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
Strongly correlated electrons with box disorder in high-dimensional lattices are investigated. We apply a statistical dynamical mean-field theory that treats local correlations nonperturbatively. The incorporation of a finite lattice connectivity allows for the detection of disorder-induced localization via the probability distribution function of the local density of states. We obtain a complete paramagnetic ground-state phase diagram and find correlation-induced as well as disorder-induced metal-insulator transitions. Our results qualitatively confirm predictions obtained by typical medium theory. Moreover, we find that the probability distribution function of the local density of states in the metallic phase strongly deviates from a log-normal distribution as found for the noninteracting case.
The interplay between Mott and Anderson routes to localization in disordered interacting systems gives rise to different transitions and transport regimes. Here, we investigate the phase diagram at finite temperatures using dynamical mean-field theory combined with typical medium theory, which is an effective theory of the Mott-Anderson metal-insulator transition. We mainly focus on the properties of the coexistence region associated with the Mott phase transition. For weak disorder, the coexistence region is found to be similar to that in the clean case. However, as we increase disorder, Anderson localization effects are responsible for shrinking the coexistence region, and at sufficiently strong disorder (approximately equal to twice the bare bandwidth) it drastically narrows, the critical temperature
Tc abruptly goes to zero, and we observe a phase transition in the absence of a coexistence of the metallic and insulating phases. In this regime, the effects of interaction and disorder are found to be of comparable importance for charge localization.
This work is devoted to the development of a systematic method for studying electron localization. The developed method is Typical Medium Dynamical Cluster Approximation (TMDCA) using the Anderson-Hubbard model. The TMDCA incorporates non-local correlations beyond the local typical environment in a self-consistent way utilizing the momentum resolved typical-density-of-states and the non-local hybridization function to characterize the localization transition. For the (non-interacting) Anderson model, I show that the TMDCA provides a proper description of the Anderson localization transition in one, two, and three dimensions. In three-dimensions, as a function of cluster size, the TMDCA systematically recovers the re-entrance behavior of the mobility edge and obtains the correct critical disorder strength for the various disorder configurations and the associated \textit{universal order-parameter-critical-exponent}
β and in lower-dimensions, the well-knowing scaling relations are reproduced in agreement with numerical exact results. The TMDCA is also extended to treat diagonal and off-diagonal disorder by generalizing the local Blackman-Esterling-Berk and the importance of finite cluster is demonstrated. It was further generalized for multiband systems. Applying the TMDCA to weakly interaction electronic systems, I show that incorporating Coulomb interactions into disordered electron system result in two competing tendencies: the suppression of the current due to correlations and the screening of the disorder leading to the homogenizing of the system. It is shown that the critical disorder strength (
WcU), required to localize all states, increases with increasing interactions (
U); implying that the metallic phase is stabilized by interactions. Using the results, a soft pseudogap at the intermediate
W close to
WcU is predicted independent of filling and dimension, and I demonstrate in three-dimensions that the mobility edge is preserved as long as the chemical potential,
μ, is at or beyond the mobility edge energy (
ωϵ). A two-particle formalism of electron localization is also developed within the TMDCA and used to calculate the direct-current conductivity, enabling direct comparison with experiments. Note significantly, the TMDCA benchmarks well with numerical exact results with a dramatic reduction in computational cost, enabling the incorporation of material's specific details as such provide an avenue for the possibility of studying electron localization in real materials.
We present a complete analytical and numerical solution of the typical medium theory (TMT) for the Anderson metal-insulator transition. This approach self-consistently calculates the typical amplitude of the electronic wave functions, thus representing the conceptually simplest order-parameter theory for the Anderson transition. We identify all possible universality classes for the critical behavior, which can be found within such a mean-field approach. This provides insights into how interaction-induced renormalizations of the disorder potential may produce qualitative modifications of the critical behavior. We also formulate a simplified description of the leading critical behavior, thus obtaining an effective Landau theory for Anderson localization.