Quantum critical phenomena, entanglement entropy and Hubbard model in 1d with the boundary site with a negative chemical potential -p and the Hubbard coupling U positive

Recently the ground state and some excited states of the half-filled case of the 1d Hubbard model were discussed exactly for an open chain with L sites. The case when the boundary site has the chemical potential -p and the Hubbard coupling U is positive was considered. We model CeAl2 nanoparticles, in which a valence of 4f electron number changes on surface Ce atoms, by this Hubbard model. A surface phase transition exists at some critical value pc3 of chemical potential (its absolute value) p in the model; when ppc3. The aim of this Letter is to find whether this surface phase transition is of the first order or of the second order. We have found that the entanglement entropy and its derivative has a discontinuity at pc3 in general and thus this transition is of the first order (with exception of two points for the probability w2 of occurrence of two electrons with opposites spins on the same site). There is a divergence in the difference of entanglement entropy for points w2=0 and w2=1/2 . The first point w2=0 corresponds to ferro- (antiferro-) magnetic state at half-filled case. The second point w2=1/2 does not correspond to any state for halffilled case. In the first case there is present the surface phase transition of the second order type