A long-wave action of spin Hamiltonians and the inverse problem of the calculus of variations

TL;DR

This paper presents a method to derive the long-wave action of spin Hamiltonians from their equations of motion, highlighting the role of potentiality and topological terms like the Berry phase.

Contribution

It introduces a novel approach linking the equations of motion of spins to the long-wave action, utilizing Vainberg's theorem and potentiality conditions.

Findings

Derived the long-wave action for Heisenberg antiferromagnets.

Identified the inclusion of the Berry phase in the action.

Established conditions for the potentiality of spin equations.

Abstract

We suggest a method of derivation of the long-wave action of the model spin Hamiltonians using the non-linear partial differential equations of motions of the individual spins. According to the Vainberg's theorem the set of these equations are (formal) potential if the symmetry analysis for the Frechet derivatives of the system is true. The case of Heisenberg (anti)ferromagnets is considered. It is shown the functional whose stationary points are described by the equations coincides with the long-wave action and includes the non-trivial topological term (Berry phase).