Topology, Crystallized (Experiments): P. Dziawa et al., arXiv:1206.1705; S.-Y. Xu et al., arXiv:1206.2088

A key difference between quantum Hall phases induced by a magnetic field and topological insulator phases induced by spin-orbit coupling is that the latter depend crucially on a symmetry, time reversal. The action of time-reversal symmetry on electrons leads to a new kind of topological invariant in twodimensional systems [1] that takes only two possible values: if this “Z2 invariant” is even, the system is an ordinary insulator and generically has no edge state, while if it is odd, the system has a protected conducting edge state. Breaking time-reversal symmetry allows these two phases to be connected adiabatically (i.e., without closing the energy gap). The interplay of symmetry and topology is more complicated in three dimensions, particularly when crystalline point or space group symmetries are considered, and a large number of phases have been conjectured theoretically. Recent experimental photemission work by two groups confirms the existence in Pb1−xSnxSe [2] and Pb1−xSnxTe [3] of one such phase, the “topological crystalline insulator”; this term was introduced by Liang Fu in a 2011 PRL [4]. In order to explain the importance of this discovery, it seems appropriate to review some prior developments in the field of three-dimensional topological invariants. With time-reversal symmetry alone, there are 4 invariants of Z2 type [5, 6, 7], of which only one is fully stable to time-reversal-symmetric disorder. The “strong topological insulator” phase described by this invariant has a surface state, and for a perfectly crystalline surface there are an odd number of “Dirac points” enclosed by the Fermi surface, as now observed in a variety of materials [8]. There is a remarkable generalization of these results [9, 10] to arbitrary dimensions and to the ten symmetry classes that result from considering only time-reversal and chiral symmetry operations (the latter appear in the Bogoliubov-de Gennes description of superconductors). This generalization provides a “no-go theorem” in a sense: without including extra symmetries such as crystalline symmetries, which are not expected to be stable to disorder, the only topological phases in any dimension are in five of the ten symmetry classes, of which three are of integer type and two of Z2 type 1 The question of what topological invariants exist when additional point or space group symmetries, possibly magnetic, are imposed, and what their physi-