Functions of two-dimensional Bravais lattices

Abstract

The symmetries of a function are considered that takes as its argument an arbitrary two-dimensional Bravais lattice, represented by a 2×2 matrix. Rotating the lattice corresponds to multiplying its matrix on the right by an element of SO(2). Matrices related by left multiplication of an element of SL(2,Z) also refer to the same lattice. Functions of matrices having these two symmetries possess an expansion known as the Roelcke–Selberg decomposition. This decomposition characterizes the function in terms of a set of coefficients (the discrete spectrum) and a function defined on a line in the complex plane (the continuous spectrum). When the function of the lattice refers to its energy, the continuous spectrum can be related to an interatomic potential between the atoms of the lattice. The decomposition can be considered a Landau theory of a strained two-dimensional Bravais lattice, valid for arbitrarily large strains, expanded about any structure. This type of theory may be useful in determining the energies of a material for many different lattice configurations from a knowledge of the energy at only a few of them.