Effect of the Local Exchange Invariance on the Magnetization Dynamics in a Ferromagnet

Abstract

The magnetism in curved geometries is attracting attention because of several theoretical predictions with possible potential applications, such as a curvature induced effective Dzyaloshinskii-Moriya like interaction [1]. Within this setting, analogously to a geometric curvature, a curvature induced by a gauge field [2] is a fruitful aspect to consider in the interaction of the magnetization with an applied static electric field.In a quantum description of the motion of a magnetic moment, an electric field does not modify the energy but it rather modifies the linear momentum, therefore introducing an additional phase for the corresponding Schrödinger wave function, an effect which is known as the Aharonov-Casher’s effect [3]. The classical analog is the spin wave of a ferromagnet.It has been shown that a spin wave, which is carrying a magnetic moment, will acquire a phase when the system is subjected to an electric field [4]. This effect, even if rather small, can be experimentally observed [5]. It is therefore of interest to extend this idea to a general micromagnetic description of the problem. The aim of this research is to pose the problem of the micromagnetics under the effect of an electric field by using the so-called gauge invariance.Briefly the spin-waves, which are small oscillations, deviate from the uniform magnetization with respect to a local reference frame and around the ez direction. They interact with an applied electric field E Fig.(1) modifying the spin wave dispersion relation.Therefore, according to the so-called local exchange invariance [6], we propose a phase shift model, in the exchange spin waves regime, in order to possibly describe and interpret the data shown in [5].More in details the local gauge symmetry is approximately a local invariance property (because of its thermodynamic origin), of the lagrangian density of the magnetization m(x, t) of a ferromagnet. It is known that it is invariant with respect to the SO(3) global symmetry [7]. In our case this symmetry reduces to a unitary representation of the SO(2) group, (i.e., U(1)). Due to the reason that this symmetry has to be local (i.e., to be position dependent) the invariance of the lagrangian density is broken unless introducing a gauge field able to solve the technical problem of the parallel transport of the magnetization vector.Therefore we show that a particular orientation of a reference frame has no physical relevance. In fact, gauge invariance should occur whenever one can define non-collinear, local reference frames. In our model, the orientation of the local non-coordinate reference frames cannot be uniquely defined because of the presence of an electric field.Summarizing in our invariant gauge formulation the reference frames’ orientation is unspecified with respect to a local phase which defines an invariance property of the exchange interaction terms in the lagrangian density of the magnetization m(x, t). The Lagrangian density of a ferromagnet, interacting with an electric field, is transformed into a new Lagrangian density upon a formal redefinition of the standard partial derivative.Moreover the transformation group, which expresses this invariance, imposes definite restrictions on the dispersion relation that at the first order in a Tylor expansion was previously found to be ω=ω(ky)-γLc-2∂ω/∂kyEx [4], with γL the gyromagnetic ratio and c the speed of light.Therefore our findings point to a more advanced theoretical interpretation with respect to previous discoveries [4] in order to drive experiments in manipulating spin waves and developing electrically tunable magnonic devices. **