Starting from 2006, one of the main focuses in material science has been on topological magnetic objects, skyrmions, that are characterized by a specific magnetization pattern in real space. The topological origin of these skyrmion spin structures protects them against temperature fluctuations and distortions of different kinds. Skyrmions have been observed in numerous bulk materials [*Science * **323**, 915–919 (2009)], surfaces [*Science* **341**, 636 (2013)] and other systems. They can be manipulated by electric and magnetic fields [*Nat. Nanotech.* **8**, 152 (2013)], which makes them very attractive for novel data storage devices functioning at room temperatures.

Up to now, the analysis of all the experimental data and most of the results of computer simulations related to magnetic skyrmions have been performed under the assumption that the local magnetization is a three-dimensional classical vector. However, from quantum theory we know that the spin is a purely quantum property of matter, and all three components of the magnetization vector cannot be measured simultaneously. In addition, the majority of novel two-dimensional magnetic systems are characterized by a small spin (*S=1/2*), which necessarily requires to take quantum effects into account. All these facts challenge the theory for a quantum description of magnetic skyrmions. Due to uncertainty principle, the standard identification of skyrmions by a magnetization pattern does not work in the quantum case since the local magnetization is simply uniform for a periodic quantum crystal. Surprisingly, this is still a common practice to avoid the discussion of this fundamental problem.

Recently, our project team that combines researchers from École Polytechnique Fédérale de Lausanne (EPFL) and Ural Federal University (UrFU) has introduced a completely new concept of a purely quantum skyrmion state (*arXiv:2004.13526*). This novel state can be seen as a quantum superposition of classical topological skyrmion configurations that can be observed as the result of independent measurements. In our work, we have proposed a theoretical framework that allows to fully characterize this novel quantum state. The key concept here is the calculation of the quantum scalar chirality, which we introduce as a quantum analog of the classical skyrmion number. Remarkably, this quantity demonstrates a nearly constant nonzero value for the quantum skyrmion phase. Moreover, we have found that this local three-spin correlation function defined on neighbouring lattice sites gives a complete information about topological properties of the infinite quantum system, something that is impossible in the classical case.

Within the project, the concept of the quantum skyrmion will be developed along two main directions. First, we will explore a connection between classical and quantum solutions of the skyrmion problem using the concept of Anderson’s tower of states. Second, we will investigate theoretically the possibility to perform experiments that will allow to probe locally the topology of the entire quantum system. Both Swiss and Russian partners will make in-kind contributions to the proposed project. Besides scientific visits, two workshops will be organized. In addition, we also plan to explore different funding options to enhance the collaboration between EPFL and UrFU.

**Participants:**

École Polytechnique Fédérale de Lausanne (EPFL):

- Prof. Frédéric Mila
- Ms. Jeanne Colbois

Ural Federal University:

- Dr. Vladimir Mazurenko
- Mr. Oleg Sotnikov
- Mr Ilia Iakovlev
- Dr. Evgeny Stepanov