin structure factor in the fully polarized phase computed with the effective spin model and the original KLM reveals a good agreement for the magnon dispersion, even though  this information was not used in the training. 

The proposed machine learning assisted protocol can be applied to a broad spectrum of problems to address not only the specific challenge posed by  KLMs, but also the more general case of fermions coupled to classical degrees of freedom. The protocol learns from a limited data set generated through the  original high-energy model and delivers an  effective low-energy model that mirrors the low-energy physics. This approach can save months of computational time, which in turn opens the door for  studying realistic materials in detail. Furthermore, the simplicity and transparency of the effective low-energy model grant new insights into the mechanisms that stabilize various field-induced magnetic orderings. Unconstrained by human bias, our technique discerns distinctive, unorthodox mechanisms that can stabilize skyrmions in real materials, and debunks prior incorrect assumptions. Besides revealing stabilization principles which remained hidden due to oversimplified assumptions, the ML derived effective model provides a fresh insight into the metallic magnets, reproduces dynamical properties that were not explicitly included in the training, and unveils new features of complex metallic magnets.

In summary, our machine learning assisted approach serves as an efficient tool to guide the search for new host materials of magnetic skyrmions by learning underlying  principles from minimal models. In essence, we are learning  from machine learning. Phenomenological approaches are  often biased towards the simplest possible explanation:  in absence of additional information, we select the simplest explanation over those that involve a larger number of assumptions and variables. While this Occam’s razor is  very useful  to guide our understanding of complex systems, it can also lead to oversimplifications caused by lack of validation of the implicit assumptions. The less biased nature of ML-assisted protocols not only allows us to correct these assumptions, but also provides enough information to infer new guiding principles applicable to broader classes of materials. In other words,  apart from expediting the computation of phase diagrams, the ML approach introduced in this study serves as a powerful learning tool, aiding in the comprehension of underlying mechanisms and accelerating the pace of discovery. Furthermore, this tool can be applied to more general models of fermions interacting with classical fields, which encompass different areas of knowledge, including quantum chemistry, condensed matter, and high-energy physics.

   

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