Topological kinematic constraints: quantum dislocations and the glide principle

TL;DR

This paper explores the topological constraints governing dislocation motion in elastic media, focusing on the climb principle, and extends the understanding of these constraints in quantum and higher-dimensional contexts.

Contribution

It provides a universal mathematical definition of climb constraints and analyzes dislocation kinematics in arbitrary space-time dimensions, including non-linear short-distance behaviors.

Findings

Climb constraint is equivalent to zero particle density condition.

Dislocations do not communicate with long-range compressional stresses.

Formalism allows analysis of non-linear short-distance effects.

Abstract

Topological defects play an important role in physics of elastic media and liquid crystals. Their kinematics is determined by constraints of topological origin. An example is the glide motion of dislocations which has been extensively studied by metallurgists. In a recent theoretical study dealing with quantum dualities associated with the quantum melting of solids it was argued that these kinematic constraints play a central role in defining the quantum field theories of relevance to the description of quantum liquid crystalline states of the nematic type. This forms the motivation to analyze more thoroughly the climb constraints underlying the glide motions. In the setting of continuum field theory the climb constraint is equivalent to the condition that the density of constituent particles is vanishing and we derive a mathematical definition of this constraint which has a universal status. This makes possible to study the kinematics of dislocations in arbitrary space-time dimensions and as an example we analyze the restricted climb associated with edge dislocations in 3+1D. Very generally, it can be shown that the climb constraint is equivalent to the condition that dislocations do not communicate with compressional stresses at long distances. However, the formalism makes possible to address the full non-linear theory of relevance to short distance behaviors where violations of the constraint become possible.