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Oberwolfach Reports

Mini-Workshop: Mathematics of Many-body Fermionic Systems

Nikolai Leopold, Phan Thành Nam, Chiara Saffirio

Published Jul 23, 2024• 0 citations
Publisher

Abstract

Fermionic quantum systems are well described by the linear many-body Schrödinger equation. For interacting systems the full Schrödinger theory is extremely complicated and theoretical as well numerical investigations are not feasible. In practice, macroscopic properties of large systems can therefore only be accessed by means of approximate theories. The intention of this workshop was to showcase the most recent advances in the mathematical study of many-body interacting fermionic systems and to stimulate discussions among different research groups.

DOI
10.4171/owr/2023/49

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Fermionic quantum systems are well described by the linear many-body Schrödinger equation. For interacting systems the full Schrödinger theory is extremely complicated and theoretical as well numerical investigations are not feasible. In practice, macroscopic properties of large systems can therefore only be accessed by means of approximate theories. The intention of this workshop was to showcase the most recent advances in the mathematical study of many-body interacting fermionic systems and to stimulate discussions among different research groups.
In the last years, substantial progress has been made in many areas of mathematical physics. The goal of this workshop was to bring together researchers working on analytic and probabilistic aspects of many-body quantum systems and quantum statistical mechanics, to discuss recent developments, exchange ideas and propose new challenges and research directions. Among the questions addressed during the workshop were the derivation of effective equations, the analysis of physically interesting nonlinear partial differential equations emerging from microscopic theories, the study of open quantum systems in and out of equilibrium, and the investigation of the ground state properties and of the dynamics of quantum spin systems.
The interaction among fundamental particles in nature leads to many interesting effects in quantum statistical mechanics; examples include superconductivity for charged systems and superfluidity in cold gases. It is a huge challenge for mathematical physics to understand the collective behavior of systems containing a large number of particles, emerging from known microscopic interactions. In this workshop we brought together researchers working on different aspects of many-body quantum mechanics to discuss recent developments, exchange ideas and propose new challenges and research directions.
The mathematical study of quantum many-body physics has seen significant breakthroughs in the past 15–20 years. Mathematical physicists have approached these systems from different perspectives with different methods. The mini-workshop aimed to connect two rapidly growing fields: the mathematical analysis of many-body systems in scaling regimes (which has led to precise asymptotic formulas of the ground state energy and time evolution, e.g., of Bose gases) and the study of strongly correlated quantum systems via Lieb-Robinson bounds (which provide fundamental insights into the spreading of quantum information).
The mathematical study of quantum many-body physics has seen significant breakthroughs in the past 15–20 years. Mathematical physicists have approached these systems from different perspectives with different methods. The mini-workshop aimed to connect two rapidly growing fields: the mathematical analysis of many-body systems in scaling regimes (which has led to precise asymptotic formulas of the ground state energy and time evolution, e.g., of Bose gases) and the study of strongly correlated quantum systems via Lieb-Robinson bounds (which provide fundamental insights into the spreading of quantum information).
The field of quantum chemistry is concerned with the modelling and simulation of the behaviour of molecular systems on the basis of the fundamental equations of quantum mechanics. Since these equations exhibit an extreme case of the curse of dimensionality (the Schrödinger equation for N electrons being a partial differential equation on \mathbb R^{3N} ), the quantum-chemical simulation of even moderate-size molecules already requires highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together selected quantum chemists and physicists, and the growing community of mathematicians working in the area, to report and discuss recent advances on topics such as coupled-cluster theory, direct approximation schemes in full configuration-interaction (FCI) theory, interacting Green’s functions, foundations and computational aspects of density-functional theory (DFT), low-rank tensor methods, quantum chemistry in the presence of a strong magnetic field, and multiscale coupling of quantum simulations.
Quantum chemistry focuses on modelling and simulating the behaviour of molecular systems using the fundamental principles of quantum mechanics. However, the underlying equations, such as the Schrödinger equation for computing the ground state of N electrons, which is a partial differential equation defined on \mathbb{R}^{3N} , suffer from the curse of dimensionality. As a result, simulating even moderately sized molecules demands advanced techniques in analysis, approximation and reduced-order modelling for overcoming the naive computational complexity. In this workshop, the rapidly growing mathematical community working in the field together with several quantum chemists and physicists were gathered to discuss recent advances in areas such as (1) the mathematical and numerical analysis of standard models used in the field, including Density Functional Theory, Coupled Cluster, and the Density Matrix Renormalization Group (2) the development of efficient numerical methods, relying e.g. on the development of error bounds and (3) the opportunities of recent data-driven methods towards the approximation of wavefunctions or density matrices.