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AIP conference proceedings

Highly Correlated Particle Systems and Composite Operator Method

Hideki Matsumoto

Published Jan 1, 2005• 0 citations
Publisher

Abstract

Since the discovery of high Tc superconductors in cuprate oxides, physics of highly correlated electron systems has become one of the central issues in condensed matter physics. We can find many examples of highly correlated particle systems not only in electron systems, such as high Tc superconductors and transition metal oxides, but also in bosonic systems, such as alkaline atomic gases in an optical lattice and mesoscopic Josephson junction arrays. In this lecture, we illustrate several models of highly correlated particle systems of fermionic and bosonic systems, and clarify common features of these systems. Then theoretical approaches to analyze these systems will be reviewed; special emphasis is on the composite operator method, which has been developed by the present author and the Salerno group.

DOI
10.1063/1.2080348

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Since the discovery of high Tc superconductors in cuprate oxides, physics of highly correlated electron systems has become one of the central issues in condensed matter physics. We can find many examples of highly correlated particle systems not only in electron systems, such as high Tc superconductors and transition metal oxides, but also in bosonic systems, such as alkaline atomic gases in an optical lattice and mesoscopic Josephson junction arrays. In this lecture, we illustrate several models of highly correlated particle systems of fermionic and bosonic systems, and clarify common features of these systems. Then theoretical approaches to analyze these systems will be reviewed; special emphasis is on the composite operator method, which has been developed by the present author and the Salerno group.
This book contains lectures on strongly correlated electron systems presented by eminent physicists. These lectures are up-to-date summaries of relevant subjects in the field of condensed matter physics intended to train students. Contributions include: strongly correlated electron behaviors and heavy Fermions in anomalous rare-earth and actinide systems; strong correlations in low dimensional systems; functional renormalization group approach to correlated electron systems; and numerical approaches to coupled quantum systems.
The discovery of high-Tc superconductivity in cuprates in 1986 moved strongly correlated systems from exotic worlds interesting only for pure theorists to the focus of solid-state research. In recent decades, the majority of hot topics in condensed matter physics (high-Tc superconductivity, colossal magnetoresistance, multiferroicity, ferromagnetism in diluted magnetic semiconductors, etc.) have been related to strongly correlated transition metal compounds. The highly successful electronic structure calculations based on density functional theory lose their predictive power when applied to such compounds. It is necessary to go beyond the mean field approximation and use the many-body theory. The methods and models that were developed for the description of strongly correlated systems are reviewed together with the examples of response function calculations that are needed for the interpretation of experimental information (inelastic neutron scattering, optical conductivity, resonant inelastic X-ray scattering, electron energy loss spectroscopy, angle-resolved photoemission, electron spin resonance, and magnetic and magnetoelectric properties). The peculiarities of (quasi-) 0-, 1-, 2-, and 3- dimensional systems are discussed.
Since the discovery of cuprate high-temperature superconductors, there has been a lot of interest in theoretical studies 011 the strongly-correlated eleetron systems.It is now accepted that the t-J model is one of the canonical models of the high-Tr.superconductors, and this model has been studied extensively b)' various methods.Most difficult aspect of the t-J model is the existence of a, local constraint 011 the physical states, excluding the double-occupancy of electrons at each site.~/Iany recent
Strongly interacting quantum systems exhibit same of the most interesting phenomena in condensed matter physi=. Until very recently the study of these systems has heen confined to either numerical simulations of rather small systems (which still have an enormous number of q“ant”m stat=) or approximate ~na. lytical methods which are uncontrolled. Mat of the effort concentrated on the fermionic systems. In the last few yearn the hmonic systems have attracted a lot of interest.l ,2,3,4,5 Example of the bmonic strongly correlated systems inc]ude ~h~rt coherence length superconductors, Jc6ephson junction arrays, the dynamics of flux lines in high temperature superconductors and the quantum hall effect.6 The ~ntial physics of all of these problems is the competition between the hopping matr~ elements and the short range strong repulsion. The minimum mdel which contains the bmicphysics of theproblem istheso called Bme Hubbard model:
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.
The study of the correlated motion of electrons in solids is of increasing importance in condensed matter physics. In the past few years, the discovery of high-temperature superconductors has stimulated an enormous theoretical effort in this area, building on earlier theories of heavy-fermion and organic superconductors, and magnetic insulators. In a separate development, the discovery of the fractional quantum Hall effect stimulated research into the behaviour of the two-dimensional electron gas in a strong transverse magnetic field. The lectures give a systematic presentation of the current status of the theory in these areas. They cover the fractional quantum Hall effect and the many-body physics of the Hubbard model and its extension, paying particular attention to the properties of doped insulators which are relevant for high-temperature superconductivity. They detail discussions of situations for which controlled calculations may be carried out - specifically infinite dimensions, one dimension, and generalized models in which the fermions have N components and N to infinity.