Since the 1950s, Heisenberg and others have addressed the problem of how to explain the appearance of countable particles in continuous fields1. Stable localized field configurations were searched for an ingredient for a general field theory of elementary particles, but the majority of nonlinear field models were unable to predict them. As an exception, Skyrme succeeded in describing nuclear particles as localized states, so-called 'skyrmions'2. Skyrmions are a characteristic of nonlinear continuum models ranging from microscopic to cosmological scales3,4,5,6. Skyrmionic states have been found under non-equilibrium conditions, or when stabilized by external fields or the proliferation of topological defects. Examples are Turing patterns in classical liquids7, spin textures in quantum Hall magnets8, or the blue phases in liquid crystals9. However, it has generally been assumed that skyrmions cannot form spontaneous ground states, such as ferromagnetic or antiferromagnetic order, in magnetic materials. Here, we show theoretically that this assumption is wrong and that skyrmion textures may form spontaneously in condensed-matter systems with chiral interactions without the assistance of external fields or the proliferation of defects. We show this within a phenomenological continuum model based on a few material-specific parameters that can be determined experimentally. Our model has a condition not considered before: we allow for softened amplitude variations of the magnetization, characteristic of, for instance, metallic magnets. Our model implies that spontaneous skyrmion lattice ground states may exist generally in a large number of materials, notably at surfaces and in thin films, as well as in bulk compounds, where a lack of space inversion symmetry leads to chiral interactions.
ler
