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The Chern-Simons term, through which the cosmic Axion-like field couples tothe electromagnetic field, has the effect to rotate CMB polarization directionsand to break the CPT symmetry. This rotation will change the CMB power spectra,no matter isotropic or anisotropic the rotation angle is. In this paper werevisit this issue by further considering the correlations between the(anisotropic) rotation angle $\alpha$ and the CMB temperature and (unrotated)$E$ polarization fields. These correlations could be generated in theAxion-like models with nonzero potential under the adiabatic initial condition.We first investigate how these correlations contribute further modifications tothe CMB power spectra, then calculate the CMB bispectra for the temperature androtated polarization fields. These bispectra would vanish if the $T\alpha$ and$E\alpha$ correlations are absent. So, they are useful in searching for CPTviolation and the $T\alpha$ and $E\alpha$ correlations arisen in the Axion-likemodels.

The effects on CMB power spectra and bispectra from the polarization rotation and its correlations with temperature and E-polarization