Why Study Metal-Insulator Transitions?

The metal-insulator transition (MIT) is one of the oldest, yet one of the fundamentally least understood problems in condensed matter physics. The subject of MITs came to a renewed focus in the two decades, with the discovery of high temperature superconductivity, which triggered much activity in the study of “bad metals”. To understand this class of systems, one has to develop a conceptual understanding of three fundamental mechanisms of electron localization:

• Anderson localization due to impurities and defects.

• Mott or Wigner localization due to strong Coulomb repulsion.

• Meta-stability and glassy dynamics, a generic feature when disorder and interactions compete.

To incorporate these very different processes in a single theoretical picture is a daunting task, practically beyond reach of standard approaches.

Dynamical Mean-Field Theory (DMFT): a new order-parameter formulation

"It is this fascination with the local, and with the failures, not successes, of band theory, that..."

P. W. Anderson, Nobel Lecture, 1978.

Over the last two decades, a new conceptual picture has been emerging, that can very naturally describe all three fundamental mechanism, by focusing on local spectral functions (or local escape rate from a given site) as a natural order parameters for the MIT. As the Heisenberg Uncertainty Principle teaches us, it is by no means obvious if the momentum or the coordinate representation is a better staring point. In good metals, where itinerant behavior prevails, the former is clearly favored. As we approach the insulator, however, the local perspective proves more natural and more profitable - as shown by much recent success and popularity of Dynamical Mean-Field Theory DMFT methods. The idea of locality, as a more useful language to describe almost-insulating regimes, has permeated the pioneering ideas of Anderson and Mott. The recent rediscovery of this conceptual framework vindicated the deep insight and the far-reaching vision of these giants, who shaped much of our thinking about condensed phases of matter.

While most DMFT theories focus on clean and homogeneous systems, the PIs work, over the last two decades, demonstrated how the DMFT language can elegantly be extended to include not only Anderson localization effects, but also the formation of Quantum Griffiths Phases, as well as describe the long-standing Electron Glass problem.

Holographic duality: The hidden meaning of DMFT?

The local perspective provided by DMFT may be a blessing or a curse. It allows a quantitative description of electronic systems even far from the Fermi liquid regime of dilute quasiparticles, and is even more accurate in the high-temperature incoherent "bad-metal" regime. By construction, it cannot properly describe long-wavelength fluctuations that dominate most examples of standard criticality. Is this perspective thus misleading or a simply wrong picture to describe quantum critical points  (QCP)? Remarkably, recent experiments provided convincing evidence that puzzling DMFT-like  "local quantum criticality" seems to permeate many QCPs of strongly correlated electrons. How can that be?

The possible answer may be coming from the place we least expected it. Very recent ideas born from superstring theories suggest that many unusual QCPs may be understood by a formal duality transformation, mapping the electronic problem to a solution of some black-hole model living in a "Anti-deSitter" space. This "Holographic Duality" scenario seems to suggest that certain new classes of QCPs may exist, characterized by diverging time scales, but NOT diverging lengthscales - in a fashion surprisingly reminiscent of the DMFT picture. Can these ideas give birth to a new conceptual picture of QCPs, and provide a hidden vindication of the DMFT methods? Time only will tell. One thing is certain: the coming years promise much excitement and fun in seeking an answer to the old but still mysterious question about the nature of the metal-insulator transition.