The Geometric Law of XY Criticality and Superfluidity

Abstract The Geometric Theory of XY Superfluids: Dimensional Collapse and Logarithmic Scaling** This paper formalizes a novel geometric derivation of the superfluid transition in liquid Helium-4 films. By applying the Dimensional Collapse theorem, we demonstrate that the transition from a 3D bulk fluid to a 2D superfluid state is governed by a fundamental shift in the thermodynamic manifold’s metric. Unlike the "hard" linear metrics characterizing Ising-like systems, XY superfluids are shown to follow a "soft" exponential response defined by the metric . Our framework successfully recovers the essential singularity of the Kosterlitz-Thouless (KT) transition, providing a geometric basis for the finite specific heat cusp and the exponential divergence of the correlation length. Most significantly, we provide the theoretical derivation for a novel, falsifiable "Smoking Gun" test: the shift in critical temperature for films of thickness follows a logarithmic scaling law, , rather than the Euclidean predicted by standard mean-field theories. Numerical verification via SageMath simulations confirms that this logarithmic signature is a direct consequence of the manifold’s topological drift. These results unify topological phase transitions under a single geometric language, proving that the superfluid state is a manifest realization of a geometric phase transition. --- Author: Heidi Peterson ORCID: 0009-0005-5630-6804 Zenodo DOI: 10.5281/zenodo.18596450 Date: February 10, 2026 Non commercial if you want licence contact me heidiswork6@gmail.com