Topological solitons in condensed matter are of particular interest for fundamental theory due to a deep connection between topology and physics manifested in these systems. At the same time, they are praised as the basis for new technologies of data storage, information processing, machine learning and neuromorphic computing. The most well-studied magnetic solitons are quasi-two-dimensional skyrmions and bubble domains. However, in recent years, attention has shifted to other 2D and even 3D localized topological structures appearing not only in magnetic materials, but also in liquid crystals, ferroelectrics and multiferroics, which expands our knowledge about topological effects in physics and possible scope of topological soliton applications (1).Although skyrmions and related objects owe their stability to topology, the topological protection is not strict in real systems due to discrete nature of condensed matter, e.g., magnetic moments localized on atomic lattices. Instead, topological solitons can be nucleated and annihilated by overcoming finite energy barriers. Such over-the-barrier transitions can be induced spontaneously by thermal fluctuations leading to a finite lifetime of the states. Successful implementation of topological solitons in technology requires their lifetime to be sufficiently long, many orders of magnitude longer than characteristic times of the microscopic dynamics. This hierarchy of the timescales makes it challenging to study the thermal stability of the solitons. Consistently with the Néel-Brown theory of thermally activated magnetization reversal (2, 3), experimental observations (4) and numerical simulations (5) of magnetic skyrmions identified Arrhenius dependency of their nucleation/annihilation rates k on temperature T :k = k 0 e −∆E/k B T .(1)However, both the energy barrier ∆E and, remarkably, the pre-exponential factor k 0 which is often taken to be a phenomenological constant, turned out to be highly sensitive to various control parameters such as an external magnetic field (4). The physics of the thermal stability of magnetic skyrmions was revealed using a statistical approach based on the rate theory for magnetic degrees of freedom (6, 7, 8, 9). Interestingly, deposited Cr layer does not introduce additional DMI interaction but leads to long-range 50 exchange interaction involving several neighbor shells.
In three-dimensional magnets, the possibility of forming even more exotic topological structures can 52 be realized. In cubic magnets with competing magnetic interactions, hopfions can be stabilized even in 53 the absence of DMI. However, theoretical estimates show that they are stable only at low temperatures of 54 a few kelvins (23, 24). In the presence of DMI and an external magnetic field, hopfions embedded in a 55 conical magnetic structure form helicnotons. Their experimental observation, however, is a challenging task.
Additional challenge lies in the interpretation of experimental data, which is far from being unambiguous, Field-controlled reorientation of metastable skyrmion lattices caused by competing anisotropies, may be 68 responsible for some features in the experimental phase diagrams of Cu 2 OSeO 3 .69
