We study the nuclear magnetic relaxation rate and Knight shift in the presence of the orbital and quadrupole interactions for three-dimensional Dirac electron systems (e.g., bismuth-antimony alloys). By using recent results of the dynamic magnetic susceptibility and permittivity, we obtain rigorous results of the relaxation rates (1/T 1 ) orb and (1/T 1 ) Q , which are due to the orbital and quadrupole interactions, respectively, and show that (1/T 1 ) Q gives a negligible contribution compared with (1/T 1 ) orb . It is found that (1/T 1 ) orb exhibits anomalous dependences on temperature T and chemical potential µ. When µ is inside the band gap, (1/T 1 ) orb ∼ T 3 log(2T/ω 0 ) for temperatures above the band gap, where ω 0 is the nuclear Larmor frequency. When µ lies in the conduction or valence bands, (1/T 1 ) orb ∝ T k 2 F log(2|v F |k F /ω 0 ) for low temperatures, where k F and v F are the Fermi momentum and Fermi velocity, respectively. The Knight shift K orb due to the orbital interaction also shows anomalous dependences on T and µ. It is shown that K orb is negative and its magnitude significantly increases with decreasing temperature when µ is located in the band gap. Because the anomalous dependences in K orb is caused by the interband particle-hole excitations across the small band gap while (1/T 1 ) orb is governed by the intraband excitations, the Korringa relation does not hold in the Dirac electron systems.

Achieving control over magnon spin currents in insulating magnets-where dissipation due to Joule heating is highly suppressed-is an active area of research that could lead to energy-efficient spintronics applications. However, magnon spin currents supported by conventional systems with uniform magnetic order have proven hard to control. An alternative approach that relies on topologically protected magnonic edge states of spatially periodic magnetic textures has recently emerged. A prime example of such textures is the ferromagnetic skyrmion crystal which hosts chiral edge states providing a platform for magnon spin currents. Here, we show, for the first time, an external magnetic field can drive a topological phase transition in the spin wave spectrum of a ferromagnetic skyrmion crystal. The topological phase transition is signaled by the closing of a low-energy bulk magnon gap at a critical field. In the topological phase, below the critical field, two topologically protected chiral magnonic edge states lie within this gap, but they unravel in the trivial phase, above the critical field. Remarkably, the topological phase transition involves an inversion of two magnon bands that at the Γ point correspond to the breathing and anticlockwise modes of the skyrmions in the crystal. Our findings suggest that an external magnetic field could be used as a knob to switch on and off magnon spin currents carried by topologically protected chiral magnonic edge states. arXiv:1910.05214v1 [cond-mat.str-el]

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The nuclear spin relaxation time T 1 is calculated taking account of the contributions from orbital currents of Dirac electrons. We consider a simple model of non-interacting Dirac electron gas in the three-dimensional bulk system. The obtained result shows T 3 dependence of 1/T 1 at temperatures T above the energy gap. This temperature dependence agrees qualitatively with the recent β-NMR experiment on the bulk of the topological insulator Bi 0.9 Sb 0.1 .