Continuum models and Morse Theory in the simulation, design and evaluation of magnetic Skyrmion devices

Nanomagnetic devices such as computer gates and memory devices based on magnetic skyrmions are close to becoming a reality. In this paper we will explore the highly nonconvex nanomagnetic energy landscape in order to draw conclusions about the complexity of magnetic phenomena. Morse theoretic arguments show that, in a bounded energy interval, the number of critical points of the energy functional grows exponentially. To show this, we introduce a hierarchy of models for the design of nanomagnetic devices to provide a solid foundation for the introduction of topological tools. To reason in terms of lattice models, one must make a distinction between two types of lattices: the quantum mechanical model of the actual physical lattice and the lattice model that can be associated with a discretization of a continuum model of the physics. By focusing on the implications of Morse theory applied to lattice systems arising from the discretization of the continuum models, and the notion of “topological frustration”, we provide a framework for understanding “complexity” in the context of nanomagnetic systems. We conclude with some suggestions for making the analysis more qualitative.