Nuclear magnetic relaxation and Knight shift due to orbital interaction in Dirac electron systems

Abstract

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.

Small Wavevector–Dependent Spin Susceptibility in Weyl Semimetals

“…Putting together the inter‐ and intraband contribution, the real part of the response function reads asfor a ≠ b . This qualitatively agrees with the result from the previous studies . We also evaluated Equation numerically and compared with Equation .…”

“…This qualitatively agrees with the result from the previous studies. [18,21,26,27] We also evaluated Equation (16) numerically and compared with Equation (27). The agreement is perfect which is shown in Figure 1.…”

“…The real part of the response scales with the logarithm of the absolute value of chemical potential and depends on a high-energy cutoff which agrees with the result from the previous works. [18,21,26,27] As μ ! 0, the response function diverges for q ¼ 0, as expected from the diamagnetic response of Weyl semimetals, and tends to a constant otherwise as we show numerically.…”

“…For Weyl semimetals, the temperature dependence of T 1 was theoretically predicted in refs. [17,18] and it was measured by Yasuoka et al The full wavenumber–dependent spin–spin correlation functions were calculated in 2D and 3D, but we are focusing on the chemical potential dependence in the limit within the framework of linear response theory. This provides us the possibility to calculate the Knight shift for 3D Weyl semimetals.…”

Dirac electron behavior and NMR evidence for topological band inversion in ZrTe5

“…A recent model of spin-orbit-based NMR relaxation in 3D Dirac and Weyl systems accounts for this behavior very well. In this theory [24,25], fluctuations in Dirac-type orbital currents are responsible for the relaxation. The orbital hyperfine interaction introduces a 1/k 2 contribution to the momentum sum determining 1/T 1 T [24,26], thus connecting to fluctuations that are more extended in space than the typical local contributions, explaining the site-independence.…”