## Periodic Reporting for period 1 - NIMBqUS (Novel Inroads into Many-Body Quantum Systems)

Reporting period: 2016-04-01 to 2018-03-31

The NIMBqUS project was concerned with the theoretical study of quantum systems consisting of many interacting particles, so-call quantum many-body system. Understanding these systems represents one of the most significant challenges for modern physics.

The overarching objective of the NIMBqUS project was to develop a better understanding of their properties. This has been achieved in three ways: First, a new numerical tool to find the configurations of lowest energy (ground state) of such systems was implemented and released as open-source software. Second, a large amount of general analytical results, that is mathematical statements, derived and proved within the framework of the theory of quantum mechanics, on the behaviour of such many-body systems was obtained. Third, the difficulty of studying certain quantum many-body systems was investigated. Here the aim is not to understand their behavior directly, but rather to find out how difficult it is to understand their behavior - which is of course again a property of the system.

While NIMBqUS was a fundamental science project, the way from pure theory to applications is often not as far as one would think. The implemented numerical method allows to find ground states of many-body systems and several of the analytical results yield insight into the behaviour of many-body systems at finite temperature. Understanding these ground and finite temperature states of concrete systems, is fundamental to many applications ranging from quantum chemistry, to material science. Currently, a major, international, and community-wide undertaking is ongoing in which quantum machines are being developed to simulate such systems in instances that cannot anymore be simulated on classical computers. The theory developed here ties in with, and can be tested on, such devices.

The overarching objective of the NIMBqUS project was to develop a better understanding of their properties. This has been achieved in three ways: First, a new numerical tool to find the configurations of lowest energy (ground state) of such systems was implemented and released as open-source software. Second, a large amount of general analytical results, that is mathematical statements, derived and proved within the framework of the theory of quantum mechanics, on the behaviour of such many-body systems was obtained. Third, the difficulty of studying certain quantum many-body systems was investigated. Here the aim is not to understand their behavior directly, but rather to find out how difficult it is to understand their behavior - which is of course again a property of the system.

While NIMBqUS was a fundamental science project, the way from pure theory to applications is often not as far as one would think. The implemented numerical method allows to find ground states of many-body systems and several of the analytical results yield insight into the behaviour of many-body systems at finite temperature. Understanding these ground and finite temperature states of concrete systems, is fundamental to many applications ranging from quantum chemistry, to material science. Currently, a major, international, and community-wide undertaking is ongoing in which quantum machines are being developed to simulate such systems in instances that cannot anymore be simulated on classical computers. The theory developed here ties in with, and can be tested on, such devices.

The work during the NIMBqUS project was both analytical and numerical. That is to say that the problems identified as interesting were attacked both with mathematical methods - here one tries to prove interesting mathematical statements about quantities in a theory, in this case quantum mechanics, that can be related to real world effects - and computer programs - here one tries to use computer simulations to gain insights into the behavior of the systems of interest.

In the beginning, the numerical work was in the foreground. For that one has to learn an appropriate programming language, familiarize oneself with the relevant software development and debugging tools. One then needs to understand the problem at hand on a sufficiently deep level to be able to break down the computation one wants to achieve into individual instructions the computer can follow and then identify interesting situations in which the produced code can be used to answer relevant science questions.

In later stages most of the work was analytical. Here one needs to first of all absorbe a great amount of knowledge about mathematical entities and techniques. One then needs to understand the essence of the problem, often by considering simplified or generalized versions or other variants of the problem one wants to study. Often numerical simulations can provide useful intuition and typically joint work of several researchers with different backgrounds and discussions during which ideas are conceived, developed, and discarded are necessary to find a proof strategy. Conferences and visits to and by researchers working on related issues are often crucial for this creative process. From that point on, the details of the proof of the results one is aiming at have to be worked out.

Once the results have been obtained, they must be written up in a form understandable for fellow researchers using precise language and the appropriate mathematical formalism. During NIMBqUS, results which, when written in a style understandable for researchers with an appropriate background, fill no less than 10 scientific papers, have been obtained in this way. All of them have in common that they advance our understanding of properties of quantum many-body systems. This constitutes the overall main result achieved.

These results have been disseminated during about a dozent conferences, discussed with colleagues during visits and invitations to give seminars. They have further been picked up by popular science and general press outlets.

