The homology of defective crystal lattices and the continuum limit

TL;DR

This paper explores how defects in crystal lattices affect the extension of fields to the continuum limit, using algebraic topology to identify obstructions caused by cohomology classes related to lattice defects.

Contribution

It introduces a cohomological framework to analyze the extension problem of fields on defective lattices, linking lattice defects to topological obstructions in the continuum limit.

Findings

Defects induce cohomological obstructions to field extension.

Examples include spin fields, vortex flows, and monopole fields on defective lattices.

Topological classes determine the possibility of smooth continuum limits.

Abstract

The problem of extending fields that are defined on lattices to fields defined on the continua that they become in the continuum limit is basically one of continuous extension from the 0-skeleton of a simplicial complex to its higher-dimensional skeletons. If the lattice in question has defects, as well as the order parameter space of the field, then this process might be obstructed by characteristic cohomology classes on the lattice with values in the homotopy groups of the order parameter space. The examples from solid-state physics that are discussed are quantum spin fields on planar lattices with point defects or orientable space lattices, vorticial flows or director fields on lattices with dislocations or disclinations, and monopole fields on lattices with point defects.