Heisenberg spin textures on a cylinder with topological defects

Abstract

The present work aims to study equilibrium configurations of spins on a cylinder with topological defects such as screw dislocation and deficit angle. By making use of elliptic-f expansion method, which in turn utilizes the Jacobi elliptic functions, we obtain exact solutions of the nonlinear sigma model in this geometry. We have significant changes in the qualitative behavior of the solutions due to the presence of the parameter k of screw dislocation. In particular, the behavior of soliton-like solutions, characteristic of a cylinder without dislocation, was not found in the model here proposed. Keywords: Heisenberg's spins -nanomagnetism The nonlinear sigma model is the continuous limit of Heisenberg's hamiltonian for spins and has gained interest recently. It allows the study of equilibrium configurations of spins on non-trivial geometries such as cylinders, tori, ellipsoids, surfaces with negative curvature and so on [1][2][3][4]. The study of the stability of spin configurations and geometric frustration on different kinds of geometry are the two effects most explored with this model [5][6][7][8]. Defects such as punctures and impurities in the structure have also played an important role related to spin topological textures. In particular, we have introduced topological defects in an ideal cylinder such as screw dislocation. This defect is obtained by producing a longitudinal displacement in the structure, distorting the lattice helically. In this way we can study how the spin textures are modified in relation to the simple cylinder without these defects.Connections with fundamental areas and important technological applications can also be made: the "similarity" between the nonlinear sigma model in 2D and Yang-Mills in (3+1)D [9]; out-of-plane vortex (its core is similar to the central region of a soliton of spins). These out-of-plane vortex are important in several mechanisms of magnetic logic (MRAM), ultra-precise magnetic sensors, magnetic recording, etc. [10,11] The Heisenberg model takes into account only the interaction of each spin of the lattice with its first neighbors. By applying an external magnetic field, the Hamiltonian of the system is thenHere h ab is the 3 × 3 dimensionless interaction law matrix (metric of the internal space of spins), J is the exchange energy, and S a i denotes the a-th component of the spin of the i-th cell of the lattice, while < i, j > includes in the sum only the first neighbor cells j of each cell i. For isotropic interaction we have h ab = diag (1, 1, 1). The parameter g is the gyromagnetic factor of the spins in the material medium, µ is their magnetic momentum and B a is the a-th component of the external magnetic field. Eq. (1) is the hamiltonian for J > 0, describing the ferromagnetic case. In the antiferromagnetic case (J < 0), the spin vector S must be replaced by the Néel vector η = 1 2 ( S 1 − S 2 ), where the index distinguish two distinct sub-lattices in the more simple case.For sufficiently small distances between neighbor cells (and large spins) we...

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