In mathematics, the Chern number is defined in terms of the Chern class of a manifold. What is the exact definition of Chern number in condensed matter physics, i.e. quantum hall system?
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1$\begingroup$ I think the definition is basically the same fundamentally. Do you mean how it's calculated? $\endgroup$– wscDec 1, 2012 at 6:23
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1$\begingroup$ @twistor59 : I'm not sure whether the Chern number explanation is right. Bellissard was able to identify the quantum hall conductance with an index of a fredholm operator. For references see Jingbo Xia, Geometric Invariants of the Quantum Hall Effect, Commun. Math. Phys. 119, 29-50 (1988). $\endgroup$– jjcaleDec 1, 2012 at 17:24
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1$\begingroup$ @jjcale I think that's a more sophisticated model of the quantum Hall effect than is usually treated (as the abstract says, it was a new interpretation - at that time). The "conventional" treatment (Thoulness), which the Xia paper references, goes something like: electron gas in 2 dimensions with periodic boundary conditions, corresponds to 2 dim parameter space T with topology S1xS1. Then the set of all phases that the wavefunctions of the energy eigenstates can have defines a U(1) bundle over T. The integral which defines the Chern class (integral of the curvature 2-form of a $\endgroup$– twistor59Dec 2, 2012 at 18:59
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1$\begingroup$ connection) corresponds to the formula for the Hall conductivity. The connection is defined by the Hilbert space inner product. I only have cursory knowledge though, not enough to provide an answer. $\endgroup$– twistor59Dec 2, 2012 at 18:59
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1$\begingroup$ @twistor59 : The Quantum Hall Effect can't be explained without taking Anderson localisation into account. The Hall conductance is only quantized if the fermi energy lies in a region of localized states. $\endgroup$– jjcaleDec 3, 2012 at 22:25
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In quantum Hall effect(QAHE), Klitzing found the Hall conductivity to be integer multiples of a fundamental constant. This effect is independent of size and impurities of the system with which we deals with. Based on it, a famous scientist R. Laughlin proposed a theory describing the integer states in terms of a topological invariant. This topological invariant is known as chern number.
For details of the cern number,there is a wikipedia link.
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$\begingroup$ Thanks for your answers! Hall conductance can be expressed as σH=νe2h, where ν is filling factor. Is filling factor ν related to the Chern number C? $\endgroup$– TimothyDec 5, 2012 at 5:12