Hopfions in Condensed Matter

Dear Colleagues,

The more complex topological cousin of a skyrmion (a noncollinear spin configuration), called a hopfion, is now beginning to be observed experimentally in many materials such as ferromagnets, ferroelectrics, liquid crystals, and even in the context of Bose‒Einstein condensates (BEC) and photonics. Micron-sized hopfions have been observed using fluorescence polarizing microscopy in a chiral ferromagnetic fluid (liquid crystal colloids in a nematic host) by way of twisting and linking the magnetization. The growing ubiquity of topological defects in materials, ranging from domain walls, vortices, and skyrmions to effective magnetic monopoles (such as in artificial spin ice) and now hopfions, points to an emerging new frontier of “topological engineering” just like band engineering in order to achieve unprecedented material properties. 

A certain field configuration defines these topological defects. The field could be magnetization (for magnetic materials), polarization (for ferroelectrics), director (for liquid crystals), or local spin rotation (for BEC). A vortex is a mapping of the field from the surface of a 2D sphere (i.e., circle, S¹) to another circle. This S¹-to-S¹ mapping defines a vortex. For skyrmions, the field can be viewed as residing on the surface of a 3D sphere (S²). It can be mapped uniquely to the surface of a physical 3D sphere. This S²-to-S² mapping defines a (baby) skyrmion. The field for a hopfion lives on the surface of a 4D sphere (S³). It can also be mapped uniquely to the surface of a physical 3D sphere. This S³-to-S² mapping characterizes a hopfion. Thus, two points on the physical sphere (P,Q) correspond to (what are called preimages) great circles on S³ and they are linked. This linking is a salient feature of the hopfion. Essentially, the hopfion is a stable three-dimensional, knotted or linked topological soliton. In a hopfion, any two field lines (e.g., of magnetization, polarization, director, etc.) are linked exactly once, which is the characteristic topological attribute of a hopfion. The S³-to-S³ mapping corresponds to the usual skyrmion.

Just as we have seen the fields of spintronics, valleytronics, skyrmionics, and twistronics during the past two decades, a new frontier of materials science is on the horizon: hopfionics. Once we fully understand the properties of a hopfion with a given Hopf index and how it interacts with other hopfions, it will likely usher in a new method of information processing and storage among other unconceived attributes as well as computing paradigms. This book purports to present the latest experimental and theoretical advances in the emerging field of hopfions in condensed matter, also including BEC and photonics.

Dr. Avadh Saxena
Guest Editor

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