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$$ \mathcal{L}_f = \psi^\dagger_R i(\partial_t + v_F \partial_x) \psi_R + \psi^\dagger_L i(\partial_t - v_F \partial_x) \psi_L $$  I agree with your equation 1.7.   $$\psi^\dagger $$\psi  = e^{-i e^{i  k_F x} \psi^\dagger_R \psi_R  + e^{i e^{-i  k_F x} \psi^\dagger_L \psi_L  $$ Just like in the perfect metal paper:  \begin{align}   \psi^\dagger_R \psi_R  & \propto & e^{-i e^{i  \phi_I} \\ \psi^\dagger_L \psi_L  & \propto & e^{i e^{-i  \phi_{N+I}} \end{align}  This implies the variations are