I want to calculate the exact eigenfunctions of Hubbard model by using Bethe ansatz. I have solved the periodic boundary condition equations (Lied-Wu equations) and have the values of the momentum "$k$"s, for $4$ sites and $2$ electrons model. Now I use the following formula for the wavefunction (attached as image):

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the amplitudes $\psi(x_1,x_2,..x_{N_e})$ are complex valued for general $k_1$ and $k_2$. But when I diagonalize the Hubbard Hamiltonian for $4$ sites and $2$ electrons, all the amplitudes of the eigenfunctions are real valued. What am I doing wrong?


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