BGU Physics Department

Colloquium, Dec. 30, 2010

Classification of Topological Insulators and Superconductors: the "Ten-Fold Way"

Andreas Ludwig, University of California at Santa Barbara
Topological Insulators and Superconductors are unusual gapped phases of non-interacting Fermions whose 'topological nature' manifests itself through the appearance of "topologically protected" gapless surface modes. We review an exhaustive classification scheme of these systems. These surface modes are in particular also robust to disorder, and completely evade the phenomenon of Anderson localization. Our approach consists in reducing the problem of classifying topological insulators (superconductors) in d spatial dimensions to a problem of Anderson localization at the (d-1)-dimensional boundary of the system. It is found that in each spatial dimension there exist previsely five distinct classes of topological insulators (superconductors). The different topological sectors within a given such class can be labeled, depending on the case, either by an integer "winding number", or by a "binary" Z_2 quantity. One of the five classes of topological insulators is the "quantum spin Hall" (or: Z_2-topological) insulator in d=2 and d=3 dimensions, recently discussed by Fu, Kane, Mele and others, and experimentally observed in HgTe/(Hg,Ce)Te semiconductor quantum wells (d=2), and in Bismuth Antimonite (Selenite) and related alloys (d=3). The other four classes of topological insulators (superconductors) are new. One of them is the B-phase of Helium-3 in d=3 dimensions. - For each spatial dimension, the five classes of topological insulators (superconductors) are shown to correspond to a certain subset of five of the ten generic symmetry classes of Hamiltonians introduced more than 15 years ago by Altland and Zirnbauer in the context of disordered systems (generalizing the three well-known, 'unitary, orthogonal, sympectic' "Wigner-Dyson" symmetry classes). We also briefly comment on the connection with the classification in terms of K-Theory. Superconductivity rests on two cornerstones: (i) all electrons in a superconductor are described by a unique macroscopic wave function and (ii) the phase of this wave function is well-defined over the whole superconducting system, i.e. superconductivity is a quantum macroscopic state maintaining the global phase coherence. The phase and the absolute value of the wave function are canonically conjugated quantum variables, thus uncertainties in the phase Dφ and in the condensed particle number, DN, are coupled by the Heisenberg relation DφD ~ 1.  This implies the existence of the superinsulating state dual to superconducting one and possessing zero conductivity at finite temperatures. We review realizations of the superinsulating state in two-dimensional disordered superconducting films and Josephson junction arrays and discuss microscopic mechanism that ensures superinsulating behavior: the cascade relaxation.