# Chern number in condensed matter physics

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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I think the definition is basically the same fundamentally. Do you mean how it's calculated? –  wsc Dec 1 '12 at 6:23
@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). –  jjcale Dec 1 '12 at 17:24
@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 –  twistor59 Dec 2 '12 at 18:59
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. –  twistor59 Dec 2 '12 at 18:59
@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. –  jjcale Dec 3 '12 at 22:25