Signed zeros of Gaussian vector fields-density, correlation functions and curvature

TL;DR

This paper derives correlation functions for the signed density of zeros in Gaussian vector fields, linking them to curvature tensors, with applications to topological defect distributions in physical systems.

Contribution

It introduces a method to express correlation functions of zeros in Gaussian vector fields using curvature tensors of Riemann-Cartan or Riemannian manifolds, providing new analytical tools.

Findings

Correlation functions expressed via curvature tensors.

Application to defect distributions in 2D systems.

Analysis of one- and two-point functions.

Abstract

We calculate correlation functions of the (signed) density of zeros of Gaussian distributed vector fields. We are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract Riemann-Cartan or Riemannian manifold. As an application, we discuss one- and two-point functions. The zeros of a two-dimensional Gaussian vector field model the distribution of topological defects in the high-temperature phase of two-dimensional systems with orientational degrees of freedom, such as superfluid films, thin superconductors and liquid crystals.