Quantum phenomena in strongly correlated electron systems

Ian P. McCulloch

Ikuo Ichinose, Tetsuo Matsui

Hiroyuki Shiba, Masao Ogata

Andreas Mielke

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

S. Zhang, E. C. Allman

M. Schaefer

Strongly correlated electron systems (SCES) represent a class of quantum materials where electron-electron interactions play a dominant role in determining physical properties, leading to a wide array of exotic phenomena such as high-temperature superconductivity, quantum spin liquids, Mott transitions, and heavy fermion behavior. Unlike weakly interacting systems described adequately by Fermi-liquid theory, SCES require more advanced theoretical frameworks due to the breakdown of single-particle descriptions. This article reviews the fundamental interactions governing SCES, emphasizing Coulomb repulsion, spin-charge separation, and entanglement-driven effects. The role of lattice geometry, orbital degeneracy, and dimensionality is explored to understand emergent behaviors. Through theoretical insights and recent experimental findings, this work outlines how these fundamental interactions shape the rich phase diagrams of SCES and their implications in quantum technologies.