# Research

Research activities focus on many-body systems and quantum-statistical methods in condensed-matter theory.

See below for a summary, for the grand research perspectives and for various long-term research projects:

## Summary

Our group studies the physical properties of quantum systems consisting of a macroscopically large number of strongly interacting fermions. These systems can show collective behavior that cannot be understood on an independent-particle level. The field covers collective magnetism, correlation-driven metal-insulator transitions, high-temperature superconductivity and unconventional states of matter in general. We are interested in classical and quantum phase transitions in low-dimensional lattice systems and nanostructures, in elementary excitation spectra and also in non-equilibrium phenomena. The employed methods range from field-theoretical techniques and exact diagonalization over dynamical mean-field theory and cluster techniques to quantum Monte-Carlo methods and the density-matrix renormalization group. An important focus is on new methodical developments.

## Grand Research Perspectives

The study of electron-correlation effects has several goals. The first is a pragmatic one. Many of the materials in condensed-matter physics studied nowadays cannot be fully understood without taking into account the effects of the Coulomb interaction among the valence electrons in an explicit way and beyond simple mean-field-type approaches. "Strongly correlated electron systems" represent a class of materials with exciting physical properties originating from the dichotomy of delocalized one-electron Bloch states and the related concept of the Fermi sea one the one hand and strong local interactions and localization or localization tendencies on the other. This "materials class" is extremely diverse, however, and covers such fundamentally different and at the same time partly overlapping sets of systems as the late 3d transition metals, many transition-metal oxides, unconventional high-temperature superconductors, Kondo systems, heavy-fermion metals, Mott insulators, low-dimensional magnetic systems, Luttinger liquids and more. Condensed-matter theory has to address those systems - they are in the focus of intense experimental activities to find and to design new materials with novel properties and functionalities, and progress in this field lives from an intense joint effort of experiment and theory.

A second goal originates from purely methodical "how to" questions. How to compute observable physical properties of a strongly interacting quantum system consisting of a macroscopically large number of degrees of freedom? How to find relevant parameters for perturbative techniques? How to devise methods and how to construct computationally efficient and reliable algorithms to treat such problems on modern computer architectures? These kinds of problems are at the heart of theoretical physics, and strongly correlated electron systems have proven themselves to represent a perfect playground for the development of new analytical concepts and numerical techniques. Their constant improvement is necessary to address ever larger systems, lower temperatures and longer time scales and to tackle systems characterized by several and emergent energy and time scales.

The dramatic progress of computer technology in the last decades had a strong impact on how we think about new methodical approaches nowadays. Pure "paper and pencil" theories are rare and have been more and more replaced by analytical concepts which from the very beginning take into account that the final numerical work can be taken over by computers. The rapid developments in the field of various wave-function-based and Green's-function-based methods represent important examples: Density-matrix renormalization-group methods, in their modern formulation in terms of matrix-product states and matrix-product operators and in all their variants, and the perspectives offered by tensor-network approaches are constantly revolutionizing the field of low-dimensional strongly correlated systems. Similarly, novel quantum-Monte-Carlo techniques for, e.g., infinite diagrammatic re-summations, have paved the way for efficient numerical solutions of highly complex impurity problems. Combined with a third line of developments, namely the various cluster and diagrammatic extensions of the dynamical mean-field theory of lattice-fermion models, this provides us with another route to the most intriguing and fascinating correlations problems, which are typically found in two-dimensional lattice models. At the same time it provides us with a new perspective for material physics beyond the paradigm of conventional band-structure theory.

Another goal, and from an intellectual point of view the maybe most honorable goal, is a conceptual one. It consists in the ambition to understand from a microscopic perspective how matter organizes itself. Here "understanding" means much more than just being able to solve some problem at hand by sheer computational power. The goal is rather to find sustainable concepts, illustrative paradigms, or, more pictorially, islands of obvious insight in the overwhelming sea of coupled microscopic degrees of freedom, from which one can start to explore the immediate surrounding and, hopefully, can extrapolate to an overall idea on the physical properties of macroscopic systems of strongly interacting constituents. Some arbitrarily chosen historical examples are given by Bethe's solution of the one-dimensional Hubbard model, by the Landau Fermi-liquid theory, by the Kosterlitz-Thouless vortex-unbinding transition, by Wilson's theory of the Kondo effect, or by the dynamical mean-field picture of the Mott transition. Continuous progress in research, however, is actually driven by numerous creative ideas, which are less far-reaching as compared to the bigger ones but necessary to pave the way for the breakthroughs.

Among the great lines, (broken) symmetry is a major concept which helps to bridge the micro and the macro world, as it is nicely expressed, for example, in P.W. Anderson's famous Science article "More is Different". But there are more and different ideas which are promising: Topological concepts, for example, play an important role to today's condensed-matter theory and, possibly, for future functional devices. This has triggered exciting novel research fields, e.g., the generalization of topological classifications of edge states and of topologically nontrivial materials to strongly correlated quantum systems, or the investigation of topologically protected excitations, such as skyrmions, and exotic particles, such as Majorana fermions. The study of fundamental mechanisms and of their potential for applications appears particularly promising at the crossroads between topological and correlated systems and, see also below, between topology and real-time dynamics. Hence, an important line of research is to search for and to explore more and more general examples and paradigms of topological behavior.

A third great line of research in strongly correlated electron systems addresses real-time dynamics: The thermodynamical equations of state must eventually emerge from the physical properties in the long-time limit as predicted by quantum-statistical theory of equilibrium. %Correlations may also trigger extremely fast thermalization. Beyond the standard expected behavior, however, a rapidly growing number of unconventional physical phenomena is being discovered, which are related, e.g., to the absence of, to a hindered, or to a strongly delayed thermalization, caused, for instance, by restrictions of the available phase space in combination with electron correlations. Concepts, such as protected metastable states, transient phases with broken symmetries, dynamical phase transitions etc. sketch some of the novel ideas helping to understand the dynamics of strongly correlated electron systems. Furthermore, the role of topologically protected excitations constraining the dynamics, "movies" of topological phase transitions or Berry-phase effects in the real-time dynamics appear as highly exciting fields of research. The search for universality in the properties of systems in steady-state equilibrium and of periodically driven systems, or ambitions to steer the systems between unconventional nonequilibrium phases represent further ultimate visions in this field. Development of new theoretical methods is necessary to make close contact with novel experimental techniques, such as pump-probe electron spectroscopies, time-dependent real-space imaging on atomic length and time scales, and the high level of control achieved in time-dependent experiments with ultra cold fermion gases trapped in optical lattices.

## Magnetic Structures in Low-Dimensional Nanosystems

## Fundamental Concepts in Real-Time Atomistic Spin Dynamics

## Theory of Electron Spectroscopies

## Dynamics and Control of Correlated Quantum Systems

## Bottom-Up Approach to High-Tc Superconductors

## Self-Energy-Functional Theory

## Topological Spin Dynamics

*feedback*of the Berry curvature in the electronic system on the (slow) dynamics of spins, including anomalous precession frequencies, novel effective spin-spin interactions, impact on spin relaxation and on the approach of thermal equilibrium.