We introduce a new expression for the Z2 topological invariant of band insulators using nonAbelian Berry's connection. Our expression can identify the topological nature of a general band insulator without any of the gauge fixing problems that plague the concrete implementation of previous invariants. The new expression can be derived from the "partner switching" of the Wannier function center during time reversal pumping and is thus equivalent to the Z2 topological invariant proposed by Kane and Mele. Using the new expression, we have recalculated the Z2 topological index for several topological insulator material systems and obtained consistent results with the previous studies.

Using self-consistent mean-field and functional renormalization-group approaches, we show that s-wave pairing symmetry is robust in the heavily electron-doped iron chalcogenides AFe 2Àx Se 2 , where A ¼ K;Cs. Recent neutron scattering experiments suggest that the effective nearest-neighbor spin exchange may be ferromagnetic in chalcogenides. This is different from the iron pnictides, where the nearest-neighbor magnetic exchange coupling is believed to be antiferromagnetic and leads to strong competition between s-wave and d-wave pairing in the electron-overdoped region. Our finding of a robust s-wave pairing in ðK;CsÞFe 2Àx Se 2 differs from the d-wave pairing result obtained by other theories where nonlocal bare interaction terms and the next-to-nearest-neighbor J 2 term are underestimated. Detecting the pairing symmetry in ðK;CsÞFe 2Àx Se 2 may hence provide important insights regarding the mechanism of superconducting pairing in iron-based superconductors.

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We give a pedagogical introduction to topologically ordered states of matter, with the aim of familiarizing the reader with their axiomatic topological quantum field theory description. We introduce basic noninteracting topological phases of matter protected by symmetries, including the Su-Schrieffer-Heeger model and the one-dimensional p-wave superconductor. The defining properties of topologically ordered states are illustrated explicitly using the toric code and -on a more abstract level -Kitaev's 16-fold classification of two-dimensional topological superconductors. Subsequently, we present a short review of category theory as an axiomatic description of topological order in two-dimensions. Equipped with this structure, we revisit Kitaev's 16-fold way.These lectures were in parts held at: