Heisenberg spins on a circular conical surface

Abstract

We investigate classical Heisenberg spins on a conical surface. The energy and configuration of non-trivial spin distributions are obtained using a non-conventional method based on Einstein's theory of gravity in lower dimensions. rThe study of low-dimensional, artificially structured materials is becoming increasingly important as we move to an era of important technological realizations. Physics in two spatial dimensions has generated a lot of information and surprises. Moreover, if the system lies on a curved surface, it adds further richness to the physical phenomena [1][2][3][4]. Recently, the fabrication of magnetic films grown on a curved substrate [5] has become possible. Researchers already discovered how to produce graphitic cones in 1997 [6,7]. In this letter we consider Heisenberg spins on a conic support. Although we do not know of physically relevant examples, it should be interesting to investigate the behavior of topological structures on exotic surfaces. After all, artificially structured materials are produced each day leading to new knowledge and technologies. Besides, the idea that topologically nontrivial excitations arise in real physical systems had a strong impact on modern physics. These structures became an object of intensive investigations in many condensed matter models and they are particularly believed to play an important role in magnetic systems [8][9][10][11][12]. Our main interest is to study the configuration and energy of such topological excitations on the surface of a cone.As it is well known, the classical two-dimensional static ferromagnetic and antiferromagnetic materials are well described, in the continuum limit, by the non-linear s model. This theory on ARTICLE IN PRESS www.elsevier.com/locate/jmmm 0304-8853/$ -see front matter r

Geometrical pinning of magnetic vortices induced by a deficit angle on a surface: Anisotropic spins on a conic space background

“…Nevertheless, based on the above discussion and on the results for soliton configurations on the conical surface (see Ref. [7]), we also expect two types of vortex solutions, which will be referred to as 'in-cone' and 'out-of-cone' vortices. Of course, it would be energetically favorable for out-of-cone vortices to nucleate preferentially around the cone apex, while, in contrast, in-cone vortices would prefer to nucleate away from this region.…”

“…Such a limit is implicit in our system as one can easily see below. First, due to the local Euclidian nature of the conical surface, it is convenient to use local flat coordinates [7,12] defined as ρ = (ρ, τ ) = (r β /β, βφ), where r = (r, φ) are assumed to be the usual polar coordinates. Thus we can rewrite Hamiltonian (3) as follows:…”

“…For example, on the surface of a magnetic cylinder, solitons appear to be sine-Gordonlike excitations [5], which tend to deform the cylindrical surface in order to relieve the geometrical frustration brought about by nonconstant curvature, anisotropies, Zeeman effect, etc [6]. Now, on a conical support their energy gets lower as long as cone is narrowed, indicating that such surfaces could be thought as pinning defects for solitons [7] (see, however [8]). In addition, transition from flower to vortex-like magnetization has been experimentally observed in ferromagnetic nanosized cones [9,10].…”

Heisenberg model on a space with negative curvature: Topological spin textures on the pseudosphere

“…Indeed, a number of works has addressed such an issue in the last years. For instance, in the context of magnetism, several aspects of solitonic solutions associated to the non-linear σ model (NLσM; which is the continuum limit of the classical isotropic Heisenberg model) have been studied in some geometries, like cylinders [2,3,4], cones [5], and so on, while a study of vortex-like excitations on a conical background has appeared in Ref. [6].…”

Heisenberg spins on a cone: an interplay between geometry and magnetism

“…In these cases, a number of new phenomena have been described, like the geometrical frustration on spin textures induced by curvature and/or by non-trivial topological aspects of the space manifold, say, angular deficit in cones, area deficit in planes with a disk cut out, and so forth (see, for example, Refs. [15,13,14,16]). Actually, the study of such systems may be of considerable importance for practical applications, for example, in connection to soft condensed matter materials [15] (deformable vesicles, membranes, etc), and also to artificially nanostructured curved objects (nanocones, nanocylinders, etc) in high storage data devices [17,18].…”