“…[9], under the assumption that the total spin along the z axis j S z j is conserved and remains a good quantum number, we have proposed a magnonic analog of the quantum spin Hall effect characterized by helical edge states and thus established a bosonic counterpart of TIs [44,45], namely the magnonic TIs in insulating AFs using the above-mentioned picture for the clean systems and following the work by Aharonov and Casher [13,46]. The proposal is built upon the fact that an electric field couples to the magnetic dipole moment σgµ B e z through the AC effect [13,[47][48][49][50][51][52][53][54][55], which is analogous to the Aharonov-Bohm (AB) effect [56][57][58] of electrically charged particles in magnetic fields. Each magnon (σ = ±1) of the insulating AF subjected to a dc electric field with E a constant gradient E(r) = E(−x, 0, 0) as a function of the position r = (x, y, 0) experiences the "electric" vector potential [9,14,49] A m (r) = E(r) × e z /c = E/c(0, x, 0).…”