Fermionic Exchange Symmetry: Consequences for the 1- and 2-Fermion Picture

The Pauli exclusion principle (PEP) is one of most prominent principles in Physics and Chemistry. It is not only fundamentally relevant but also very elegant and can therefore easily be taught already at the high school level: Each energy shell in atoms cannot be occupied by more than two electrons. This significant restriction on occupation numbers manifests itself in the Aufbau principle explaining the structure of atoms and is consequently at the heart of our periodic table. PEP is not only relevant on this atomic scale but also on much larger scales. For instance, it is responsible for the stability of neutron stars, built up from billions of billions of neutrons: All those neutrons would like in principle to collapse to a tiny spatial region due to their attractive gravitational attraction which, however, is prohibited by the PEP not allowing more than two electrons to sit at the same place.
Despite success as an isolated concept, PEP as a restriction on the way how electrons occupy energy shells is de facto a consequence of the more substantial fermionic exchange symmetry. This mathematical property of the many electron system follows from the quantum character of electrons: They are identical and the swapping of any two of them must not change the theoretical description of that physical system. In in a ground-breaking work by the geometer Alexander Klyachko it was recently shown that this more substantial fermionic exchange symmetry implies further restrictions on occupation numbers. These so-called generalized Pauli constraints (GPC) are more restrictive than the famous PEP and make the latter one obsolete.

Now, with all these GPC at hand it is possible and timely as well to develop a comprehensive understanding of the influence of the more abstract fermionic exchange symmetry on the behaviour of electrons occupying energy shells. This is particularly important for our understanding of Quantum Physics since the description of physical effects is made much easier by exploiting an occupation number picture rather than studying the full mathematical properties of the complex many-electron wave function.

In my proposed project I will study and eventually confirm the physical relevance of these generalized Pauli constraints. Since the electrons are `fighting' for the lowest energy shells these GPC are expected to be relevant: At least some electrons need to occupy less preferable (i.e. higher) energy shells to not violate any GPC. This expected conflict between electrons shall be explored and quantified in my project. In particular, I will develop a heuristically motivated principle, the `Pauli pressure', explaining how some electrons are pressed into the lowest few energy shells and how some other electrons desperately try to occupy lower shells. Besides these more fundamental and mathematically concise concepts I will study concrete systems of a few electrons to test my proposed concept. For which systems is this `Pauli pressure' particularly dominant and how does it simplify the theoretical description of quantum systems? How can we measure the `Pauli pressure' and can we find an elegant way to even visualize it in experiments?

The GPC and their potential relevance described in the form of an emerging `Pauli pressure' will also affect our understanding of entanglement. This concept of a spooky action between quantum particles describes some additional correlations between different electrons even if they are spatially well-separated. Yet, if the `Pauli pressure' is sufficiently strong it can freeze electrons in specific energy shells and therefore reduce the total entanglement. This shall be studied and eventually be understood by concise mathematical means.

Since my proposed project is at the heart of Many-Body Quantum Physics it can lead to a very broad academic impact in several related disciplines, as Computational Physics, Quantum Chemistry, Material Science, Condensed Matter Physics and Quantum Information Theory. Since all those academic beneficiaries significantly contribute to the development of quantum technologies my project will have a long-term societal and economic impact.

The expected future quantum revolution will induce transformative advances to industry, science and society. It will create new commercial opportunities that address global challenges and can provide new capabilities for security. This development will also create a lucrative knowledge-based industry, leading to long-term economic, scientific and societal benefits. It will result in a more sustainable, more productive and more secure United Kingdom.

Just to name a few concrete examples, new ideas as secure communication networks, sensors of high sensitivity for biomedical imaging and new paradigms of computation will increase UK's quality of life and will stimulate economic progress, helping to maintain or even strengthen UK's strong position in the world's economy. In particular, quantum computers, due to their capabilities will lead to breakthroughs in the design of chemical processes, new materials, such as higher temperature superconductors, and new paradigms in artificial intelligence and machine learning. Each of these quantum concepts or technologies could result in tremdous improvement in terms of capacity, sensitivity and speed, and will be one of the decisive factors for future success in many industries. Additionally, such quantum applications and technologies are of strategic importance to UK's independence and safety, for instance in the form of secure information storage and transmission and in creating new materials for medicine and energy solutions.