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I encounter a problem in finding the many-body ground state wave-function of the tight-binding model in python. The Hamiltonian of tight-binding model is follow: \begin{equation} H = -t \sum_{<ij>} c^{\dagger}_{i} c_{j} + \mu \sum_{i}c^{\dagger}_{i}c_{i} \end{equation} Suppose I want to solve the 1D Tight-binding model, I can re-write the Hamiltonian in the following matrix form: \begin{equation} H = \begin{bmatrix} \mu & -t & 0 & 0 & \dots & -t \\ -t & \mu & -t & 0 &\dots & 0 \\ 0 & -t & \mu & -t &\dots & 0 \\ 0 & 0 & -t & \mu &\dots & 0 \\ \vdots & \vdots & \vdots &\ddots & \ddots & \ddots \\ -t & 0 & 0& \dots & -t & \mu \end{bmatrix} \end{equation}

Therefore, I can directly solve the ground-state eigenvalue and the eigenvector. However, the ground state that I solve is the ground state correspond to single particle case. In tight-binding model, we know that the many-body ground state should be the Fermi sea: \begin{equation} | \Psi_{GS} \rangle = \prod_{k \leq k_{F}} c^{\dagger}_{k} | 0 \rangle \end{equation} Therefore, my problem is that how can I calculate the many-body ground state wave function once I solve all the eigenvectors and eigenvalues of the matrix $H$? Besides, how can I relate real-space eigenvectors to momentum space eigenvector like the $|\Psi_{GS} \rangle$?

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  • $\begingroup$ Do you have periodic boundary conditions? From your matrix it looks like the answer is maybe no? $\endgroup$
    – jacob1729
    Apr 22 at 15:15
  • $\begingroup$ I forgot to impose the period boundary condition for the Hamiltonian. Thanks for reminding. $\endgroup$
    – Ricky Pang
    Apr 22 at 16:13

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