BGU Physics Department
# Colloquium, Dec. 30, 2010

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

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 D*φ *
and in the condensed particle
number, D*N*,
are coupled by the Heisenberg relation D*φ*D*N *
~ 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.