In addition to the scientific work, a plethora of service activities to the scientific community have been carried out, ranging from acting as a referee to knowledge transfer and outreach activities. A very significant amount of time also had to be invested in applications for follow-up grants and academic positions, as well as the preparations for the respective job interviews. Especially during the second year, this has consumed a very substantial part of the working hours.

In the beginning, the numerical work was in the foreground. For that one has to learn an appropriate programming language, familiarize oneself with the relevant software development and debugging tools. One then needs to understand the problem at hand on a sufficiently deep level to be able to break down the computation one wants to achieve into individual instructions the computer can follow and then identify interesting situations in which the produced code can be used to answer relevant science questions.

In later stages most of the work was analytical. Here one needs to first of all absorbe a great amount of knowledge about mathematical entities and techniques. One then needs to understand the essence of the problem, often by considering simplified or generalized versions or other variants of the problem one wants to study. Often numerical simulations can provide useful intuition and typically joint work of several researchers with different backgrounds and discussions during which ideas are conceived, developed, and discarded are necessary to find a proof strategy. Conferences and visits to and by researchers working on related issues are often crucial for this creative process. From that point on, the details of the proof of the results one is aiming at have to be worked out.

Once the results have been obtained, they must be written up in a form understandable for fellow researchers using precise language and the appropriate mathematical formalism. During NIMBqUS, results which, when written in a style understandable for researchers with an appropriate background, fill no less than 10 scientific papers, have been obtained in this way. All of them have in common that they advance our understanding of properties of quantum many-body systems. This constitutes the overall main result achieved.

These results have been disseminated during about a dozent conferences, discussed with colleagues during visits and invitations to give seminars. They have further been picked up by popular science and general press outlets.

In addition to the scientific work, a plethora of service activities to the scientific community have been carried out, ranging from acting as a referee to knowledge transfer and outreach activities. A very significant amount of time also had to be invested in applications for follow-up grants and academic positions, as well as the preparations for the respective job interviews. Especially during the second year, this has consumed a very substantial part of the working hours.

During NIMBqUS many results that go very significantly beyond the state of the art were obtained. The publications in highly selective journals that only publish works which make such contributions are an obvious proof of this fact. The results however are numerous and an explanation in layman's terms that would at least convey the main ideas would fill many pages. As an example, one of the results that was obtained is a very general bound that tells us how so-called correlation functions of so-called fermionic operators decay with the distance between the regions on which these operators act. Ultimately, this bound allows us to say something about which type of phase (this word is here meant as a generalization of the widely known concept of phases such as liquid, gaseous, solid) certain systems consisting of many quantum particles, that interact with a strength that decays with the distance of the particles in a way similar to how the Coulomb interaction between charged particles does, can be in.

NIMBqUS was a fundamental science project during which such mathematical statements of lasting value were proved rigorously. As such, it will have a lasting impact on the relevant fields of research and it will eventually have wider socio-economic implications and wider societal implications as it has improved our understanding of quantum many-body systems, which form the basis of current and future technologies. This process hower happens on a time scale of decades and it was clear from the start that no such impacts could be expected to be visible only weeks after the grant period.

On a short time scale, this project does have a very immediate educational value for society. Not only has the fellow been able to widen his own horizon (both scientifically and personally through interactions with researchers from around the world) but also has he been teaching and supervising PhD students.

A further very immediate socio-economic impact this project had is through the community service activity of creating Quantum - the open journal for quantum science. The roughly 60 papers published to the present date, have saved the scientific community around 120.000€ in article processing charges compared to similarly selective journals run by commercial publishers.

NIMBqUS was a fundamental science project during which such mathematical statements of lasting value were proved rigorously. As such, it will have a lasting impact on the relevant fields of research and it will eventually have wider socio-economic implications and wider societal implications as it has improved our understanding of quantum many-body systems, which form the basis of current and future technologies. This process hower happens on a time scale of decades and it was clear from the start that no such impacts could be expected to be visible only weeks after the grant period.

On a short time scale, this project does have a very immediate educational value for society. Not only has the fellow been able to widen his own horizon (both scientifically and personally through interactions with researchers from around the world) but also has he been teaching and supervising PhD students.

A further very immediate socio-economic impact this project had is through the community service activity of creating Quantum - the open journal for quantum science. The roughly 60 papers published to the present date, have saved the scientific community around 120.000€ in article processing charges compared to similarly selective journals run by commercial publishers.