Methods to determine the Hausdorff dimension of vortex loops in the three-dimensional XY model

TL;DR

This paper introduces methods to determine the Hausdorff dimension of vortex loops in the 3D XY model, providing insights into the geometric nature of critical fluctuations in superfluid systems.

Contribution

The paper presents a direct evaluation method for the Hausdorff dimension of vortex loops and offers analytical reasoning for the scaling relation involving this dimension.

Findings

Hausdorff dimension D_H of vortex loops is evaluated directly.

Analytical arguments show the scaling exponent  in the relation  + D_H = 2 +  must be zero.

Provides a link between geometric properties and critical scaling in the 3D XY model.

Abstract

The geometric properties of critical fluctuations in the 3D XY model are analyzed. The 3D XY model is a lattice model describing superfluids. We present a direct evaluation of the Hausdorff dimension D_H of the vortex loops which are the critical fluctuations of the 3D XY model. We also present analytical arguments for why \vartheta in the scaling relation \eta_{\phi} + D_H = 2 + \vartheta between D_H and the anomalous scaling dimension of the corresponding field theory, must be zero.