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Different $K$ matrices can represent the same topological state if they are related by a similarity transformation (Wen and Zee, PRB 1992). In your case, as long as they give the same filling fraction (or the same $\sigma_{xy}$), the same charge and statistics for the excitations, then they represent the same incompressible QH state. To be more precise: $$\...


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Would suggest, if fractional charges were in Nature. Fractional charges are the result of error of James Chadwick in 1934. He identified (with no real proof) that neutron is particle similar to proton. If you trust that positive proton and neutral neutron as similar particles, you have to trust in fractional charges. But analysis shows that alternative is ...


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