Superdiffusive two-dimensional superconductors

We formulate a non-local, field-theoretic description of phase fluctuations and topological transitions in two-dimensional (2D) superconductors by generalizing the local only-phase Popov action to a framework utilizing non-local fractional operators. By replacing the standard spatial Laplacian with the Riesz fractional Laplacian $-(-\nabla^2)^{α/2}$ (where $1<α<2$), we prove that the vortex-antivortex interaction transitions from logarithmic confinement to a stronger power-law confinement of the form $V(r)\propto r^{2-α}$. Through a generalized Kosterlitz-Thouless energetic-entropic analysis, we demonstrate that this non-local confinement strictly suppresses thermal vortex proliferation. Because the non-local operator safely places the system outside the strict domain of validity of the Mermin-Wagner restriction, this power-law confinement natively stabilizes true long-range order at finite temperatures. Finally, by constructing a gauge-invariant fractional action, we formulate a fractional London equation. This approach yields a real-space power-law kernel that maps directly onto the anomalous Pippard limit of superconductors with long coherence lengths, providing a unified phenomenologically motivated framework for non-local electrodynamics.