  We discuss the ground state and some excited states of the half-filled
Hubbard model defined on an open chain with L sites, where only one of the
boundary sites has a different value of chemical potential. We consider the
case when the boundary site has a negative chemical potential -p and the
Hubbard coupling U is positive. By an analytic method we show that when p is
larger than the transfer integral some of the ground-state solutions of the
Bethe ansatz equations become complex-valued. It follows 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 the half-filled band, while there exists a
massless charge mode when p>p_c.
