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 for an open chain with L sites. Authors considered the case when the boundary site has a negative chemical potential -p and the Hubbard coupling U is positive. They have shown by an analytic method that when p is larger than the transfer integral some of the ground-state solutions of the Bethe ansatz equations become complex-valued. They have found that there is a surface phase transition at some critical value p_c; when p < p_c all the charge excitations have the gap for this case, while there exists a massless charge mode when p > p_c. To find whether this surface phase transition is of the first order or of the second order we have used the entanglement entropy concept. The entropy and its derivative has a discontinuity there, so this transition is of the first order.