Magnon edge modes

An exact result is obtained for the dispersion relation for magnons localized at the apex of a semi-infinite, right-angle wedge of a simple-cubic, Heisenberg ferromagnet, with nearest- and next-nearest-neighbor exchange interactions, formed by the intersection of two (100) surfaces. The modes studied are wavelike in the direction parallel to the edge of the wedge, and they are characterized by a one-dimensional wave vector $q$. Their amplitudes decay exponentially with increasing distance into the wedge from its apex. The finite difference equation of motion for the magnon creation operators is solved by expanding the latter in a double series of Gottlieb functions. The dispersion relation for these edge localized modes is obtained for all values of $q$ in the one-dimensional first Brillouin zone of the wedge. In the long-wavelength limit it agrees with the result obtained recently in this limit by Sharon and Maradudin.