A new thermodynamic approach to bulk-boundary correspondence

01/09/2021 to 31/08/2023

Topological insulators are special materials that are insulating in their bulk but can sustain dissipation-less currents on their boundaries. Their theoretical discovery and experimental realization have inspired solid state physics research in the last decades. Therefore, mathematical physics faced the challenge to put these new physics phenomena on a firm mathematical ground.

The fingerprint of topological insulators is their remarkable correspondence between bulk and boundary quantum transport coefficients. This correspondence is nowadays known as bulk-boundary correspondence and it has been intensively studied in independent electron models at zero-temperature. However, no mathematical results are known for positive temperature.

The aim of this research project is to develop a new approach to the analysis of bulk-boundary correspondence that is capable to go beyond the well-established zero-temperature setting. The core of this approach is based on careful analysis of the magnetic response of the integrated density of states, which is a thermodynamic quantity that can be defined also at positive temperature.

We plan to perform the research in three main steps: first, by using our new approach, we extend the zero-temperature bulk-boundary result on the half-plane to positive temperature. Then, we address the problem of more general domains at positive temperature, which is relevant in modelling the experimental setup of finite samples of topological insulators. In this context, we intend to recast the bulk-boundary correspondence in a mathematically precise thermodynamic framework, making a rigorous connection between the boundary conductance and the bulk magnetization. Eventually, we plan to explore how this thermodynamic approach can be extended to interacting electron models.