Topological solitons and geometrical frustration

Villain-Guillot, Simon; Dandoloff, Rossen; Saxena, Avadh; Ar, Bishop

doi:10.1103/physrevb.52.6712

Cited by 38 publications

(53 citation statements)

References 24 publications

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“…The regions where the domain-walls are localized are deformed perpendicularly to the axis of the elastic cylinder. It is clear that these deformations may also change the solutions in order to eliminate the geometrical frustration [34]. From Figs.…”

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confidence: 97%

Heisenberg spin textures on a cylinder with topological defects

Paula¹

2009

Braz. J. Phys.

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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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confidence: 97%

Heisenberg spin textures on a cylinder with topological defects

Paula¹

2009

Braz. J. Phys.

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“…Conversely, when the shape tensor violates the local property of self-duality/anti-selfduality then the bending Hamiltonian is frustrated, i.e. an extra energy contribution is spontaneously created [12,[26][27][28][29] that tends to vanish globally in the system. Before following this matter any further, we turn our attention to the spherical vesicles of revolution which are of pertinent interest.…”

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confidence: 99%

“…As a matter of fact, topology gives rise to new physics [24,25] with some of the following relevant features: (i) metastable configurations (mostly solitons) fall into distinct classes, of which the trivial configuration (vacuum) belongs only to one class; (ii) there exists no superposition principle, i.e. resultant configurations form complicated ones; (iii) frustration phenomena may arise from topological obstruction leading to both global and localized effects [26][27][28][29].…”

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confidence: 99%

Self-dual bending theory for vesicles

2003

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Abstract. We present a self-dual bending theory that may enable a better understanding of highly nonlinear global behavior observed in biological vesicles. Adopting this topological approach for spherical vesicles of revolution allows us to describe them as frustrated sine-Gordon kinks. Finally, to illustrate an application of our results, we consider a spherical vesicle globally distorted by two polar latex beads.

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“…However, it was limited to a system of masses coupled by nonlinear springs. Some studies have also been performed in a special case of the continuum limit of classical spin systems (such as the Heisenberg chain) coupled to the curvature; geometric frustration was found to arise in such settings [10].About twenty years ago, a problem similar to the theme of our study was examined in the context of molecular dynamics studies of structural transitions in crystals. In particular, in a series of papers [11], Rahman and Parrinello introduced a new idea for studying such transitions by means of an augmented Lagrangian that would account for the degrees of freedom of the "box" (the cell) in which the MD particles lie.…”

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Coupling fields and underlying space curvature: An augmented Lagrangian approach

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We demonstrate a systematic implementation of coupling between a scalar field and the geometry of the space (curve, surface, etc.) which carries the field. This naturally gives rise to a feedback mechanism between the field and the geometry. We develop a systematic model for the feedback in a general form, inspired by a specific implementation in the context of molecular dynamics (the so-called Rahman-Parrinello molecular dynamics, or RP-MD). We use a generalized Lagrangian that allows for the coupling of the space's metric tensor (the first fundamental form) to the scalar field, and add terms motivated by RP-MD. We present two implementations of the scheme: one in which the metric is only time-dependent [which gives rise to ordinary differential equation (ODE) for its temporal evolution], and one with spatio-temporal dependence [wherein the metric's evolution is governed by a partial differential equation (PDE)]. Numerical results are reported for the (1+1)-dimensional model with a nonlinearity of the sine-Gordon type.Recently, much attention has been focused on softcondensed-matter objects, such as vesicles, microtubules, and membranes [1][2][3][4]. Many nanoscale physical systems, including nanotubes and electronic and photonic waveguide structures [5,6], have nontrivial geometry and are influenced by substrate effects. These classes of systems, many of which are inherently nonlinear, raise the question of the interplay between nonlinearity and a substrate with variable curvature. Of particular interest is a possibility of developing curvature in the substrate due to forces generated by the nonlinear field. The resulting curvature can in turn affect the field.There is an increasing body of literature dealing with the interplay of nonlinearity and a curved substrate. Usually, however, the substrate geometry is assumed to be fixed, see, e.g., [7]. Nevertheless, for many applications, ranging from condensed matter to optics to biophysics, it is relevant to introduce models that admit a flexible substrate, which is affected by the field(s) that it carries, as well as feeding back into the field dynamics. In this situation, equations for the fields in a nonlinear system abutting on the flexible substrate should include both the field dynamics proper and the feedback coupling to the substrate. Equations for the evolution of the substrate should in turn be affected by the evolution of the field. A prototypical physical example of this type is Euler buckling [8], where the evolution of a thermal profile causes the underlying surface to buckle (and hence locally modify its curvature).In a discrete setting, a model of this type has recently been presented in [9]. However, it was limited to a system of masses coupled by nonlinear springs. Some studies have also been performed in a special case of the continuum limit of classical spin systems (such as the Heisenberg chain) coupled to the curvature; geometric frustration was found to arise in such settings [10].About twenty years ago, a problem similar to the theme of our...

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Topological solitons and geometrical frustration

Cited by 38 publications

References 24 publications

Heisenberg spin textures on a cylinder with topological defects

Heisenberg spin textures on a cylinder with topological defects

Self-dual bending theory for vesicles

Coupling fields and underlying space curvature: An augmented Lagrangian approach

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