A Solution of the Hubbard Model

We report a ground-state solution for the two-dimensional fermionic Hubbard model, which is obtained via a numerical variational method. The two ingredients in this approach are tensor network states and the time-evolving block decimation. We easily handle the horizontal hopping in the Hamiltonian, and we proceed further to observe the fermion-exchange effect caused by the vertical hopping. By requiring no divergence and no convergence to zero for the ground state, we successively determine the ground-state energy per site as a function of the chemical potential and the lattice length. In addition, we observe saturation in the behavior of the ground-state energy as the lattice length increases.