Spin-wave softening and Hund's coupling in ferromagnetic manganites

Using one-orbital model of hole-doped manganites, we show with the help of Holstein-Primakov transformation that finite Hund's coupling is responsible for the spin-wave softening in the ferromagnetic $B$-phase manganites. We obtain an analytical result for the spin-wave spectrum for $\JH\gg t$. In the limit of infinte Hund's coupling, the spectrum is the conventional nearest-neighbor Heisenberg ferromagnetic spin-wave. The $o(t/\JH)$-order correction is negative and thus accounts for the softening near the zone boundary.