**P1** **Quantum phase transitions and collective modes**

*Silke Bühler-Paschen (TU Wien) | Karsten Held (TU Wien) | Jan Kunes (TU Wien) | Alessandro Toschi (TU Wien)*

Quantum phase transitions (QPTs) and collective modes are a prime example where spatial *and* temporal correlations must be taken into account on an equal footing. Beyond weak coupling perturbation theory, spatio-temporal electronic correlations are, however, largely uncharted territory hitherto. Recent methodological advances, together with the developments planned within this project, will allow us to address QPTs and collective modes in strongly correlated systems in a more controlled way. Here we refer to dynamical mean-field theory (DMFT) and its diagrammatic extensions such as the dynamical vertex approximation (DΓA), which include spatial and temporal correlations at all length and time scales.

The **P1** program ranges from investigation of simple Hubbard and periodic Anderson models to material-specific multi-orbital calculations, which will be directly compared to the planned experiments. The **P1** “anchor” material of the research unit **QUAST** is Ce_{3}Bi_{4}Pd_{3}. It exhibits not only signatures of quantum criticality but also evidence for Weyl-Kondo semimetal behavior. Hence fathoming how strong correlations can produce such an unconventional topological state through both additional experiments and theoretical calculations is a further aspect of our project.

**P3** **Correlated topological quantum matter**

*Claudia Felser (MPI-CPfS Dresden) | Johannes Gooth (MPI-CPfS Dresden) | Ronny Thomale (U Würzburg) | Titus Neupert (U Zürich)*

Since the discovery of new topological insulators, the continuously refined study of topological band structures has been extraordinarily fruitful, with a tight interplay of theory and experiment. Weyl, Dirac and nodal line semimetals, topological crystalline insulators, and higher-order topological insulators are just a few examples. The field has so far been dominated by materials for which electron correlations and interactions are claimed to be negligible, while a comprehensive understanding of interactions and topology is still missing.

We will investigate novel topological phenomena in materials that bear signatures of correlations, such as magnetism, charge density waves, Mott and Kondo physics and superconductivity. Our research agenda is oriented along two directions: (i) from weak to strong

correlations, (ii) from two-dimensional (2D) systems to three-dimensional (3D) bulk crystals.

**P4 ****Modeling the interplay of non-locality and topology in correlated systems**

**Modeling the interplay of non-locality and topology in correlated systems**

*Roser Valentí (GU Frankfurt) | Maia G. Vergniory (MPI-CPfS Dresden)*

Understanding topological properties in materials has been of major interest in the last decade with much focus being placed on non-interacting systems. For correlated materials, the developed concepts are, however, often inapplicable due to their reliance on band theory only. This project will address the interplay of correlations, topology and non-locality.

We plan a microscopic modeling of strongly correlated electronic systems with a focus on building tools for the identification of their topological properties. In particular, we want to investigate the effect of non-local (k-dependent) correlations on topology. We will proceed along three lines: (i) we will explore exemplary many-body models to test and develop methods beyond a topological Hamiltonian framework to understand and describe topologically nontrivial states, (ii) we will apply these concepts to analyze the topological properties of relevant materials in **QUAST**, in particular WTe_{2} and TaX_{2} (X = S, Se) in various doping and multilayer arrangements in collaboration with **P1**, **P3**, **P5**, **P6**, **P7** and **P8**. For this purpose we will combine *ab initio* density functional theory (DFT) and projective Wannier functions with many-body techniques (TPSC+DMFT, CPT, DCA) that include momentum-dependent self-energies and correlation functions. (iii) As a complementary explorative long-term goal we will investigate possible routes for constructing topological models for interacting systems by a combination of statistical methods and machine learning techniques. This will contribute to the search, diagnosis and description in **QUAST** of correlated topological models and materials.

