Crystal symmetries in charge-orbital order and topological band theory

The present thesis analysis two main topics: first charge and orbital order systems, and second the link between the emergence of edge states in 1D crystals and the symmetry based topological classification of crystalline insulators. Although the two topics can be studied completely independently, both share a common theme: symmetry.
For the charge and orbital ordered systems studied here, symmetry is analysed through the role of interactions. In more precise terms, the chiral trigonal crystal structure of Se and Te is modelled as the result of microscopic interactions, like the electron-phonon coupling and the Coulomb repulsion, which favour the formation of three charge density waves in the ground state of the crystal. Because each of these three charge density waves is associated with the charge distribution along a particular p-orbital sector, the optimal charge distribution of the three dimensional crystal sets in by requiring that charge ordering occurs side-by-side with orbital order. The origin of the crystal structure of Po is then considered by adding spin-orbit coupling to the model.
The second topic analysis what are the minimal requirements needed in order to have edge states in 1D crystals. It is shown that these are exactly the same requirements that dictate certain bulk states to no longer be eligible solutions, and thus that the two come hand-in-hand. The relation with the symmetry based topological classification of crystalline insulators is then made clear by forcing two crystals to share an interface with each other.