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Recently the ground state and some excited states of the half-filled case ofthe 1d Hubbard model were discussed for an open chain with L sites. Authorsconsidered the case when the boundary site has a negative chemical potential -pand the Hubbard coupling U is positive. They have shown by an analytic methodthat when p is larger than the transfer integral some of the ground-statesolutions of the Bethe ansatz equations become complex-valued. They have foundthat 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 amassless charge mode when p > p_c. To find whether this surface phasetransition is of the first order or of the second order we have used theentanglement entropy concept. The entropy and its derivative has adiscontinuity there, so this transition is of the first order.

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