Zero modes in magnetic systems: General theory and an efficient computational scheme

Abstract

The presence of topological defects in magnetic media often leads to normal modes with zero frequency (zero modes). Such modes are crucial for long-time behavior, describing, for example, the motion of a domain wall as a whole. Conventional numerical methods to calculate the spin-wave spectrum in magnetic media are either inefficient or they fail for systems with zero modes. We present a new efficient computational scheme that reduces the magnetic normal-mode problem to a generalized Hermitian eigenvalue problem also in the presence of zero modes. We apply our scheme to several examples, including two-dimensional domain walls and Skyrmions, and show how the effective masses that determine the dynamics can be calculated directly. These systems highlight the fundamental distinction between the two types of zero modes that can occur in spin systems, which we call special and inertial zero modes. Whereas the inertial modes are generic Goldstone modes related to a broken continuous symmetry, the special modes arise naturally when two broken continuous symmetries coexist. Our method is suitable for both conservative and dissipative systems. For the latter case, we present a perturbative scheme to take into account damping, which can also be used to calculate dynamical susceptibilities.

Stability of a skyrmion and interaction of magnons

“…This slightly artificial case allows us to fully employ analytical approach in the semiclassical 1/s expansion and explicitly find the leading contributions to the self-energy corrections and the two-particle magnon Green's function. The additional interest in the case of one BP skyrmion is related to the presence of three zero modes associated with three broken conformal symmetries of the problem 10,[19][20][21] , see also 18 . Any second-order quantum correction to the energy is non-positive and a strictly negative correction to zero modes would make the sys-tem unstable.…”

Theoretical formalism for collective spin-wave edge excitations in arrays of dipolarly interacting magnetic nanodots

“…Thus, below, in our attempt to describe the general properties of collective edge modes in magnetic dot arrays, we decided to use approximate analytical methods based on the Fourier transform of the mutual demagnetization tensor of individual array's elements 17,33,36,52,53 and an operator form of the linearized Landau-Lifshitz equation 45 . The analytical approach developed by Verba et al 17 calculates the spin wave spectra in spatially infinite periodic arrays of magnetic nanodots using the fundamental tensorF k of the array.…”

“…Later, several theoretical approaches were developed to describe spin-wave excitations in systems where magnetic properties are spatially periodic. The developed approaches include the method of plane wave expansion (PWE) [37][38][39][40] , which was adopted from the theory of periodic dielectric 41 and acoustic 42 structures, the dynamic matrix method 12,[43][44][45] , the transfer matrix method 46,47 , the multiple scattering theory 32 , and several other. The above mentioned methods have proven their applicability and convenience for the analysis of infinite periodic magnetic systems.…”

“…Dynamics of spin wave excitations in a finite array of dipolarly coupled magnetic dots, in principle, can be simulated using one of the available micro-magnetic numerical techniques 43,45,50 . However, using only the results of micromagnetic simulations, that provide the frequencies and profiles of the system's eigenmodes, it is, sometimes, difficult to extract the symmetry properties of the system 19,51 or to describe the system's behavior in a critical regime, when the stable static state of the system is changed 18,24 .…”

Motion of Domain Walls and the Dynamics of Kinks in the Magnetic Peierls Potential

“…(4) with y 0 ¼ 0. An inertial zero-frequency normal mode [26] is associated with the collective coordinate y 0 . We have ∂xðyÞ=∂y 0 ¼ −ða=wÞsechðπy=wÞ and ∂ϑðyÞ=∂y 0 ¼ 0.…”