  The block entanglement entropy and fluctuations are investigated in one
dimension in finite size correlated electron systems using the Gutzwiller wave
function as a prototype correlated electron state. Entanglement entropy shows
logarithmic divergence for all values of the correlation projection parameter
$g$, as predicted by conformal field theories for critical systems, but the
central charge requires finite size corrections. There is an infinite
correlation length corresponding to correlation between same kinds of spins,
for all values of $g$. A scaling form for the block entropy, as a function of
$g$ and the system size $N$, is proposed which predicts a metal-insulator
crossover at $N^{1/3} g\approx 0.24$. Bipartite fluctuations in the number of
particles in a block, and the spin fluctuations also obey an approximate
scaling. A relation is found between the block entropy and the bipartite spin
fluctuations. Our results show some correspondence with an experiment on Ni
nanochains.
