Beyond conventional topology

Abstract

Materials are commonly classified according to their electronic properties, for example: conductors, semiconductors, or insulators. However, this classification does not tell the whole story. Since the discovery of the quantum Hall effect, it is known that two insulators with similar energy spectra can differ fundamentally because of the structure of their electronic wave functions. In a topological insulator, the bulk of the material remains insulating, while conducting states may occur along its edges or surfaces. This behavior is captured by a topological invariant, which remains unchanged as long as the energy gap (band gap) stays open and the protecting symmetry is preserved. As a result, the edge states can be robust against certain perturbations. When interactions between particles are neglected, the tenfold-way classification describes which topological phases are possible in different spatial dimensions, depending on time-reversal, particle-hole, and chiral symmetry. However, many modern systems do not fit within this framework. Their topology may be related to crystalline or non-Hermitian properties, an irregular geometry, or a dimension that depends on the scale at which the system is examined. Moreover, the relevant symmetry does not always need to be apparent at the level of the full microscopic Hamiltonian. This raises two central questions: can a hidden symmetry nevertheless have topological consequences, and what happens to topology when a system changes dimension or acquires a fractal structure? This thesis aims to adress these questions by considering systems where the tenfold way no longer applies. To this end, two closely related directions are investigated. Part I examines symmetries that are hidden, emergent, or only partially present. Part II explores how topological properties change in fractals, thin films, and nanoribbons, where the effective dimension is not fixed. Together, the two parts demonstrate that such simplified descriptions can reveal new forms of topological protection.