In mathematical terms, topological vortices are based on polar coordinate transformations of complex fields, using Euler's formula to express the field as ϕ(x) = r_ϕ(x) e^(iθ_ϕ(x)), where the winding number n in θ_ϕ(x) = nθ_x + φ determines the topological properties of the vortex—when the spatial coordinate angle θ_x rotates by 2π, the field angle rotates n times. This structure appears in the potential energy landscape as a minimum-value ring of the "Mexican hat" potential, corresponding to continuous variations of the ground state in field theory.
Although Derrick's theorem indicates the instability of steady-state topological objects in higher dimensions, the introduction of gauge fields can counteract the energy divergence caused by derivative terms, theoretically stabilizing the vortex configuration.
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