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 MielkeStrong 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

Jan 1, 20150 citations

Self-Generated Gauge Fields in Quantum Matter (SGGF)

Jincheng Zhang

In many-body quantum systems, especially those with strong electronic correlations, the collective behavior of quasiparticles can give rise to emergent gauge degrees of freedom that are not imposed externally but generated intrinsically by the dynamics of the system. This paper proposes a unified theoretical framework for describing such self-generated gauge fields, here denoted as $A_{\mu}^{\text{eff}}$ , that arise naturally in quantum matter. We formulate the conditions under which the internal degrees of freedom of a material can spontaneously form a gauge structure, and we derive the effective action for the emergent gauge field in terms of the microscopic variables. The formalism is applied to a variety of strongly correlated systems, including spin liquids, fractional quantum Hall states, and high-temperature superconductors, illustrating how $A_{\mu}^{\text{eff}}$ modifies the low-energy excitations and leads to novel observable phenomena. Furthermore, we discuss the potential of these emergent interactions to serve as platforms for discovering new artificial fundamental forces, thereby opening a route toward Nobel Prize–level physics.In many-body quantum systems, especially those with strong electronic correlations, the collective behavior of quasiparticles can give rise to emergent gauge degrees of freedom that are not imposed externally but generated intrinsically by the dynamics of the system. This paper proposes a unified theoretical framework for describing such self-generated gauge fields, here denoted as $A_{\mu}^{\text{eff}}$, that arise naturally in quantum matter. We formulate the conditions under which the internal degrees of freedom of a material can spontaneously form a gauge structure, and we derive the effective action for the emergent gauge field in terms of the microscopic variables. The formalism is applied to a variety of strongly correlated systems, including spin liquids, fractional quantum Hall states, and high-temperature superconductors, illustrating how $A_{\mu}^{\text{eff}}$ modifies the low-energy excitations and leads to novel observable phenomena. Furthermore, we discuss the potential of these emergent interactions to serve as platforms for discovering new artificial fundamental forces, thereby opening a route toward Nobel Prize–level physics.

Aug 17, 20260 citations

Quantum phenomena in strongly correlated electron systems

Pavel V. Shevchenko

Sep 1, 19990 citations

Quantum many-body physics : a perspective on strong correlations

義夫倉本

Springer eBooksJan 1, 20203 citations

Lars Onsager Prize Talk: A New Challenge for Cold Atom Physics: Achieving the Strongly Correlated Regimes for Cold Atoms in Optical Lattices.

Tin-Lun Ho

Bulletin of the American Physical SocietyMar 11, 20080 citations

A_{\mu}^{\text{eff}}

, that arise naturally in quantum matter. We formulate the conditions under which the internal degrees of freedom of a material can spontaneously form a gauge structure, and we derive the effective action for the emergent gauge field in terms of the microscopic variables. The formalism is applied to a variety of strongly correlated systems, including spin liquids, fractional quantum Hall states, and high-temperature superconductors, illustrating how

A_{\mu}^{\text{eff}}

modifies the low-energy excitations and leads to novel observable phenomena. Furthermore, we discuss the potential of these emergent interactions to serve as platforms for discovering new artificial fundamental forces, thereby opening a route toward Nobel Prize–level physics.