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Analysis of quantum many-body systems

Periodic Reporting for period 4 - AQUAMS (Analysis of quantum many-body systems)

Reporting period: 2021-04-01 to 2022-03-31

The main focus of this project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g. Bose-Einstein condensation and superfluidity, which makes them interesting both from a physical and a mathematical point of view.

The goal of this project is the development of new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and thus increase our understanding of physical systems. From the point of view of mathematical physics, there has been significant progress in the last few years in understanding the interesting phenomena occurring in quantum gases, and the goal of this project is to investigate some of the key issues that remain unsolved. Due to the complex nature of the problems, new mathematical ideas and methods will have to be developed for this purpose.

During the course of this project, several important open problem could be solved. These include the existence of Bose-Einstein condensation at positive temperature for dilute trapped gases, the stability analysis of an impurity in a Fermi gas, or the confirmation of the validity of the Gellmann-Bruckner formula for the correlation energy of a weakly interacting Fermi gas, just to mention a few. The tools and methods developed in the course of these projects can be expected to to trigger further advances in the field in the future.
This research project is concerned with the mathematical analysis of quantum many-body systems. Important progress was already made during the course of the project. In a joint work [1] with Andreas Deuchert and Jakob Yngvason (University of Vienna), dilute trapped Bose gases were investigated at non-zero temperature. For gases in a harmonic trap, occurrence of Bose-Einstein Condensation was rigorously proved to occur in the Gross-Pitaevskii limit, generalizing previous results applicable only to zero temperature. This result was then extended to homogeneous systems [19], again in joint work with A. Deuchert. The Statistical Mechanics of a uniform electron gas was investigated in joint work [3] with Mathieu Lewin (Paris) and Elliott Lieb (Princeton University), establishing the existence of the thermodynamic limit in a very general sense. The validity of the local density approximation was then proved in the later work [preprint, arXiv:1903.04046]. In joint work with Niels Benedikter, Phan Thanh Nam (LMU Munich), Marcello Porta (Tübingen) and Benjamin Schlein (Zürich), the correlation energy of weakly interacting Fermi gases was investigated [23], and the Gellmann-Bruckner formula was shown to be asymptotically valid as an upper bound. The stability of fermionic systems interacting with point interactions is the subject of the joint work [2,5,6] with Thomas Moser (also at IST Austria). The validity of the Lieb-Thirring inequality for an ideal anyon system in two dimensions was proved in joint work [4] with Douglas Lundholm (KTH Stockholm). Moreover, Andreas Deuchert, Nikolai Leopold and myself have contributed in various ways [7,8,9,15] to the analysis of the angulon model describing rotating impurities in a quantum environment. The strong coupling limit of the polaron is the subject of publications [18,25,26]. The low-density asymptotics of the free energy of a two-dimensional Bose gas was established in [20,31]. Finally, a rigorous derivation of Haldane pseudo-potentials for dilute quantum gases in a magnetic field was obtained in [14].
All the results described above required the development of new mathematical methods and techniques for their successful resolution, hence can be considered going beyond the state-of-the-art of current methods. I will here only mention a couple of important points. To obtain results on Bose-Einstein condensation at non-zero temperature, a key novel feature is the proof of a coercivity estimate for the (bosonic) relative entropy. To investigate the weakly interacting Fermi gas, a rigorous and novel version of bosonization was introduced, which promises to shed new light on correlations in fermionic systems.