**P5 Modeling non-local interaction phenomena in real materials: electrons, lattice & topology**

*Tim Wehling (U Hamburg) | Giorgio Sangiovanni (U Würzburg)*

The coupling of electronic and lattice degrees of freedom is a central motif of many-body physics and defines a formidable challenge to quantitative modeling of quantum matter. In this project, our goal is to develop an *ab initio* downfolding-based theory of coupled electronic and lattice degrees of freedom in correlated electron materials, which accounts for atomistic material details and realistic non-local effects. On this basis, we will address two overarching questions: (i) How are lattice stability and dynamics affected by electronic correlations? We pursue a realistic description of phenomena beyond Born Oppenheimer such as dissipation and Berry curvature as well as electronic non-equilibrium effects on the dynamics of the atomic nuclei. (ii) How do lattice distortions and electron-phonon coupling affect the interplay of electronic correlations and topology? We aim to study electronic vertex functions for the description of non-local correlations, for their impact on electron topology and for their role as central quantities determining the electron-phonon interplay.

Close cooperations within **QUAST** are planned focusing, in particular, on two-particle formalisms for non-local correlation effects (**P1**), Kondo systems (**P2**), interaction effects in WTe_{2} (**P3**), TaX_{2} heterostructures (**P4**), excited state structural dynamics in TaX_{2} (**P6**) and on geometric phases in adiabatic molecular dynamics (**P8**).

**P6 Non-local correlations out of equilibrium**

*Uwe Bovensiepen (U Duisburg-Essen) | Martin Eckstein (U Erlangen-Nürnberg) | Philipp Werner (U Fribourg)*

Correlated materials often exhibit competing low-energy states and complex free energy landscapes. This results in complicated phase diagrams and large responses to external fields. A new perspective on these correlation phenomena can be obtained by driving the materials out of their equilibrium state by laser pulses. For example, the study of the relaxation dynamics of photo-excited charge carriers reveals information on the electron-electron, electron-spin, and electron-phonon interactions in the solid.

The quasi-two-dimensional dichalcogenides 1*T*-TaS_{2} and 1*T*-TaSe_{2} provide and ideal testbed to explore non-equilibrium phenomena on all timescales, ranging from the femtosecond dynamics of correlated electrons close to a Mott transition via the intertwined motion of electrons, lattice, and the charge-density wave (CDW) order on the picosecond timescale to long-lived hidden quantum states which are accessible only via non-thermal pathways.

This project will connect advanced modeling and semi-realistic non-equilibrium materials simulations to ultrafast time-resolved spectroscopy experiments. We will build on our successful recent collaboration, which combined two state-of-the-art tools: time- and angle-resolved photoemission spectroscopy (trARPES) and nonequilibrium dynamical mean-field theory (DMFT). By improving the time-resolution of the experiments and the description of local and nonlocal correlations in the theoretical calculations, we aim at an accurate description of the photo-induced dynamics in the target systems 1*T*-TaS_{2} and 1*T*-TaSe_{2}, which are small-gap Mott insulators of central importance for **QUAST**. The systematic study of photo-excitations in these two materials as a function of static doping, as proposed here, can give insight into the interplay between nonlocal correlations and nonthermal populations, and will provide important benchmarks for the development of an *ab initio* electronic structure theory for strongly-correlated nonequilibrium systems.

In the theory part of the project, we will approach the non-equilibrium dynamics from two opposite limits: In addition to the ultrafast real-time simulations, we will develop extensions of nonequilibrium DMFT to track the evolution of quasi-steady electronic states on the picosecond timescale, much slower than intrinsic processes such as electron tunneling and scattering. This will allow to examine the role of strong correlations in the coupled electron-lattice dynamics of 1*T*-TaS_{2} on the timescale of the CDW order parameter, as well as possibilities to control two-dimensional correlated surfaces via photo excitation of the substrate.

**P7 Coherent propagation and decay of quasi-particles**

*Maurits W. Haverkort (U Heidelberg)*

We will develop a real-space / momentum, real-time / frequency Green’s function method to describe the decay of quasi-particles in the dynamics of realistic materials as seen in local and non-local *n*-point correlation functions. The focus will be on the decay due to charge scattering for either low energy excitations (magnons) or high energy excitations (x-ray induced excitons and resonances). Project **P7** directly contributes to thread 3 on dynamical properties of correlated models and materials in excited states.

