# Numerical Approach to find the Many-Body Ground State Wavefunction of Tight-Binding model

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: $$$$H = -t \sum_{} c^{\dagger}_{i} c_{j} + \mu \sum_{i}c^{\dagger}_{i}c_{i}$$$$ Suppose I want to solve the 1D Tight-binding model, I can re-write the Hamiltonian in the following matrix form: $$$$H = \begin{bmatrix} \mu & -t & 0 & 0 & \dots & -t \\ -t & \mu & -t & 0 &\dots & 0 \\ 0 & -t & \mu & -t &\dots & 0 \\ 0 & 0 & -t & \mu &\dots & 0 \\ \vdots & \vdots & \vdots &\ddots & \ddots & \ddots \\ -t & 0 & 0& \dots & -t & \mu \end{bmatrix}$$$$

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: $$$$| \Psi_{GS} \rangle = \prod_{k \leq k_{F}} c^{\dagger}_{k} | 0 \rangle$$$$ Therefore, my problem is that how can I calculate the many-body ground state wave function once I solve all the eigenvectors and eigenvalues of the matrix $$H$$? Besides, how can I relate real-space eigenvectors to momentum space eigenvector like the $$|\Psi_{GS} \rangle$$?

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