# A7 Low-Energy States and Spectra of Quantum Hall Systems

**Project Leaders**

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Prof. Dr. Marius Lemm, Prof. Dr. Simone Warzel

**Researchers**

Dr. Chokri Manai, Anouar Kouraich

**Summary**

The fractional quantum Hall (FQH) effect is a result of the collective behavior of interacting charge degrees of freedom in a two-dimensional geometry with perpendicular magnetic field. Two hallmarks characterize the remarkable properties of this quantum state of matter: the incompressibility of the liquid into which the charge carriers condense and the existence of an energy gap to excitations with fractional charge. Theoretical models start from Laughlin’s famous ansatz for the many-body correlated ground state wave function. A Hamiltonian approach to the observed features as well as the properties of excitations above the ground state is based on Haldane pseudo-potentials, i.e., short-range repulsive interactions projected onto the lowest Landau level. Both fermionic and bosonic pseudo-potentails are of interest. The latter capture the properties of rapidly rotating Bose gases.

In this project, we will analyze the fundamental properties of Haldane pseudo-potentials and related models for both fermions and bosons. This includes a mathematically complete and effective description of their ground states. The long-term goal is to prove Haldane’s conjecture on the spectral gap for pseudo-potentials, and investigate properties of the low-energy excitations. This would put these Hamiltonians on par with the Affleck–Kennedy–Lieb–Tasaki (AKLT) model as a reference point where a solid understanding can serve as a touchstone for further investigations. One of them is the question of the stability of the spectral gaps with regard to perturbations in these models. In particular, we aim to develop many-body gap estimation techniques which due to the long-range nature of the interactions in those systems are currently beyond the state of the art. Ultimately, the proof of Haldane’s conjecture is an important element in the explanation of the FQH effect.