Equation of motion for the Heisenberg spin chain

Abstract

A systematic treatment is given of the equation of motion of the classical anisotropic Heisenberg spin chain, both in the discrete case and in the continuum limit in which the spins S m ( t) associated with the lattice sites m are replaced by a spin density S( x, t), which is a function of the time t and the position x on the chain. In the case of axial symmetry the equation of motion for the spins is shown to be equivalent to a new equation in terms of one real variable, i.e. q m ( t) in the discrete case q( x, t) in the continuum limit. (From the treatment by A.E. Borovik it follows that the new equation of motion for q( x, t) is completely integrible in the special case of quadratic anisotropy.) Explicit expressions are given for the Lagrangians, both in the ferromagnetic and in the antiferromagnetic case. The relation with the nonlinear Schrödinger equation on the one hand and the sine-Gordon equation on the other hand is discussed in some detail.