^{1}Faculty of Physics, ITMO University, St. Petersburg, Russia^{2}School of Science and Technology, Örebro University, Örebro, Sweden^{3}Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden^{4}Institute of Physics, Martin Luther University Halle-Wittenberg, Halle, Germany^{5}Department of Physics and Electrical Engineering, Linnaeus University, Kalmar, Sweden^{6}Science Institute, University of Iceland, Reykjavík, Iceland

**Editorial on the Research Topic ****Nucleation and stability of exotic solitons in condensed matter**

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 two-dimensional and even three-dimensional 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*:

However, both the energy barrier Δ*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 [6–9]. Calculations of minimum energy paths (MEPs) connecting the skyrmion state with topologically trivial background state have uncovered skyrmion collapse mechanisms [10–12], some of which were confirmed experimentally [13]. The rate theory has made it possible to identify, in a definite way, both the energy barrier and the Arrhenius pre-exponential factor. In particular, the unexpected variations of the pre-exponential factor were explained by large entropy difference between the skyrmion state and the transition state–the bottleneck for the skyrmion collapse [11, 14–16]. Overall, recent developments of theoretical and computational methods for the rate theory [17–20] have made it possible to establish a coherent picture about thermal stability of magnetic skyrmions. At the same time, the theoretical framework is quite general and can be applied to solitons beyond magnetic skyrmions.

Two-dimensional magnetic films with Dzyaloshinskii–Moriya interaction (DMI) can host, together with axisymmetric skyrmions, other locally stable configurations even with the same topological charge. Among them are so-called tailed skyrmions Kuchkin et al.. They have an elongated shape and can exist in a narrow range of fields near the transition from spin spirals to a uniform ferromagnetic state. “Growing a tail” is an additional mechanism for obtaining new solitons. There is a continuous transition (homotopy) between such structures and usual skyrmions. The homotopies can be efficiently found by calculating MEPs using the geodesic nudged elastic band method [17]. The discovery of tailed skyrmions extends the range of already known soliton solutions.

Additional possibilities appear in multilayer systems due to controlled modification of the interlayer exchange coupling (IEC). For example, synthetic antiferromagnet can be obtained by establishing an antiferromagnetic (AFM) IEC between the ferromagnetic (FM) layers through a non-magnetic spacer. DMI-stabilized ferromagnetic skyrmions in each layer can couple with each other in these systems thus forming composite AFM skyrmions [21]. AFM skyrmions can also be created intrinsically in AFM materials. Ab initio calculations predict that this can be done, for example, by depositing a row-wise AFM Cr layer on the PdFeIr(111) structure hosting FM skyrmions [22]. In this case, only exchange interactions may be required to form a complex AF structure. Aldarawsheh et al. investigate this system using the Heisenberg model, which includes basic magnetic interactions necessary to form AFM skyrmions on a triangular lattice. Interestingly, deposited Cr layer does not introduce additional DMI interaction but leads to long-range exchange interaction involving several neighbor shells.

In three-dimensional magnets, the possibility of forming even more exotic topological structures can be realized. In cubic magnets with competing magnetic interactions, hopfions can be stabilized even in the absence of DMI. However, theoretical estimates show that they are stable only at low temperatures of a few kelvins [23, 24]. In the presence of DMI and an external magnetic field, hopfions embedded in a conical magnetic structure form heliknotons. Their experimental observation, however, is a challenging task. Additional challenge lies in the interpretation of experimental data, which is far from being unambiguous, especially in three-dimensional systems. Therefore, it is necessary to use mutually complementary techniques and explore different interpretation options. Savchenko et al. show this using magnetic bubbles with alternating chirality in domain walls as an example. There, mathematical modeling of the system response, obtained in the framework of different experimental methods in combination with various theoretical approaches to the study of dynamics and stability assessment, is very useful. Kuchkin et al. discuss the stability of heliknotons and conditions of their detection based on micromagnetic modeling, rate theory, and stochastic spin dynamics simulations.

In three-dimensional samples with chiral interactions, delocalized states are possible in addition to localized topological structures. Leonov and Pappas carried out a systematic study of the states of an inclined spiral arising due to competition of cubic and exchange anisotropies inherent to cubic helimagnets. Field-controlled reorientation of metastable skyrmion lattices caused by competing anisotropies, may be responsible for some features in the experimental phase diagrams of Cu_{2}OSeO_{3}.

## Author contributions

VU: Writing–original draft, Writing–review and editing. DT: Writing–original draft, Writing–review and editing. BG: Writing–original draft, Writing–review and editing. PB: Writing–original draft, Writing–review and editing.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

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Keywords: soliton, condensed matter physics, skyrmion, thermal stability, magnetic material

Citation: Uzdin VM, Thonig D, Göbel B and Bessarab PF (2023) Editorial: Nucleation and stability of exotic solitons in condensed matter. *Front. Phys.* 11:1275990. doi: 10.3389/fphy.2023.1275990

Received: 11 August 2023; Accepted: 15 August 2023;

Published: 23 August 2023.

Edited and reviewed by:

Matthias Eschrig, University of Greifswald, GermanyCopyright © 2023 Uzdin, Thonig, Göbel and Bessarab. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Pavel F. Bessarab, pavel.bessarab@lnu.se