Introduction How a substance evolves from a metallic to a non metallic state is one of the most fundamental and richest problems in condensed matter physics. In general, there are several mechanisms at play. In particular, electron-electron interactions can drive a metal to insulator transition in a pure substance (Mott, 1990). This transition is named after Sir Nevill Mott who laid down the foundations for the physical understanding of this phenomenon. Another route was discovered by Anderson (1958) who realized that sufficiently strong disorder can drive a metal-insulator transition even in systems of non interacting electrons. The theoretical description of the situation when both effects are present is a central unsolved problem. For recent reviews see Lee & Ramakrishnan (1983) and Belitz & Kirkpatrick (1995). Early treatments (Abrahams et al. 1979, Wegner 1976, Schaffer & Wegner 1980, Finkelshtein 1983, 1984) used direct analogies from the theory of magnetism and the scal
Introduction How a substance evolves from a metallic to a non metallic state is one of the most fundamental and richest problems in condensed matter physics. In general, there are several mechanisms at play. In particular, electron-electron interactions can drive a metal to insulator transition in a pure substance (Mott, 1990). This transition is named after Sir Nevill Mott who laid down the foundations for the physical understanding of this phenomenon. Another route was discovered by Anderson (1958) who realized that sufficiently strong disorder can drive a metal-insulator transition even in systems of non interacting electrons. The theoretical description of the situation when both effects are present is a central unsolved problem. For recent reviews see Lee & Ramakrishnan (1983) and Belitz & Kirkpatrick (1995). Early treatments (Abrahams et al. 1979, Wegner 1976, Schaffer & Wegner 1980, Finkelshtein 1983, 1984) used direct analogies from the theory of magnetism and the scal
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 Mott and the Anderson routes to localization have long been recognized as the two basic processes that can drive the metal-insulator transition (MIT).Theories separately describing each of these mechanisms were discussed long ago, but an accepted approach that can include both has remained elusive.The lack of any obvious static symmetry distinguishing the metal from the insulator poses another fundamental problem, since an appropriate static order parameter cannot be easily found.More recent work, however, has revisited the original arguments of Anderson and Mott, which stressed that the key diference between the metal end the insulator lies in the dynamics of the electron.This physical picture has suggested that the "typical" (geometrically averaged) escape rate τ -1 typ = exp ln τ -1 esc from a given lattice site should be regarded as the proper dynamical order parameter for the MIT, one that can naturally describe both the Anderson and the Mott mechanism for localization.This article provides an overview of the recent results obtained from the corresponding Typical-Medium Theory, which provided new insight into the the two-fluid character of the Mott-Anderson transition. I. FROM METAL TO INSULATOR: A NEW PERSPECTIVETo understand many, if not most exotic new materials, one has to tackle the difficult problem of understanding the metal-insulator quantum phase transition, as driven by the combined effects of strong electronic correlations and disorder.Traditional approaches to the problem, which emerged in the early 1980s, have focused on examining the perturbative effects of disorder within the Fermi liquid framework.Despite their mathematical elegance, these theories, unfortunately, prove ill-suited to describe several key physical processes, such as tendency to local magnetic moment formation and the approach to the Mott insulating state.In addition, such weak-coupling theories cannot easily describe strongly inhomogeneous phases, with behavior often dominated by broad distributions and rare disorder configurations.This new insight, which is largely driven by experimental advances, seems to suggest that an alternative theoretical picture may provide a better starting point.In this article we describe recent advances based on a new theoretical method, which offers a complementary perspective to the conventional weak-coupling theories.By revisiting the original ides of Anderson and Mott, it examines the typical escape rate from a given site as the fundamental dynamical order parameter to distinguish between a metal and an insulator.This article describes the corresponding Typical-Medium Theory (TMT) and discusses some of its recent results and potential applications.We fist discuss discuss, in some detail, several experimental and theoretical clues suggesting that a new theoretical paradigm is needed.The formulation of TMT for Anderson localization of noninteracting electrons is then discussed, with emphasis on available analytical results.Finally, we review recent progress in applying TMT to the Mott-Anderson transition for disordered Hubbard models, and discuss resulting the two-fluid behavior at the critical point. II. THEORETICAL CHALLENGES: BEYOND CINDERELLA'S SLIPPER?The existence of a sharp metal-insulator transition at T = 0 has been appreciated for many years [1].Experiments on many systems indeed have demonstrated that a well defined critical carrier concentration