Quantization of Hall Conductance and Stabilization of Flux State in Noncommutative Space

We studied the quantization of Hall conductance and generalized Peierls instability in two-dimensional noncommutative space under a constant magnetic field. We show that the Hall conductance is quantized and the quantization is stable against a change of the noncommutative parameter θ. For a rational magnetic flux φ = p/q and a rational θ = r/s, the flux state is stabilized at a filling ν = p(4qs − pr)/4q 2 s. The condition is different from that of commutative space. If we regard θ as a dynamical parameter, we propose a mechanism for how the model determine the value of θ.