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• A great variety of topologically protected defects exist in ordinary and exotic states of matter. Some of them are promising candidates for technological applications such as magnetic memory or quantum computers. Others possess some properties of elementary particles that could lead to potential applications in quantum fields. Nematic liquid crystals provide a convenient platform to study the physics of such defects. A specific form of topological defects in nematics are ring disclinations. Here we studied this object in the form of a torus-shaped hole in the nematic field with a $\pi$-twist through the center. Vector models of liquid crystal energy cannot describe these objects as they lead to nonphysical discontinuities and spikes in energy density. Therefore, the tensor model of liquid crystals has been adopted for the energy model. An energy minimization algorithm was developed to calculate the minimum energy for a fixed torus size and free boundary conditions. The optimized defect energy was found to scale linearly with the torus size, in agreement with a scaling argument based on qualitative dimensional analysis. Potential relevance of ring disclinations beyond liquid crystals and specifically to classical field theories has been studied by including a fourth-order gradient term in the energy model. It was found that the optimized fourth-order configuration energy has a local minimum as a function of torus size. So, the global minimization path of the fourth-order energy model with respect to torus size was found and mapped.
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