Mathematics

In this talk, I will present some new results regarding topological quantum phases of matter, in the context of time-reversal symmetric (TRS) topological insulators in 2 and 3 dimensions. I will first review how geometric information can be extracted from periodic gapped quantum systems, modelling e.g. electrons in crystalline solids. This information is encoded in integer-valued topological invariants, namely Chern numbers of the so-called Bloch bundle. In 2d systems, the Chern number is known to measure the obstruction to the existence of a smooth periodic Bloch frame for an isolated set of energy bands of the Hamiltonian, and is related to physical observables such as the transverse conductivity in the Quantum Hall Effect. I will then explain how the presence of further symmetries, such as time-reversal, leads to a "refinement" of the geometry. This is relevant in the framework of 2d and 3d TRS topological insulators, whose quantum phases are labelled by integers mod 2 and are distinguished by the presence of edge spin currents. Our result allows to identify these Z_2-valued invariants again as geometric obstructions.The results were obtained in collaboration with D. Fiorenza and G. Panati ("La Sapienza" Università di Roma).