Screening of charged singularities of random fields

Wilkinson, Michael (2004). Screening of charged singularities of random fields. Journal of Physics A: Mathematical and General, 37 pp. 6763–6771.

DOI: https://doi.org/10.1088/0305-4470/37/26/012

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.