Magnon spectrum in ferromagnets with a skyrmion

Abstract

The analysis of the spin wave excitations in two-dimensional isotropic Heisenberg ferromagnet is performed with a single skyrmion in the ground state. We employ the ideas of semiclassical quantization method, duly modified for the use of the lattice model and Maleyev-Dyson boson representation of spin operators. The resulting Schrödinger equation for magnons describes the dispersion and wave functions of spin-wave excitations with strictly non-negative spectrum. In contrast to usual ferromagnet, we demonstrate the existence of three zero modes, corresponding to conformal symmetries spontaneously violated by the skyrmion configuration.1. Topological defects play an important role in condensed matter physics. The first and the most famous example is vortex lines, defining critical properties of type II superconductors in the external magnetic field [1]. In two dimensions, the role of defects is even more noticeable. So an interaction of vortices in the O(2) model leads to emergence of a quasi-long-range order and a Berezinskii-Kosterlitz-Thouless (BKT) transition [2]. In O(2) symmetric systems with the additional twofold degeneracy of the ground state, such as Josephson junction arrays in the magnetic field or XY helimagnets, vortex excitations with fractional charges lead to a phase transition on domain walls, and as a consequence to separation of a BKT and Ising (chiral) transitions [3,4]. The appearance of so-called Z 2 -vortices corresponds to exceptional thermal properties of twodimensional frustrated magnets with isotropic spins (see [5] and Refs. therein). The superlattice structure observed in magnets [6,7,8,9, 10] and multiferroics [11] with the Dzyaloshinskii-Moria (DM) spin-orbit interaction in the magnetic field is believed to be related to vortex-like excitations, called skyrmions. The similar skyrmion structures appear in the quantum Hall systems [12,13,14,15].In this paper we discuss topological defects in twodimensional quantum ferromagnets. It is known that the usual O(N ) model, describing ferromagnets, has different types of topological defects. The case N = 1 corresponds to the Ising model, where line-like defects are domain walls. The case N = 2 has been mentioned above in a context of point-like vortices and a BKT transition. At N = 3, defects of another type are present. They can be obtained as static classical solution of the O(3) sigma model [16], describing lowtemperature properties of ferromagnets,Taking into account the isotropic condition at spatial infinity ϕ(∞) = ϕ 0 , the field ϕ becomes a map ϕ : R 2 ∪ {∞} S 2 → S 2 , which is characterized by an integer number Q, the topological degree of the map ϕ.The families of solutions consist of configurations related to each other by global field-rotations and coordinate transformations. The latter symmetry includes rescalings (dilatations), that is specific to the two-dimensional sigma model, which is conformal invariant. As a consequence, a size of defects is not defined by the energy minimum conditions, in accordance with the Derrick the...

Stability of a skyrmion and interaction of magnons

“…(2) with r 0 a implied. Knowing the explicit expressions for matriceŝ U (r),R(r, n) (see 10 ) and using (4) we represent the appearing bosonic Hamiltonian as a formal expansion in small parameter 1/s :…”

“…10 The lattice model with localized moments allows the well-known Maleyev-Dyson representation for spin operators, preserving the spin commutation relations, [S a , S b ] = i abc S c :…”

“…This statement becomes rather obvious when we represent the LSWT Hamiltonian in the form H (2) = A † rÂr with r given in (10) and † r its Hermitian conjugate. 10,21 To calculate the matrix elements of the interaction operators, we write…”

“…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.…”

“…This means that the BP skyrmion is a metastable solution, but is not unstable one because the spectrum of small fluctuations is non-negative. A number of theoretical works suggested various ways to lower the classical energy of the skyrmion state and thus make it energetically favorable [5][6][7][8][9] (see also 10 and references therein). This process of adding weaker interactions to the main Heisenberg Hamiltonian is called stabilization.…”