Abstract
We analyze spin transport in insulating antiferromagnets described by the $XXZ$ Heisenberg model in two and three dimensions. Spin currents can be generated by a magnetic-field gradient or, in systems with spin-orbit coupling, perpendicular to a time-dependent electric field. The Kubo formula for the longitudinal spin conductivity is derived analogously to the Kubo formula for the optical conductivity of electronic systems. The spin conductivity is calculated within interacting spin-wave theory. In the Ising regime, the $XXZ$ magnet is a spin insulator. For the isotropic Heisenberg model, the dimensionality of the system plays a crucial role: In $d=3$ the regular part of the spin conductivity vanishes linearly in the zero frequency limit, whereas in $d=2$ it approaches a finite zero frequency value.
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