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:
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 hrt 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:
Abstract: The metal–insulator transition (MIT) constitutes one of the most profound and persistent problems in condensed matter physics, sitting at the intersection of electronic structure theory, many-body physics, and emergent quantum phenomena. Conventional band theory, which successfully classifies materials as metals or insulators according to the filling of their electronic bands, fails dramatically when confronted with a class of compounds transition metal oxides chief among them that ought to be metallic by band-counting arguments yet are robustly insulating in the laboratory. The resolution of this paradox required abandoning the independent-electron picture entirely and recognizing that strong Coulomb repulsion between electrons can, by itself, drive localization. This insight, formalized by Mott and later distilled into the Hubbard model, established correlation-driven insulation as a phenomenon distinct from band insulation. The present paper offers a theoretical exploration of the MIT problem, tracing its conceptual development from the Mott criterion through the Hubbard Hamiltonian, the Brinkman–Rice scenario of quasiparticle mass divergence, and the watershed development of dynamical mean-field theory (DMFT), which rendered the Mott transition quantitatively tractable for the first time. We argue that the competition between the electronic bandwidth and the on-site interaction encoded in the ratio U/W provides the unifying organizing principle for the field. Building toward the modern frontier, we examine how the Mott transition connects to quantum phase transitions and quantum criticality, where zero-temperature instabilities governed by quantum rather than thermal fluctuations produce non-Fermi-liquid behavior and anomalous scaling. Case studies of vanadium oxides, rare-earth nickelates, and doped cuprates illustrate the diversity and subtlety of real correlated materials. We conclude that the MIT remains a central and unresolved problem whose continued study is increasingly entangled with the development of quantum materials and quantum technologies. Keywords: Metal–Insulator Transition; Mott Insulator; Strongly Correlated Electrons; Hubbard Model; DMFT; Quantum Criticality; Quantum Materials; Condensed Matter Physics
The first reliable analytic calculation of the phase diagram of the bose gas on a d-dimensional lattice with on-site repulsion is presented. In one dimension, the analytic calculation is in excellent agreement with the numerical Monte Carlo results. In higher dimensions, the deviations from the Monte Carlo calculations are larger, but the correct shape of the Mott insulator lobes is still obtained. Explicit expressions for the energy of the Mott and the “defect” phase are given in a strong-coupling expansion. Typeset using REVTEX 1 Strongly interacting quantum systems continue to challenge theoretical physics. Fermionic examples of strong correlations include high temperature superconductors, heavy fermion materials and metal-insulator transitions. The study of these systems has been confined to either numerical simulations which are plagued by finite-size effects (with the notable exception of the infinite-dimensional expansion) or uncontrollable approximations. Similar problems arise in strongly interacting bosonic systems which have attracted a lot of
Introduction In this chapter we study metal–insulator transitions which occur due to correlations between the charge carriers. This quantum phase transitions results from the competition between kinetic energy and Coulomb repulsion where the latter has the tendency to localise the electrons. The simplest model which describes this competition is the one-band Hubbard model with its two energy scales, the bandwidth W and the Coulomb repulsion U between electrons of opposite spins on the same site. For a fixed density of electrons, n = 1; as the ratio U/W increases, we expect to find a zero-temperature phase transition from a metallic state to an insulator where the electrons are frozen at the lattice sites. The method employed here to investigate this transition is the Gutzwiller approach (Gutzwiller, 1965). This is a variational method which played an important role in providing a framework for the ideas of Mott on the problem of the metal–insulator transition (Mott, 1974). On the metallic side of the transition, this approach describes a metal with renormalised Fermi liquid parameters, essentially with enhanced effective mass and uniform susceptibility. This is the so-called highly correlated electron gas (Brinkman and Rice, 1970) for which the metal oxide V 2 O 3 is the prototype system. In the insulating side, Gutzwiller's approximation yields a noninteracting system of localised electrons. This insulating phase is incompressible , i.e. has zero compressibility and infinite zero temperature susceptibility. The latter is not a consequence of interactions but simply due to the divergence of the Curie susceptibility, χ ∝ 1 /T , of non-interacting moments at T = 0. Then, in Gutzwiller's approach the transition is from a paramagnetic metal to a paramagnetic insulator. We will continue using the designation Mott transition to describe this type of phase transition where the insulating phase has no long-range magnetic order. In real systems, most frequently, localisation due to interactions is accompanied by the appearance of antiferromagnetic order.
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.
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.