Critical Phenomena in Nonlinear Sigma Models

TL;DR

This paper investigates critical phenomena in nonlinear sigma models, identifying boundary behaviors between singular and nonsingular solutions, and drawing parallels to black hole critical phenomena.

Contribution

It reveals the existence of self-similar and static solutions at the boundary between different solution regimes in nonlinear sigma models.

Findings

Identification of critical behavior separating singular and nonsingular solutions.

Discovery of self-similar solutions at the boundary for localized families.

Observation of static solutions at the boundary for other families.

Abstract

We consider solutions to the nonlinear sigma model (wave maps) with target space S^3 and base space 3+1 Minkowski space, and we find critical behavior separating singular solutions from nonsingular solutions. For families of solutions with localized spatial support a self-similar solution is found at the boundary. For other families, we find that a static solution appears to sit at the boundary. This behavior is compared to the black hole critical phenomena found by Choptuik.