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Abstract |
Topological states of matter are characterised by a gap in the bulk of the system
referring to an
insulator or a superconductor and topological edge modes as well which find
various applications in
transport and spintronics. The bulk-edge correspondence is associated to a
topological number. The table
of topological states include the quantum Hall effect and the quantum anomalous
Hall effect, topological
insulators and topological superconductors in various dimensions and lattice
geometries. Here, we discuss
classes of states which can be understood from mapping onto a spin-1/2 particle in
the reciprocal space
of wave-vectors. We develop a geometrical approach on the associated Poincare-
Bloch sphere, developing
smooth fields, which shows that the topology can be encoded from the poles only.
We show applications for
the light-matter coupling when coupling to circular polarisations and develop a
relation with quantum
transport and the quantum Hall conductivity. The formalism allows us to include
interaction effects. We
show our recent developments on a stochastic approach to englobe these interaction
effects and discuss
applications for the Mott transition of the Haldane and Kane-Mele models. Then, we
develop a model of
coupled spheres and show the possibility of fractional topological numbers as a
result of interactions
between spheres and entanglement allowing a superposition of two geometries, one
encircling a topological
charge and one revealing a Bell or EPR pair. Then, we show applications of the
fractional topological
numbers C=1/2 in bilayer honeycomb models describing topological semi-metals
characterised by a quantised
$\pi$ Berry phase at one Dirac point.
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