Exploiting $\vartheta -$functions for the identification of topological materials

An exact analytical expression is derived for Bloch states in three dimensions, based on the only assumption that the electronic wavefunction can be expanded in terms of Gaussian type orbitals. The resulting expression features multidimensional $\vartheta -$functions (and their derivatives) on which the action of discrete space group symmetries is evaluated analytically and contrasted against the symmetry transformations proper of modular forms. We integrate group theoretical arguments with continuity requirements of the Bloch states to produce a viable algorithm for the determination of band inversions in materials with a non-trivial topological electronic band structure; the proposed methodology is then applied to two simplified materials models.