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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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.