  Entanglement related properties work as nice fingerprint of the quantum
many-body wave function. However, those of fermionic models are hard to
evaluate in standard numerical methods because they suffer from finite size
effects. We show that a so-called density embedding theory (DET) can evaluate
them without size scaling analysis in comparably high quality with those
obtained by the large-size density matrix renormalization group analysis. This
method projects the large scale original many-body Hamiltonian to the small
number of basis sets defined on a local cluster, and optimizes the choice of
these bases by tuning the local density matrix. The DET entanglement spectrum
of one-dimensional interacting fermions perfectly reproduces the exact ones and
works as a marker of the phase transition point. It is further shown that the
phase transitions in two-dimension could be determined by the entanglement
entropy and the fidelity that reflects the change of the structure of the wave
function.
