Triangle lattice Green functions for vector fields

Abstract

All the identities and integral theorems of vector calculus are contained in the calculus of differential forms. The analogy between the exterior calculus of forms and the homology theory of a cell complex yields discrete lattice models for an array of interesting physical phenomena. These models, based on arbitrary combinations of coupled scalar and polar or axial vector field quantities, can be manipulated as conveniently as the standard scalar tight-binding models. We develop Green functions (GFs) for an infinite hierarchy of such models expressed using the four fundamental operators of the triangle lattice. The triangle lattice is distinguished among two-dimensional grid types for having the highest possible isotropy. Closed formulae are derived for GFs of the scalar and vector models that belong to a hierarchy generated by the four fundamental operators of the triangle family of lattices. The particular example of lattice electromagnetism coupled to an elastic distortion field is treated in detail. Topological properties not dependent upon symmetry split the response functions into plasmon- and polariton-like parts. Since the fundamental vector operators of the triangle lattice are related simply to adjacency matrices of the Kagome lattice, scalar GFs for this lattice are found also as a byproduct.