YM-Thermodynamics 2024, 40: Evolving Center-Vortex Loops, n = 1

For n = 1, assume that curve-shrinking for closed curves of the figure-eight type is only relevant, if immersed in a 2D plane. This ensures the existence of a planar limit. Some condensed matter systems at low resolution sees environments confining valence electrons to spatial planes. The standard QM point-particle associates strong correlations with electrons in these systems. At n = 1, the location of the selfintersection point is a single point on the curve. If both wings of the center flux are of finite size, the selfintersection point can be shifted with almost no energy cost. The transition from nonintersecting to selfintersecting center-vortex loop is a twist of nonintersecting curves. The localized (anti)monopole in is due to oppositely directed center fluxes in the intersection region. Rotation of one half-plane by π each wing forms a closed flux loop by itself. Topologically, it's equivalent to the untwisted case. Another rotation introduces an intermediate loop, which can be shrunk to an isolating, spinning (anti)monopole due to oppositely directed center fluxes. In the last stage of such a shrinking process, under dual gauge modes, there is Biot-Savart repulsion, so energy in form of the monopole mass is required in the center. There doesn't need to be a distinction between relative direction of the center flux within two curve segments, as it's irrelevant for the microscopic evolution described by the same curve-shrinking with n = 0. The curve may be treated as a smooth immersion into the plane with exactly one double point, and a total rotation number 0. For n = 0, a smooth embedded curve shrinks to a point under the flow approaching finite critical flow parameter, and the isoperimetric ratio approaches 4π from above. At critical flow parameter, a physical singularity terminates the flow. Since the rate of area change is constant, the derivative by flow-parameter for the area is set to -2π. For n = 1, the difference between the disjointed wing areas is constant instead. If this is 0, the system is extremely fine-tuned. As a Wilsonian renormalization-group flow,

When all curves are normalized to have the same initial area (over both wings, summed) the intersection point is decided by the critical value of the flow parameter. Ordering a maximal-size ensemble of n = 1 curves by critical flow parameters (T-ordered) gives rise to qualitative analysis through second-order partial derivative under periodic boundary conditions. This is a numerical process, which yields a fictitious singularity away from the expectation, at which point the evolution may be stopped.