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arXiv:1411.4558 · Physical Review B

Quantum criticality at the Anderson transition: A typical medium theory perspective

Samiyeh Mahmoudian, Shao Hui Tang, V. Dobrosavljević

Published Oct 30, 2015• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.92.144202

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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.
We present a detailed analysis of the critical behavior close to the Mott-Anderson transition. Our findings are based on a combination of numerical and analytical results obtained within the framework of typical-medium theory-the simplest extension of dynamical mean field theory capable of incorporating Anderson localization effects. By making use of previous scaling studies of Anderson impurity models close to the metal-insulator transition, we solve this problem analytically and reveal the dependence of the critical behavior on the particle-hole symmetry. Our main result is that, for sufficiently strong disorder, the Mott-Anderson transition is characterized by a precisely defined two-fluid behavior, in which only a fraction of the electrons undergo a "site selective" Mott localization; the rest become Anderson-localized quasiparticles.
The Mott-Anderson transition in the disordered charge-transfer model displays several new features in comparison to what is found in the disordered single-band Hubbard model, as recently demonstrated by large-scale computational (statistical dynamical mean-field theory) studies. Here we show that a much simpler typical medium theory approach (TMT-DMFT) to the same model is able to capture most qualitative and even quantitative aspects of the phase diagram, the emergence of an intermediate electronic Griffiths phase, and the critical behavior close to the metal-insulator transition. The conceptual and mathematical simplicity of the TMT-DMFT formulation thus makes it possible to gain useful new insight into the mechanism of the Mott-Anderson transition in these models.
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 self-consistent theory of Anderson localization that yields a simple algorithm to obtain the typical local density of states as an order parameter, thereby reproducing the essential features of a phase diagram of localization-delocalization quantum phase transition in the standard lattice models of the disordered electron problem. Due to the local character of our theory, it can easily be combined with dynamical mean-field approaches to strongly correlated electrons, thus opening an attractive avenue for a genuine non-perturbative treatment of the interplay of strong interactions and strong disorder.
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=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 Mott-Anderson transition at a fixed electron density, as the disorder strength is increased. Surprisingly, we find two critical values of disorder Wnfl{W}_{\mathrm{nfl}}Wnfl​ and Wc{W}_{c}Wc​. For W>WnflW>{W}_{\mathrm{nfl}}W>Wnfl​, the system enters a ``Griffiths'' phase, displaying metallic non-Fermi liquid behavior. At even stronger disorder, W=Wc>Wnfl{W\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}W}_{c}>{W}_{\mathrm{nfl}}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.
The DOS, the dynamic (optical) conductivity, and the phase diagram of a strongly correlated and strongly disordered paramagnetic Anderson-Hubbard model are analyzed within the generalized dynamical mean field theory (DMFT + Σ approximation). Strong correlations are taken into account by the DMFT, and disorder is taken into account via an appropriate generalization of the self-consistent theory of localization. The DMFT effective single-impurity problem is solved by a numerical renormalization group (NRG); we consider the three-dimensional system with a semielliptic DOS. The correlated metal, Mott insulator, and correlated Anderson insulator phases are identified via the evolution of the DOS and dynamic conductivity, demonstrating both the Mott-Hubbard and Anderson metal-insulator transition and allowing the construction of the complete zero-temperature phase diagram of the Anderson-Hubbard model. Rather unusual is the possibility of a disorder-induced Mott insulator-to-metal transition.