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# Strongly Correlated Matter

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Currently, so-called strongly correlated many-body systems constitute one of the richest playgrounds in condensed matter physics. As is clear by now, the incorporation of strong interactions can lead to a wide variety of fascinating phenomena ranging from the paradigmatic fractional quantum hall effect, the first system naturally exhibiting topological properties, to the physics of transition metal oxides, where a rich variety of fascinating phases have been either observed or conjectured to arise, and ultra-cold atomic ensembles.

The group of high-temperature superconductors consists of cuprates, which are layered compounds of copper and oxygen, and of several iron-based materials. It is believed that the superconducting mechanism is based on electron-electron interactions, however a full understanding of the topic is missing. Some insight was obtained by cluster quantum theories (DCA, DCA^{+}) using 2-dimensional Hubbard model as a simplification of the complicated layered structure of cuprates. Further research is motivated by a large application potential of superconductors working in ideal case at a room temperature.

One of the main driving forces behind our research in the field of transition metal oxides stems from the remarkable features arising as a result of a fine interplay between electron correlations and the relativistic spin-orbit coupling, leading to a rich variety of phenomena like intricate spin-orbital ordering as well as the much-sought-after quantum spin liquid phases. In this area we are currently developing a series of first-principles-based methods relying on well-tested dynamical mean field theory (DMFT) and density functional theory (DFT), which we have succesfully applied to the study of the 5d pyrochlores.

An interesting route we have been recently pursuing is connected with the simulation of ultra-cold atomic gases (UAG). UAG research has gained immense momentum over the past few decades as one of the main roads towards the elucidation of strongly-correlated phenomena owing to the high tunability and ultra-cleanness reachable in these experiments. Working closely with world leading experimentalists, our studies focus on the various aspects of cold atomic systems where we adopt unbiased numerical simulations to benchmark and validate experiments at equilibrium. In this context we have also recently proposed a series of new experimental set-ups aimed towards the study of novel non-equilibrium dynamics as well as exotic topological phenomena.

Indeed, the fractional quantum hall effect (FQHE) can only be understood as a result of strong electron-electron interactions. Thus, no small parameter expansion is possible making it intractable for analytical approaches and, moreover, posing serious constraints on the application of approximate numerical methods. Quite remarkably, FQHE is also interesting from the topological quantum computation point of view since, under certain conditions, it hosts quasi-particle excitations exhibiting non-Abelian anyonic statistics. A key feature potentially allowing for robust quantum computations. Here we have developed state-of-the-art exact diagonalization codes in order to study FQHE under experimentally relevant conditions.

There exists an important intermediate regime where warm states of matter are inadequately described by traditional theoretical frameworks. More precisely, warm-dense matter (WDM) is defined as the region of phase space where the Coulomb coupling parameter and the electron degeneracy parameter are both approximately unity. This implies that the electronic correlation of the system is on par with the relevant thermal effects. Likewise, quantum statistics play an important role, since here the average interparticle spacing is on the same order as the thermal DeBroglie wavelength. This places the WDM regime exactly in between weakly coupled plasma physics and condensed matter physics. The microscopic description of WDM poses a particular difficulty. Since WDM operates around the Fermi temperature and at high densities, classical mechanics is far from adequate. It is not convenient to describe the system as a perturbation from the ground state, i.e. as a sum over electronic excitations, since at this temperature there are so many states.

In the real universe, WDM is predicted to occur in planetary and stellar interiors. Moreover, WDM has recently been shown to be accessible in several experimental setups. Within the condensed matter physics community, interest has been stimulated by the emergence of new techniques for generating strong shock waves in materials via large-scale lasers, heavy ion beams, or energetic materials. Within the plasma physics community, the development of novel sources enabling experimental access to plasma-like states of matter at low temperature and high density has lead to emerging research opportunities at the forefront of the field. All of these methods have the capability of being used in conjunction with one another, e.g. an explosively generated shock wave may be probed by an x-ray source, etc. Additionally short pulse lasers have recently been shown to be powerful tools in WDM research as they can be used to both produce and diagnose WDM. In the figure below, we highlight the region of the temperature-density phase space occupied by the WDM regime.