The homology of defective crystal lattices and the continuum limit

Details The homology of defective crystal lattices and the continuum limit The homology of defective crystal lattices and the continuum limit

2010-10-06

arxiv.org/abs/1010.1218v1

Annalen Phys.522:874-903,2010

Physics

Mathematical Physics

38 pages, 8 figures, 1 table

Scientific paper

10.1002/andp.201010461

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.

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