  Using the Aharonov-Casher (A-C) phase, we present a microscopic theory of the
Josephson and persistent spin currents in quasi-equilibrium Bose-Einstein
condensates (BECs) of magnons in ferromagnetic insulators. Starting from a
microscopic spin model that we map onto a Gross-Pitaevskii Hamiltonian, we
derive a two-state model for the Josephson junction between the weakly coupled
magnon-BECs. We then show how to obtain the alternating-current (ac) Josephson
effect with magnons as well as macroscopic quantum self-trapping in a
magnon-BEC. We next propose how to control the direct-current (dc) Josephson
effect electrically using the A-C phase, which is the geometric phase acquired
by magnons moving in an electric field. Finally, we introduce a magnon-BEC ring
and show that persistent magnon-BEC currents flow due to the A-C phase.
Focusing on the feature that the persistent magnon-BEC current is a steady flow
of magnetic dipoles that produces an electric field, we propose a method to
directly measure it experimentally.
