Attractive Inverse Square Potential, U(1) Gauge, and Winding Transitions

TL;DR

This paper links the conformality-breaking in the inverse square potential to a topological winding transition in a classical one-dimensional system, revealing a Kosterlitz-Thouless law and applying to polymer tangling and diffusion kinetics.

Contribution

It introduces a topological transition framework for the inverse square potential using a U(1) gauge theory, connecting quantum phenomena to classical polymer and diffusion models.

Findings

Winding number exhibits a Kosterlitz-Thouless transition law.

Polymer tangling transitions depend on torque and temperature.

Model unifies quantum inverse square potential with classical winding phenomena.

Abstract

The inverse square potential arises in a variety of different quantum phenomena, yet notoriously it must be handled with care: it suffers from pathologies rooted in the mathematical foundations of quantum mechanics. We show that its recently studied conformality-breaking corresponds to an infinitely smooth winding-unwinding topological transition for the {\it classical} statistical mechanics of a one-dimensional system: this describes the the tangling/untangling of floppy polymers under a biasing torque. When the ratio between torque and temperature exceeds a critical value the polymer undergoes tangled oscillations, with an extensive winding number. At lower torque or higher temperature the winding number per unit length is zero. Approaching criticality, the correlation length of the order parameter---the extensive winding number---follows a Kosterlitz-Thouless type law. The model is described by the Wilson line of a (0+1) $U(1)$ gauge theory, and applies to the tangling/untangling of floppy polymers and to the winding/diffusing kinetics in diffusion-convection-reactions.