TRR 352:  Mathematics of Many-Body Quantum Systems and Their Collective Phenomena

• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 470903074

Understanding the extremely rich macro- and mesoscopic phenomenology of matter in terms of microscopic physical laws has always been one of the most fascinating endeavors of science. Improvements in this understanding based on quantum mechanical laws have led to undreamt-of technological advances in many areas, and in recent years even former visions such as quantum computing seem to have come within reach. The mission of the proposed CRC is to contribute to this scientific enterprise at the level of its mathematical foundations. The proposed CRC is focused on the mathematical analysis of models from condensed matter physics, where the collective behavior of a large number of interacting components such as particles or spins leads to a diverse and technologically relevant phenomenology. While the mathematical equations that make up the microscopic quantum models are mostly well understood, the task of extracting useful information from them and linking them to collective phenomena at a meso- or macroscopic level still pushes the limits of current mathematical technology. The mathematical understanding and harnessing of correlations is the unifying theme of the proposed CRC. Different facets and types of correlations and entanglement in many-body quantum systems will be addressed from different perspectives. Each project either contributes to the analysis of correlations in specific classes of models, or proves the emergence of collective phenomena of interest and develops improved numerical algorithms based on a thorough understanding of the correlations, or both. The planned CRC is a joint activity of the Universities of Munich (LMU and TUM) and Tübingen and is further strengthened by the external members from the Institute of Science and Technology Austria (Klosterneuburg) and the associate members from the Universities of Copenhagen and Zurich.

• DFG Programme: CRC/Transregios
• International Connection: Austria, Denmark, Switzerland

Current projects

• A01 - Large Quantum Coulomb Systems (Project Heads Frank, Rupert ;  Nam, Phan Thanh ;  Solovej, Jan Philip )
• A02 - Rigorous Approximations for Fermionic Kinetic Energies (Project Head Frank, Rupert )
• A03 - Decay of Correlations in Continuum Particle Systems (Project Heads Heydenreich, Markus ;  Jansen, Sabine )
• A04 - The Correlation Energy of Many-Body Bose Systems (Project Heads Hainzl, Christian ;  Schlein, Benjamin ;  Triay, Arnaud )
• A05 - The Correlation Energy of Many-Body Fermi Systems (Project Heads Hainzl, Christian ;  Nam, Phan Thanh ;  Seiringer, Robert )
• A06 - Mathematical Challenges in the BCS Theory of Superconductivity (Project Heads Frank, Rupert ;  Hainzl, Christian ;  Seiringer, Robert )
• A07 - Low-Energy States and Spectra of Quantum Hall Systems (Project Heads Lemm, Marius ;  Warzel, Simone )
• A08 - Finite-Size Criteria for Spectral Gaps in Quantum Lattice Systems (Project Heads Lemm, Marius ;  Warzel, Simone )
• A09 - Localization for Very Sparse Erdös-Rényi Random Graphs (Project Heads Erdös, Ph.D., Laszlo ;  Heydenreich, Markus ;  Müller, Peter )
• B01 - Norm Approximation for Interacting Many-Body Fermi Gases (Project Heads Pickl, Peter ;  Schlein, Benjamin )
• B02 - Derivation of Nonlinear Quantum Boltzmann Equation (Project Heads Nam, Phan Thanh ;  Spohn, Herbert )
• B03 - Hydrodynamic Equations for Integrable Many-Body Systems (Project Heads Mendl, Christian ;  Spohn, Herbert )
• B04 - Numerical Gaussian-Based and Tensor-Network Methods in Quantum Dynamics (Project Heads Lasser, Caroline ;  Lubich, Christian )
• B05 - Numerics for Wigner-Weyl methods in Many-Body Quantum Dynamics (Project Heads Lasser, Caroline ;  Mendl, Christian )
• B06 - Complexity in Many-Body Quantum Systems (Project Heads Christandl, Ph.D., Matthias ;  König, Robert ;  Wolf, Michael )
• B07 - Markovian Dynamics in Many-Body Quantum Systems (Project Heads Capel, Angela ;  Rouzé, Cambyse )
• B08 - Local Stable Gaps and Response in Interacting Many-Body Quantum Systems (Project Heads Capel, Angela ;  Teufel, Stefan )
• Z - Central Task (Project Head Hainzl, Christian )

• Applicant Institution: Ludwig-Maximilians-Universität München
• Co-Applicant Institution: Eberhard Karls Universität Tübingen; Technische Universität München (TUM)
• Participating University: Københavns Universitet; Universität Zürich
• Participating Institution: Institute of Science and Technology Austria
• Spokesperson: Professor Dr. Christian Hainzl