# 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

## 1 Answer

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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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? – Timothy Dec 5 '12 at 5:12