We incorrectly used the Weber-Minnhagen finite-size scaling analysis to estimate the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature [see Eq. (21)]. First, for systems with anisotropic exchange interactions, this procedure is valid only for the geometric mean helicity modulus ϒ gm = √ ϒ a ϒ b , because it is the geometric mean helicity that has the universal jump predicted by the theory. Second, the universal value of the jump is not renormalized, as we assumed in the paper, despite the fact that we measure the temperature in units of J 1 rather than in units of J eff .Then, the BKT transition temperature should be estimated using the equationrather than Eq. (21). As a result, one obtains Fig. 1 instead of Figs. 11 and 13 from our paper. The root-mean-square error χ 2 of the least-squares fit of our numerical data for ϒ gm (T ,L) is shown in Fig. 2, which should replace Figs. 12 and 14 in our paper. As a result, we find for the BKT transition temperature at J 2 = 0.5,Note that T (ϒ gm ) BKT is in better agreement with the estimate of T BKT obtained from another universal criterion η(T = T BKT ) = 0.25 [see Eq. (25)] than the incorrect estimations given by Eqs. (22) and (23). FIG. 1. FIG. 2. 059904-1 1098-0121/2012/86(5)/059904(2)

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...

Critical behavior of three-dimensional classical frustrated antiferromagnets with a collinear spin ordering and with an additional twofold degeneracy of the ground state is studied. We consider two lattice models, whose continuous limit describes a single phase transition with a symmetry class differing from the class of non-frustrated magnets as well as from the classes of magnets with non-collinear spin ordering. A symmetry breaking is described by a pair of independent order parameters, which are similar to order parameters of the Ising and O(N) models correspondingly. Using the renormalization group method, it is shown that a transition is of first order for non-Ising spins. For Ising spins, a second order phase transition from the universality class of the O(2) model may be observed. The lattice models are considered by Monte Carlo simulations based on the Wang-Landau algorithm. The models are a ferromagnet on a body-centered cubic lattice with the additional antiferromagnetic exchange interaction between next-nearest-neighbor spins and an antiferromagnet on a simple cubic lattice with the additional interaction in layers. We consider the cases N=1,2,3 and in all of them find a first-order transition. For the N=1 case we exclude possibilities of the second order or pseudo-first order of a transition. An almost second order transition for large N is also discussed.

We consider multiskyrmion configurations in 2D ferromagnets with Dzyaloshinskii-Moriya (DM) interaction and the magnetic field, using the stereographic projection method. In the absence of DM interaction, D, and the field, B, the skyrmions do not interact and the exact multiskyrmion solution is a sum of individual projections. In certain range of B, D = 0, skyrmions become stable and form a hexagonal lattice. The shape of one skyrmion on the plane is fully determined by D and B. We describe multiskyrmion configurations by simple sums of individual skyrmion projections, of the same shape and adjusted scale. This procedure reveals pairwise and triple interactions between skyrmions, and the energy of proposed hexagonal structure is found in a good agreement with previous studies. It allows an effective theory of skyrmion structures in terms of variables, referring to individual skyrmions, i.e., their position, size and phase, elliptic distortions etc. arXiv:1811.01883v1 [cond-mat.str-el]