Quantum geometry in 2D materials

Thesis project under the supervision of Dr. Pierre Anthony Pantaleón, researcher at IMDEA Nanoscience in Spain.

About

Quantum geometry studies the geometrical properties of wave functions in Hilbert space. The best known property is the Pancharatnam-Berry phase which is fundamental to explain phenomena such as the "(spin) quantum Hall effect". This phase is a consequence of a geometrical property called metric tensor, whose real part is the quantum metric and the imaginary part the Berry phase. In two-dimensional materials, such as graphene, the Berry phase is used to characterize edge states in finite systems and to describe topological currents. The quantum metric describes the shape of the momentum space in which wave functions are defined and is used to analyze the distribution of wave functions in space. The curvature of such a space is characterized by the Riemman curvature.

In this work, we study the geometry of quantum systems with exact analytical solutions, such as the Haldane model (2-dimensional), the Kane Mele model (4-dimensional), graphene multilayers with simple interactions, among others. The low dimensionality of these systems will allow obtaining analytical forms for the metric tensor, Berry phase and Riemman curvature, and with this, characterize the different topological phases. It is worth mentioning that, although the mentioned systems are relatively simple, there is no detailed description of their geometrical-quantum properties in the literature.