Spontaneous Breaking of Rotational Symmetry with Arbitrary Defects and a Rigidity Estimate

TL;DR

This paper extends a geometric rigidity theorem to 1-forms with non-zero exterior derivatives and applies it to demonstrate spontaneous rotational symmetry breaking in crystal models with various defects.

Contribution

It introduces a generalized rigidity estimate for 1-forms and uses it to analyze symmetry breaking in defective crystal structures.

Findings

Rigidity estimate for 1-forms with non-vanishing exterior derivative

Spontaneous breaking of rotational symmetry in crystal models

Applicability to models with diverse defects including dislocations

Abstract

The goal of this paper is twofold. First we prove a rigidity estimate, which generalises the theorem on geometric rigidity of Friesecke, James and M\"uller to 1-forms with non-vanishing exterior derivative. Second we use this estimate to prove a kind of spontaneous breaking of rotational symmetry for some models of crystals, which allow almost all kinds of defects, including unbounded defects as well as edge, screw and mixed dislocations, i.e. defects with Burgers vectors.