We have investigated the effect of unidirectional periodic potential modulation on the fractional quantum Hall (FQH) states located within the filling factor range $1< u<2$. We find that odd-numerator FQH states are strongly suppressed, while the even-numerator ones are less affected or even enhanced, by the introduction of the modulation. In the picture of the composite fermions (CFs), the behaviors are equivalent to the suppression of the spin splitting of the CFs by the modulation. We discuss the origin of the suppression and the possible modulation-induced phase transition from the FQH state to the stripe state.
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