Numerical Modeling of Interaction and Decay Processes of Topological Vortices in Reversed Time

Abstract

The paper studies T-symmetry properties of the well-known nonlinear sigma model (NSM), a class of quantum field systems in which physical fields are regarded as the coordinates of a certain manifold. Considers a symmetry property of the (2+1)-dimensional anisotropic O(3) NSM (n-field) with respect to time reversal. From the geometric point of view, O(3) NSM describes the dynamics of a single three-component isovectornthat takes on values in the Bloch sphere. The isovectorncan be identified with a point on the surface of a two-dimensional sphere and, therefore, corresponds to the element of a rotation group O(3). The anisotropy is selected in thez-axis direction and thus, the zero-energy states (vacuum states) of the model under investigation are equivalent to the point (the end of the isovectorn) at the poles of the Bloch sphere.О(3) NSМ has an exact solution in the form of topological solitons (vortices, quasi-particles) possessing a topological charge (the Hopf index). It was shown in [8] that, for certain rate values of their motion, the interacting topological solitons decay into localized perturbations. In this case, the property of preserving the total sum of the topological charge is revealed.In the present work, numerical modeling of interaction and decay processes of topological solitons of O(3) NSM in reversed time are carried out. It is shown that when there is time reversal, a complete restoration process of the initial state of the field of topological solitons by combining individual localized perturbations is observed. Thus, the paper confirms the property of T-invariance of O(3) NSM. To construct numerical models, are used methods of the theory of finite difference schemes, based on a specially developed algorithm for applying the properties of the stereographic projection of the Bloch sphere onto the complex plane. The developed method allows accurate calculations of the energy density of interacting vortex fields at each point of the stratified space. The paper proposes a software complex that allows us to conduct numerical studies of interaction processes of localized solutions of non-linear field-theoretical models in class O(3) NSM in reversed time.