# Research

Materials with strong electronic correlations are amongst the most interesting topics at the forefront research in physics. The reason for this is that they exhibit a vast variety of fascinating phenomena with a high potential for applications, but they still are poorly understood by theory. One prototypical example is superconductivity, a pure quantum mechanism causing the electric resistance of a material to vanish upon cooling. In a certain class of superconductors, the so called cuprates, this transition from the metallic to the ideally conducting state can happen at temperatures of 135 K. This is quite remarkable as this temperature is above the boiling point of liquid nitrogen (78 K), resulting in this class of materials being a perfect candidate for future technological applications. On the other hand, the theoretical description of these systems is immensely challenging, so that no consistent theory could be established since their first discovery in 1987. This research group aims at the description of systems with unconventional superconductivity (like cuprates and nickelates) with cutting-edge quantum field theoretical techniques. Complementary techniques (multi-method) are applied for obtaining various oservables (multi-messenger) at the one- and two particle level. This agenda comprises the calculation of spectral functions as well as susceptibilities in several regimes of the phase diagrams of these materials (superconductivity, pseudogap, magnetism and Fermi liquid regimes) in order to create comprehensive physical insight.

M. Klett, N. Wentzell, T. Schäfer, et al., Phys. Rev. Research

**2**, 033476 (2020)M. Kitatani, T. Schäfer, et al., Phys. Rev. B

**99**, 041115(R) (2019)
The phenomenon of quantum criticality in magnetic systems, on the one hand, exhibits fascinating physics, e.g., the breakdown of the standard theory of metals, Landau’s Fermi liquid theory. On the other hand it provides a big challenge for the theory, mainly because several of the experimentally analyzed materials exhibiting quantum phase transitions are strongly correlated, such as the so-called heavy fermion compounds. The difficulty of describing strongly correlated electron systems arises as in these materials the Coulomb interaction between the electrons is poorly screened and, hence, it is not possible to understand their physics in terms of a non-interacting electron gas. This implies that standard methods such as conventional perturbation theories are no longer applicable. In fact, also in the specific case of quantum phase transitions, the Hertz-Millis-Moriya theory, is expected to break down if correlations become dominant. In the state-of-the-art theory, dynamical mean field theory (DMFT), strong correlations can be treated. DMFT neglects all spatial correlations but takes all temporal correlations into account. However, for the description of low dimensionality (e.g. three or even two dimensions) as well as the highly non-local quantum criticality, also non-local (i.e. spatial) correlations have to be included in the theory on top of the purely temporal ones. This can be achieved by extending the DMFT in certain ways. One possible extension, which includes spatial correlations on every length scale, is the so-called dynamical vertex approximation (DΓA), which has the potential to properly describe a magnetic quantum phase transition at zero temperature.

**122**, 227201 (2019)

T. Schäfer, et al., Phys. Rev. Lett.

**119**, 046402 (2017)

One way of dealing with the complexity inherent to strongly correlated materials is found in abstraction: what is the simplest Hamiltonian incorporating the basic physics of the problem at hand? The Hubbard model represents, alike fruit flies for biology,

*the*fundamental modellization for such interacting quantum systems: electrons can hop from one crystal site to another and the electron-electron interaction is only present, when two electrons occupy the same site. Despite its simplicity, the Hubbard model has not been solved in finite dimensions larger than one and recent years saw a strong increase in the number of computational approaches aiming at the determination its properties. Within this research group, the fundamental properties of the Hubbard model are being investigated, both on the physical as well as on the methodological level.T. Schäfer et al., arXiv:2006.10769 (2020)

T. Schäfer et al., Phys. Rev. Lett.

**110**, 246405 (2013)

In order to gain physical insight into the strongly correlated systems described above, state-of-the-art numerical many body techniques are not only applied but also developed in this research group. One of the big steps forward in the description of strong correlations was the dynamical mean field theory (DMFT), that incorporates all temporal fluctuations. However, if spatial correlations are becoming important (as in the vicinity of second order phase transitions), one has to go one step beyond DMFT. Within the DMFT framework, there are two paths to achieve this inclusion of spatial correlations: its cluster and diagrammatic extensions, respectively. Both are applied and developed further in this research group.

O. Gunnarsson, T. Schäfer, et al., Phys. Rev. Lett.

**114**, 236402 (2015)

G. Rohringer et al., Rev. Mod. Phys.

**90**, 025003 (2018)

T. Maier et al., Rev. Mod. Phys.

**77**, 1027 (2005)