Novel physics arises when strongly correlated system is driven out of equilibrium by external fields. Dramatic changes in physical properties, such as conductivity, are empirically observed in strongly correlated materials under high electric field. In particular, electric-field driven metal-insulator transitions are well-known as the resistive switching effect in a variety of materials, such as VO
2, V
2O
3 and other transition metal oxides. To satisfactorily explain both the phenomenology and its underlying mechanism, it is required to model microscopically the out-of-equilibrium dissipative lattice system of interacting electrons. In this thesis, we developed a systematic method of modeling non-equilibrium steady states for dissipative lattice systems by means of Non-equilibrium Green's function and Dynamical Mean Field Theory. We firstly establish a "minimum model" to formulate the strong-field transport in non-interacting dissipative electron lattice. This model is exactly soluble and convenient for discussing energy dissipation and steady-state properties. The formalism is then combined with Dynamical Mean Field Theory to provide a systematic framework describing the nonequilibrium steady-state of correlated materials. We use the formalism to study the strong-field transport properties of correlated materials, Mott insulators and Dirac electrons in graphene. We concentrate on the microscopic description of resistive switching. Of particular interest is the filament formation during the dynamical phase transition, which has been interpreted as a result of the delicate interplay between dissipation and Mott physics. We will also examine
IV characteristics and particularly the current saturation of Dirac electrons in graphene. The arXiv version has been updated with minor modifications and corrections.
Strong correlations are important for the understanding of many phenomena in modern condensed matter physics. Examples are high temperature superconductivity (Nobel prize 1987 http://www.nobel.se/physics/ laureates/1987/index.html) and the fractional quantum Hall effect (Nobel prize 1998 http://www.nobel. se/physics/laureates/1998/index.html). The theory for strong correlations needs new concepts and methods to describe such phenomena and to solve the corresponding models. The aim of this course is to provide these methods, to introduce the models, and to help understanding the phenomena in strongly correlated systems. Not the only one but the most important model to describe strongly correlated Fermions is the Hubbard model. In the modern theory of strong correlations it plays the same role as the Ising model in statistical physics: It serves as a standard model to describe and understand most of the phenomena in strongly correlated systems. The Hubbard model was initially introduced to describe the physical behaviour of transition metals, to understand magnetic phenomena in itinerant electron systems (ferromagnetism, anti-ferromagnetism, ferrimagnetism), to describe the Mott transition, and to describe π-electron systems in quantum chemistry. In one space dimension it describes a Luttinger liquid. Although the model has a very simple structure, the behaviour depends strongly on the parameters, the interaction strength, the density, and the underlying lattice. In this course we try to give an overview over the physics of strongly correlated electrons and the Hubbard model. The students are expected to know quantum mechanics and statistical physics. Some knowledge in condensed matter physics is helpful as well. Acknowledgement I am grateful to Johannes Holck. He carefully read the manuscript, found lots of errors, and had several ideas for improvements. About this text The manuscript is available as pdf. Links for citations are coloured in dark red, links within the text in dark blue, links to external material in dark green. Copyright c ©2015ff Andreas Mielke
The Kagome lattice Fermi-Hubbard model is one of the most physically rich, and at the same time most challenging, models to study in strongly-correlated physics. Among its special features are geometric frustration and a flat energy band that create conditions favorable to ferromagnetism near the band insulating limit. Here, we utilize two exact finite-temperature methods, the numerical linked-cluster expansion and the determinant quantum Monte Carlo, to study the extent as well as doping and interaction dependence of ferromagnetic correlations and other thermodynamic properties of the model. We find that repulsive interactions enhance ferromagnetic correlations at high electron densities and that increasing the interaction strength, extends the region with strong ferromagnetic correlations towards half filling, smoothly connecting it to Nagaoka ferromagnetism near the Mott insulating region. We also use the charge compressibility to obtain an accurate estimate for the critical interaction strength for the metal-insulator transition at half filling. These results improve our understanding of the magnetic tendencies of the model away from half filling and pave the way for further studies, including with ultracold atoms in optical lattices.