Correlated electron systems present a profound challenge in condensed matter physics due to the intricate interplay between charge, spin, orbital, and lattice degrees of freedom. Understanding their rich macroscopic phase diagrams, which include phenomena like high-temperature superconductivity, colossal magnetoresistance, and Mott insulation, necessitates a bridge between the microscopic quantum interactions and emergent collective behavior. This paper introduces and formalizes the concept of quantum fugacity as a critical thermodynamic parameter capable of establishing this connection. Extending the classical notion of fugacity—a measure of a real gas's effective partial pressure—to the quantum many-body realm, quantum fugacity accounts for both particle density and the profound influence of quantum statistics and strong electron correlations. We develop a theoretical framework to calculate quantum fugacity from microscopic Hamiltonians, demonstrating its utility in mapping quantum phase transitions and discerning the driving forces behind symmetry-breaking phases. Through this approach, we show how variations in quantum fugacity serve as indicators for the onset of novel phases, providing a unified perspective on the phase behavior of diverse correlated electron materials. This concept offers a powerful new tool for designing and predicting materials with desired quantum functionalities.
Correlated electron systems present a profound challenge in condensed matter physics due to the intricate interplay between charge, spin, orbital, and lattice degrees of freedom. Understanding their rich macroscopic phase diagrams, which include phenomena like high-temperature superconductivity, colossal magnetoresistance, and Mott insulation, necessitates a bridge between the microscopic quantum interactions and emergent collective behavior. This paper introduces and formalizes the concept of quantum fugacity as a critical thermodynamic parameter capable of establishing this connection. Extending the classical notion of fugacity—a measure of a real gas's effective partial pressure—to the quantum many-body realm, quantum fugacity accounts for both particle density and the profound influence of quantum statistics and strong electron correlations. We develop a theoretical framework to calculate quantum fugacity from microscopic Hamiltonians, demonstrating its utility in mapping quantum phase transitions and discerning the driving forces behind symmetry-breaking phases. Through this approach, we show how variations in quantum fugacity serve as indicators for the onset of novel phases, providing a unified perspective on the phase behavior of diverse correlated electron materials. This concept offers a powerful new tool for designing and predicting materials with desired quantum functionalities.
The exploration of quantum phases in strongly correlated electron systems represents a frontier in condensed matter physics, offering insights into phenomena such as high-temperature superconductivity, Mott insulation, and quantum spin liquids. Traditional theoretical approaches, relying on perturbative or mean-field approximations, often fall short in capturing the intricate interplay of electron correlations that drive these exotic phases. This paper delves into the critical role of exact and highly accurate diagrammatic expansions as powerful, non-perturbative tools for unraveling the emergence and characterization of quantum phases. We review the theoretical foundations of these methods, including advanced techniques like Dynamical Mean-Field Theory extensions and the Functional Renormalization Group, and discuss their ability to systematically incorporate non-local correlations. By examining their application to canonical models, we illustrate how these expansions provide a robust framework for constructing phase diagrams, predicting novel quantum states, and discerning the mechanisms underlying quantum phase transitions. We highlight the computational advancements, such as diagrammatic Monte Carlo, that enable the practical implementation of these complex theories. Finally, we discuss current challenges and future prospects, emphasizing the potential of exact diagrammatic expansions to deepen our understanding of quantum matter and guide the discovery of new materials with tailored functionalities.
The study of strongly correlated materials remains one of the most challenging and active frontiers in condensed matter physics. In these systems, the intricate interplay between electrons' kinetic energy and their mutual Coulomb repulsion gives rise to a plethora of exotic phenomena, from high-temperature superconductivity to giant magnetoresistance, which cannot be explained by conventional one-electron theories. This necessitates the development and application of advanced theoretical and computational methods capable of capturing the essential physics of strong electronic correlations. A diverse array of numerical and analytical techniques has been developed to tackle the many-body problem. However, no single method serves as a panacea; each comes with its own set of strengths and weaknesses, and their domains of applicability vary significantly. The choice of an appropriate approximation is therefore highly dependent on the specific physical system and the scientific question at hand. Different problems may benefit from different, and sometimes complementary, methodological perspectives. This thesis focuses primarily on the application and development of local or quasi-local approaches. The rationale for this focus is rooted in the physical reality of many key correlated materials, such as transition metal oxides, heavy-fermion systems or molecular solids like alkali-doped fullerenes, where the local on-site electronic interaction is the dominant energy scale. By prioritizing the accurate treatment of local physics, these methods provide a robust and computationally efficient approximation for understanding a wide range of material properties. Recent experimental and theoretical advancements—spanning the rich physics of multiorbital compounds, the precise control over interactions in ultracold atomic gases, and the complex interplay in electron-phonon coupled systems—have created an urgent need for theoretical frameworks that can deliver reliable and rapid insights. The methods explored in this work are meant to address these challenges, offering a balance between accuracy and computational feasibility. Given the variety of topics addressed, from methodological developments to their application in diverse physical contexts, this thesis will try to be as detailed and self-contained as necessary, providing a clear exposition of the underlying concepts and technical aspects.
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.
Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.
In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d -wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high-level theoretical presentations and research papers specialized in certain classes of materials. It will give an overview on a variety of different quantum transitions, critically discuss open theoretical questions, and frequently make contact with recent experiments in condensed matter physics.
Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.