Screening of charged singularities of random fields

TL;DR

This paper investigates the topological charges of point singularities in random fields, demonstrating a universal screening effect where positive and negative charges cancel each other, with a specific counterexample where this screening fails.

Contribution

It provides a simple, general derivation of the screening effect of singularities in random fields and presents a counterexample where screening is incomplete.

Findings

Singularities in many random fields are perfectly screened by opposite charges.

A counterexample shows screening can be incomplete under certain conditions.

The derivation is simple and broadly applicable.

Abstract

Many types of point singularity have a topological index, or 'charge', associated with them. For example the phase of a complex field depending on two variables can either increase or decrease on making a clockwise circuit around a simple zero, enabling the zeros to be assigned charges of plus or minus one. In random fields we can define a correlation function for the charge-weighted density of singularities. In many types of random fields, this correlation function satisfies an identity which shows that the singularities 'screen' each other perfectly: a positive singularity is surrounded by an excess of concentration of negatives which exactly cancel its charge, and vice-versa. This paper gives a simple and widely applicable derivation of this result. A counterexample where screening is incomplete is also exhibited.