  We study the effects due to limited entanglement in the one-dimensional
Hubbard model by representing the ground states in the form of the matrix
product states. Finite-entanglement scaling behavior over a wide range is
observed at half-filling. The critical exponents characterizing the length
scale in terms of the size of matrices used are obtained, confirming the
theoretical prediction that the values of the exponents are solely determined
by the central charge. The entanglement spectrum shows that a global double
degeneracy occurs in the ground states with a charge gap. We also find that the
Mott transition, tuned by changing the chemical potential, always occurs
through a first-order transition and the metallic phase has a few conducting
states, including the states with the mean-field nature close to the critical
point, as expected in variational matrix product states with a finite amount of
entanglement.
