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Oberwolfach Reports

Many-Body Quantum Systems and Effective Theories

Christian Hainzl, Benjamin Schlein, Robert Seiringer

Published Apr 22, 2017• 0 citations
Publisher

Abstract

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.

DOI
10.4171/owr/2016/43

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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.
During the last decade a host of experiments has established the existence of highly unconventional states of matter in a variety of low dimensional quantum systems. These include equilibrium states characterized by topological properties as well as stationary states in out of equilibrium situations. Both of these give access to hitherto unexplored areas of many-body quantum physics. The key aim of the workshop was to bring together experts in several different research areas revolving around novel quantum states of matter.
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.
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.
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 NNN electrons, which is a partial differential equation defined on R3N\mathbb{R}^{3N}R3N, 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.