Nuclear Spin Relaxation Time due to the Orbital Currents in Dirac Electron Systems

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 $\beta$-NMR experiment on the bulk of the topological insulator $\mathrm{Bi}_{0.9}\mathrm{Sb}_{0.1}$.