Sublattice

A sublattice is a concept in lattice theory, crystallography, and related fields. A lattice Λ in R^n is a discrete subgroup that spans the ambient space, usually described as all integer linear combinations of a chosen basis v1, …, vn. A sublattice Λ' of Λ is a subgroup that is itself a lattice of full rank, meaning Λ' ⊆ Λ and span(Λ') = span(Λ). Equivalently, Λ' can be generated by vectors w1, …, wn, where each wj is an integer linear combination of the vi’s. The matrix that expresses the wj in the basis {v1, …, vn} has integer entries and is non-singular.

The index of a sublattice is a key measure of its size relative to Λ. If Λ' is a sublattice of Λ, the quotient Λ/Λ' is a finite abelian group, and the index [Λ:Λ'] equals the absolute value of the determinant of the integer change-of-basis matrix linking the bases of Λ and Λ'. In practical terms, the index counts how many Λ points lie in a fundamental cell of Λ'. For example, in Z^2, the sublattice generated by (2,0) and (0,1) has index 2, while the sublattice obtained by scaling all vectors by 3 has index 9 in Z^2.

Sublattices have broad applications. In crystallography, they describe ways to choose a larger primitive cell that still tiles space with the same lattice points. In number theory, sublattices of Z^n correspond to finite-index subgroups, with the index related to solving linear Diophantine equations. The concept also appears in physics, where bipartite lattices and intervalley phenomena arise from sublattice structures in materials like graphene.