- #141
ZombieFeynman
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atyy said:I'm hardly an expert, since I am the true non-rigourous guy here, not you. But roughly, there are two different sorts of topology in condensed matter physics.
(1) There is the topology of the integer quantum hall effect, involving Chern numbers. Topological insulators are generalizations of this idea.
http://www.physics.upenn.edu/~kane/pedagogical/WindsorLec2.pdf
http://physics.princeton.edu/~haldane/talks/dirac.pdf
http://www.bioee.ee.columbia.edu/downloads/2013/nature12186.pdf
(2) Then there is the topology of the fractional quantum hall effect, one sign of which is that the ground state degeneracy depends on the topology on which the Hamiltonian is placed. A proposed use of this sort of topology is in Kitaev's topological quantum computation.
http://stationq.cnsi.ucsb.edu/~freedman/publications/96.pdf
http://www.simonsfoundation.org/quanta/20140515-forging-a-qubit-to-rule-them-all/
From the Haldane slides above:
"The moral of this long story: suggests three distinct ingredients for success.
• Profound, correct, but perhaps opaque formal topological results (Invariants, braid group, etc)
• Profound, simple and transparent “toy models” that can be explicitly treated (The honeycomb Chern Insulator, the Kitaev Majorana chain, etc)
• Understanding the real materials needed for “realistic” (but more complex) experimentally achievable systems that can bring “toy model results” to life in the hands of experimentalist colleagues."
This is strictly not all of the topology that lies in modern Condensed Matter Physics. The QH effect and its Chern number is rather different than Topological Insulators and the Z2 topological QSH effect in 2D. This is generalized to a whole family of 3D Topological Materials. See the excellent review by Qi and Zhang.
I truly don't mean to quibble but this is an extremely exciting area of physics to me!
The points made by Haldane above are brought together in a very harmonious way in the original BHZ paper I cited above.
EDIT: Perhaps at a very rough approximation, I agree with your division.
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