Lokalstabile Spektrallücken und Antworttheorie in wechselwirkenden Vielteilchenquantensystemen

01.01.2023 bis 31.12.2026

The theoretical understanding of the quantum Hall effect has driven important developments in mathe- matics and mathematical physics over the past three decades and continues to do so. The most important aspects are the quantization of Hall conductivity, the validity of the linear response formalism, the role of random impurities, and the correspondence between bulk and boundaries. Only recently have important advances concerning quantization and linear response been made specifically for models of interacting fermions at zero temperature either on torus geometries or in the thermodynamic limit.
For systems without boundaries the starting point for the mathematical analysis of quantum Hall systems is a many-body Hamiltonian with a gapped ground state. In particular, such systems display exponential decay of correlations [HK06]. While establishing a spectral gap for a given many-body Hamiltonian is an important and difficult problem in itself (see for example Projects A7 and A8), for us the central question will be the local stability and the local response of gapped ground states resp. thermal states: starting from an extended fermionic gapped system describing the electrons in a Hall insulator on a torus geometry, the introduction of edges can be viewed as a local perturbation of the Hamiltonian near the new edges. Because of the appearance of edge states, such a perturbation closes the spectral gap above the ground state energy and introduces long-range correlations along the boundary.
In our project, we aim to further develop the mathematical tools necessary to understand adiabatic and linear response for systems with boundaries. To this end, we explore mathematical questions whose relevance is not limited to quantum Hall systems, but extends, for example, to quantum information. Conversely, we also intend to use new approaches recently developed in quantum information to solve these problems motivated from mathematical physics.