These authors contributed equally to this work. S1: Nanoskyrmion lattice solution to the Heisenberg modelSince the spin-spiral state is the exact solution of the classical Heisenberg model for a periodic arrangement of atoms, it is an excellent starting point to expand the solution due to further but smaller interactions such as the four-spin interaction. Therefore, we consider magnetic structures obtained from the superposition of spin spirals. These lead in general to two-dimensional magnetic structures.In order to be in accordance with the experimentally observed magnetic unit cell, the superposition or multi-Q state must include the Q-vectors determined experimentally from the Fourier transform of the spin-polarized STM image (Fig. 1c). In general, it is impossible to find a superposition state for two arbitrary but symmetry equivalent Q-vectors � and � in the 2D Brillouin zone, if we require the magnetic moment to be constant at all lattice sites, i.e. |� , in order to prevent a large loss of

Topologically nontrivial spin textures have recently been investigated for spintronic applications. Here, we report on an ultrathin magnetic film in which individual skyrmions can be written and deleted in a controlled fashion with local spin-polarized currents from a scanning tunneling microscope. An external magnetic field is used to tune the energy landscape, and the temperature is adjusted to prevent thermally activated switching between topologically distinct states. Switching rate and direction can then be controlled by the parameters used for current injection. The creation and annihilation of individual magnetic skyrmions demonstrates the potential for topological charge in future information-storage concepts.

Chirality is a fascinating phenomenon that can manifest itself in subtle ways, for example in biochemistry (in the observed single-handedness of biomolecules) and in particle physics (in the charge-parity violation of electroweak interactions). In condensed matter, magnetic materials can also display single-handed, or homochiral, spin structures. This may be caused by the Dzyaloshinskii-Moriya interaction, which arises from spin-orbit scattering of electrons in an inversion-asymmetric crystal field. This effect is typically irrelevant in bulk metals as their crystals are inversion symmetric. However, low-dimensional systems lack structural inversion symmetry, so that homochiral spin structures may occur. Here we report the observation of magnetic order of a specific chirality in a single atomic layer of manganese on a tungsten (110) substrate. Spin-polarized scanning tunnelling microscopy reveals that adjacent spins are not perfectly antiferromagnetic but slightly canted, resulting in a spin spiral structure with a period of about 12 nm. We show by quantitative theory that this chiral order is caused by the Dzyaloshinskii-Moriya interaction and leads to a left-rotating spin cycloid. Our findings confirm the significance of this interaction for magnets in reduced dimensions. Chirality in nanoscale magnets may play a crucial role in spintronic devices, where the spin rather than the charge of an electron is used for data transmission and manipulation. For instance, a spin-polarized current flowing through chiral magnetic structures will exert a spin-torque on the magnetic structure, causing a variety of excitations or manipulations of the magnetization and giving rise to microwave emission, magnetization switching, or magnetic motors.

The atomic-scale spin structure of individual isolated Skyrmions in an ultrathin film is investigated in real space by spin-polarized scanning tunneling microscopy. Their axial symmetry as well as their unique rotational sense is revealed by using both out-of-plane and in-plane sensitive tips. The size and shape of Skyrmions change as a function of the magnetic field. An analytical expression for the description of Skyrmions is proposed and applied to connect the experimental data to the original theoretical model describing chiral Skyrmions. Thereby, the relevant material parameters responsible for Skyrmion formation can be obtained.

In the Kondo effect, a localized magnetic moment is screened by forming a correlated electron system with the surrounding conduction electrons of a non-magnetic host 1 . Spin S = 1/2 Kondo systems have been investigated extensively in theory and experiments, but magnetic atoms often have a larger spin 2 . Larger spins are subject to the influence of magnetocrystalline anisotropy, which describes the dependence of the magnetic moment's energy on the orientation of the spin relative to its surrounding atomic environment 3,4 . Here we demonstrate the decisive role of magnetic anisotropy in the physics of Kondo screening. A scanning tunnelling microscope is used to simultaneously determine the magnitude of the spin, the magnetic anisotropy and the Kondo properties of individual magnetic atoms on a surface. We find that a Kondo resonance emerges for large-spin atoms only when the magnetic anisotropy creates degenerate ground-state levels that are connected by the spin flip of a screening electron. The magnetic anisotropy also determines how the Kondo resonance evolves in a magnetic field: the resonance peak splits at rates that are strongly direction dependent. These rates are well described by the energies of the underlying unscreened spin states.A low density of magnetic impurities in a non-magnetic host metal can have dramatic effects on the magnetic, thermodynamic and electrical properties of the material owing to the Kondo effect, a many-body interaction between the metal's conduction electrons and the electron spin of the localized magnetic impurity 5 . This interaction gives rise to a narrow, pronounced peak in the density of states close to the Fermi energy 1 . The past decade has seen a surge of interest in the Kondo screening of individual atomic spins, as a result of experimental advances in probing individual magnetic atoms by using scanning tunnelling microscopes 6,7 and single-molecule transistors 8,9 . When magnetic atoms are placed on metal surfaces the Kondo interaction between the localized spin and the conduction electrons is very strong, leading to high Kondo temperatures T K , in the range from 40 to 200 K (ref. 10). In order to probe the Kondo physics it is desirable to reduce this interaction so that the Kondo screening competes on an equal footing with external influences such as magnetic fields. It was shown recently that this can be achieved by incorporating a decoupling layer between the atomic spin and the screening conduction electrons 11 .