Application of matrix product states to the Hubbard model in one spatial dimension

We investigate the application of matrix product states to the Hubbard model in one spatial dimension with both of open and periodic boundary conditions. We develop the variatinal method that the optimization of the variational parameters is carried out locally and sequentially in the framework of matrix product operators (MPO) by including the sign, due to the anti-commutation relation of fermion operators, in the matrix element of MPO. The numerical accuracy of the ground state energy is examined.