can easily be identified.On the theoretical side, ambiguities on how to describe or even think of the metal-insulator transition have made it difficult to directly address the nature of the critical region.In practice, one often employs the theoretical tools that are available, even if possibly inappropriate.Even worse, one often focuses on those systems and phenomena that fit an available theoretical mold, ignoring and brushing aside precisely those features that seem difficult to understand.This "Cinderella's slipper" approach is exactly what one should not do; unfortunately it happens all too often.A cure is, of course, given by soberly confronting the experimental reality: what seems paradoxical at first sight often proves to be the first clue to the solution.A. Traditional approaches to disordered interacting electrons
We present a theory for disordered interacting electrons that can describe both the Mott and Anderson transitions in the respective limits of zero disorder and zero interaction. We use it to investigate the T=0 Mott-Anderson transition at a fixed electron density, as the disorder strength is increased. Surprisingly, we find two critical values of disorder Wnfl and Wc. For W>Wnfl, the system enters a ``Griffiths'' phase, displaying metallic non-Fermi liquid behavior. At even stronger disorder, W=Wc>Wnfl the system undergoes a metal-insulator transition, characterized by the linear vanishing of both the typical density of states and the typical quasiparticle weight.
Quantum materials exhibit a rich dynamic of physical parameters, which, when combined, can lead to entirely different behaviors. These parameters constantly compete with each other, with the most influential parameters determining the state of the system. For example, in the case of metal–insulator transitions, electron–electron interactions compete with the kinetic energy of the electrons and disorder. Understanding these complex dynamics is crucial for both fundamental physics and the development of novel technological applications, particularly given the role of disorder in tuning critical temperatures, a property with significant potential benefit in the fabrication of new devices where temperature requirements are still the bottleneck. In this article, properties of the Mott metal–insulator transition within disordered electron systems are explored using the disordered Hubbard model, the simplest Hamiltonian for capturing the metal–insulator transition. The model solutions are obtained using the self-consistent statistical dynamical mean-field theory (statDMFT). statDMFT incorporates local electronic correlation effects while allowing for Anderson localization due to disorder.
Correlated electronic systems involve strong shortrange repulsion. At integer filling, the primary effect of correlation is the emergence of an insulating state where band theory predicts a metal. Understanding the transition from a metal to this ‘Mott insulator’, as interaction strength is increased, and the effect of doping the Mott state, are classic problems in quantum many-body physics. They have been explored for decades, starting with the approximate solutions of Hubbard. The real breakthrough came in the 1990s with the advent of dynamical mean field theory, which mapped the correlated system to an embedded single site, and described the detailed spectral evolution across the Mott transition. The only deficiency in this approach is the neglect of spatial correlations, and, possibly, the lack of some ‘visual intuition’ about the transition. We discuss a complementary approach, which provides a reasonable description of Mott physics in a real-space setting and allows a certain degree of visualization. This article provides a pedagogical review of our approach and presents some illustrative results.
The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions, driven by experiments on cuprate superconductors, heavy fermion materials, organic conductors, and related compounds. Although quantum phase transitions in simple spin systems, like the Ising model in a transverse field, were studied in the early 1970s, much of the subsequent theoretical work examined a particular example: the metal–insulator transition. While this is a subject of considerable experimental importance, the greatest theoretical progress was made for the case of the Anderson transition of non-interacting electrons, which is driven by localization of the electronic states in the presence of a random potential. The critical properties of this transition of noninteracting electrons constituted the primary basis upon which most condensed matter physicists have formed their intuition on the behavior of the systems near a quantum phase transition. However, it is clear that strong electronic interactions play a crucial role in the systems of current interest noted earlier, and simple paradigms for the behavior of such systems near quantum critical points are not widely known. It is the purpose of this book to move interactions to center stage by describing and classifying the physical properties of the simplest interacting systems undergoing a quantum phase transition. The effects of disorder will be neglected for the most part but will be considered in the concluding chapters. Our focus will be on the dynamical properties of such systems at nonzero temperature, and it will become apparent that these differ substantially from the noninteracting case.