Lepton Mixing in $A_5$ Family Symmetry and Generalized CP

We study lepton mixing patterns which can be derived from the $A_5$ family symmetry and generalized CP. We find five phenomenologically interesting mixing patterns for which one column of the PMNS matrix is $(\sqrt{\frac{5+\sqrt{5}}{10}},\frac{1}{\sqrt{5+\sqrt{5}}},\frac{1}{\sqrt{5+\sqrt{5}}})^{T}$ (the first column of the golden ratio mixing), $(\sqrt{\frac{5-\sqrt{5}}{10}},\frac{1}{\sqrt{5-\sqrt{5}}},\frac{1}{\sqrt{5-\sqrt{5}}})^{T}$ (the second column of the golden ratio mixing), $(1,1,1)^{T}/\sqrt{3}$ or $(\sqrt{5}+1,-2,\sqrt{5}-1)^{T}/4$. The three lepton mixing angles are determined in terms of a single real parameter $\theta$, and agreement with experimental data can be achieved for certain values of $\theta$. The Dirac CP violating phase is predicted to be trivial or maximal while Majorana phases are trivial. We construct a supersymmetric model based on $A_5$ family symmetry and generalized CP. The lepton mixing is exactly the golden ratio pattern at leading order, and the mixing patterns of case III and case IV are reproduced after higher order corrections are considered.