We will extend the methods in our program package QUANTY to calculate the dynamics of x-ray excited core states, and low energy magnetic and orbital excitations to include a material specific self energy describing the decay. We will focus on correlated metals relevant within **QUAST** such as, Ce_{3}Bi_{4}Pd_{3}, CeRu_{4}Sn_{6}, Co_{3}Sn_{2}S_{2}, CeBa_{7}Au_{4}Si_{40} and TaS_{2} experimentally investigated within **P1**, **P3** and **P6**. We will use LDA+DMFT as a starting point for the electronic structure calculations.

We will implement methods to calculate the change in the dynamics of x-ray core excited states as a result of the Auger-Meitner decay due to Coulomb scattering of local electrons into the continuum. In order to evade excessively large Hilbert spaces we will implement the Coulomb scattering into non-local states as a self energy for the electronic propagation of the quasi-particle. The propagating quasi-particle in the case of x-ray excitations is a core-hole – valence-electron exciton. Having a non-zero self energy makes the description of the dynamics easier as the system forgets long-time behaviour. We can test our methods by comparing 2 and 4 point correlation functions to the x-ray absorption and resonant inelastic x-ray scattering of the model correlated metals measured (**P1**) and calculated (**P1**, **P5**) within **QUAST**.

After we understand how to treat the self energy of core excited states we will implement the self energy of low energy magnetic excitations due to charge scattering. In the frequency domain this allows one to not only obtain an energy momentum dispersion of magnetic excitations in materials like Ce_{3}Bi_{4}Pd_{3}, CeRu_{4}Sn_{6}, Co_{3}Sn_{2}S_{2} and CeBa_{7}Au_{4}Si_{40} but also their line widths. In the time domain the self energy sets a time scale for the transfer of excitation energy between the spin and charge degrees of freedom of the material. Thereby setting time scales for the equilibration of the slow spin dynamics (**P8**).

All methods will be implemented in QUANTY, a freely available software package which provides a script language able to solve problems in quantum mechanics based on second quantization and a variety of many body techniques.

**P8 Correlation effects in adiabatic spin dynamics**

*Michael Potthoff (U Hamburg) | Alexander I. Lichtenstein (U Hamburg) | Georg Rohringer (U Hambur*g)

This project explores novel physics at the crossroads between topologically nontrivial electronic structure, electron correlations and slow real-time dynamics in condensed-matter systems. Strong Coulomb interaction features the formation of local magnetic moments with a dynamics on a much longer time scale as compared to the fast femtosecond scale of conduction electrons. This time-scale separation, in the extreme adiabatic limit, is exploited to advance the formulation and the application of an effective low-energy theory, called adiabatic spin dynamics (ASD). The ASD incorporates an important topological twist: While the slow spin degrees of freedom generate the well-known Berry phase in the electronic quantum system, there is an additional *feedback* of Berry physics on the slow spin dynamics, which expresses itself as geometrical spin torques.

Geometrical torques result from a nonzero spin-Berry curvature, which in turn is closely related to the nonlocal magnetic response of a quantum system. We analyze the largely universal curvature tensor and study its dependencies on the dimensionality, the symmetries and the topological properties of the underlying system. Considering prototypical models with topologically nontrivial electronic structure, which are relevant for experimental studies in **P1**, **P3**, **P6**, we explore the impact of geometrical torques on the real-time dynamics of magnetic impurities, on magnon spectra, and on thermodynamical properties.

We have a clear focus on electron correlations and their twofold role, namely as a cause for the formation of local magnetic moments and for strong qualitative renormalizations of nonlocal response functions, including the spin-Berry curvature. Corresponding numerical studies require advanced diagrammatic techniques beyond dynamical mean-field theory and profit from know-how exchange with all projects of the research unit.

In a close cooperation with **P1** we advance the ASD approach in the parametric vicinity of a quantum-critical point in the two-dimensional Hubbard model. Effects of electron-phonon interaction in correlated

Chern insulators are jointly studied with **P5**, by elaborating a concept of correlated adiabatic molecular dynamics, which is largely analogous to ASD but accounts for effects of geometrical *forces*. Together with **P4** we explore the potential of generalized Chern numbers derived from the spin-, charge- or pairing-Berry curvature for finding new topological phases and classification schemes.