Skyrmions represent topologically stable field configurations with particle-like properties. We used neutron scattering to observe the spontaneous formation of a two-dimensional lattice of skyrmion lines, a type of magnetic vortex, in the chiral itinerant-electron magnet MnSi. The skyrmion lattice stabilizes at the border between paramagnetism and long-range helimagnetic order perpendicular to a small applied magnetic field regardless of the direction of the magnetic field relative to the atomic lattice. Our study experimentally establishes magnetic materials lacking inversion symmetry as an arena for new forms of crystalline order composed of topologically stable spin states.

Skyrmion Lattice in a Chiral Magnet

Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

Magnetic skyrmions are particle-like nanometre-sized spin textures of topological origin found in several magnetic materials, and are characterized by a long lifetime. Skyrmions have been observed both by means of neutron scattering in momentum space and microscopy techniques in real space, and their properties include novel Hall effects, current-driven motion with ultralow current density and multiferroic behaviour. These properties can be understood from a unified viewpoint, namely the emergent electromagnetism associated with the non-coplanar spin structure of skyrmions. From this description, potential applications of skyrmions as information carriers in magnetic information storage and processing devices are envisaged.

Topological properties and dynamics of magnetic skyrmions

Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs

Crystal order is not restricted to the periodic atomic array, but can also be found in electronic systems such as the Wigner crystal or in the form of orbital order, stripe order and magnetic order. In the case of magnetic order, spins align parallel to each other in ferromagnets and antiparallel in antiferromagnets. In other, less conventional, cases, spins can sometimes form highly nontrivial structures called spin textures. Among them is the unusual, topologically stable skyrmion spin texture, in which the spins point in all the directions wrapping a sphere. The skyrmion configuration in a magnetic solid is anticipated to produce unconventional spin-electronic phenomena such as the topological Hall effect. The crystallization of skyrmions as driven by thermal fluctuations has recently been confirmed in a narrow region of the temperature/magnetic field (T-B) phase diagram in neutron scattering studies of the three-dimensional helical magnets MnSi (ref. 17) and Fe(1-x)Co(x)Si (ref. 22). Here we report real-space imaging of a two-dimensional skyrmion lattice in a thin film of Fe(0.5)Co(0.5)Si using Lorentz transmission electron microscopy. With a magnetic field of 50-70 mT applied normal to the film, we observe skyrmions in the form of a hexagonal arrangement of swirling spin textures, with a lattice spacing of 90 nm. The related T-B phase diagram is found to be in good agreement with Monte Carlo simulations. In this two-dimensional case, the skyrmion crystal seems very stable and appears over a wide range of the phase diagram, including near zero temperature. Such a controlled nanometre-scale spin topology in a thin film may be useful in observing unconventional magneto-transport effects.

Real-space observation of a two-dimensional skyrmion crystal

Charge-neutral bilayer graphene has an energy gap of 2 meV that can be reduced by an electric field and increased by a magnetic field.

Transport spectroscopy of symmetry-broken insulating states in bilayer graphene

MnSi is an itinerant magnet which at low temperatures develops a helical spin-density wave. Under pressure it undergoes a transition into an unusual partially ordered state whose nature is debated. Here we propose that the helical spin crystal (the magnetic analog of a solid) is a useful starting point to understand partial order in MnSi. We consider different helical spin crystals and determine conditions under which they may be energetically favored. The most promising candidate has bcc structure and is reminiscent of the blue phase of liquid crystals in that it has line nodes of magnetization protected by symmetry. We introduce a Landau theory to study the properties of these states, in particular, the effect of crystal anisotropy, magnetic field, and disorder. These results compare favorably with existing data on MnSi from neutron scattering and magnetic field studies. Future experiments to test this scenario are also proposed.

Theory of the helical spin crystal: a candidate for the partially ordered state of MnSi.

Weyl semimetals are three-dimensional analogs of graphene where the energy of the excitations is a linear function of their momentum. Pyrochlore iridates $({A}_{2}{\text{Ir}}_{2}{\text{O}}_{7}$ with A yttrium or a lanthanide element) are conjectured to be examples of such a system, with the low-energy physics described by 24 Weyl nodes. An intriguing possibility is that these materials provide a physical realization of the Adler-Bell-Jackiw anomaly. In this Rapid Communication we investigate the properties of pyrochlore iridates in an applied magnetic field. We find that the dispersion of the lowest Landau level depends on the direction of the applied magnetic field. Consequently, the velocity at low energies can be manipulated by changing the direction of the applied field. The resulting anisotropy in longitudinal conductivity is investigated.

Adler-Bell-Jackiw anomaly in Weyl semimetals: Application to pyrochlore iridates

Weyl semimetals are three dimensional analogs of graphene where the energy of the excitations are a linear function of their momentum. Pyrochlore Iridates $(A_{2}Ir_{2}O_{7})$ with A =yttrium or lanthanide element) are conjectured to be examples of such a system, with the low energy physics described by twenty four Weyl nodes. An intriguing possibility is that these materials provide a physical realization of the Adler-Bell-Jackiw anomaly. In this letter we investigate the properties of pyrochlore iridates in an applied magnetic field. We find that the dispersion of the lowest landau level depends on the direction of the applied magnetic field. As a consequence the magneto-conductivity in an electric field, applied parallel to the magnetic field is highly anisotropic, providing a detectable signature of the semi-metallic state.

Adler-Bell-Jackiw anomaly in Weyl semi-metals: Application to Pyrochlore Iridates

Weyl semimetals are examples of a new class of topological states of matter, which are gapless in the bulk with protected surface states Their low-energy sector is characterized by massless chiral fermions, which are robust against translationally invariant perturbations A variant of these systems has two nondegenerate bands touching along lines rather than points A proposal to realize such a phase involves alternating layers of topological insulators and magnetic insulators, where the magnetization lies perpendicular to the symmetry axis of the heterostructure The shape, size, and the dispersion in the vicinity of the nodal lines varies with the strength of the magnetization, offering a new knob to control the properties of the system In this paper, we map out the evolution of the nodal lines and the dependence of the conductivity on magnetization and identify signatures of the low-energy sector in quantum oscillation measurements

Vivek Aji

Papers

Transport spectroscopy of symmetry-broken insulating states in bilayer graphene

Theory of the helical spin crystal: a candidate for the partially ordered state of MnSi.

Adler-Bell-Jackiw anomaly in Weyl semimetals: Application to pyrochlore iridates

Adler-Bell-Jackiw anomaly in Weyl semi-metals: Application to Pyrochlore Iridates

Tunable Line Node Semimetals