Exact surface energy of the Hubbard model with nonparallel boundary magnetic fields

In this study, we explore the precise physical quantities in the thermodynamic limit of the one-dimensional Hubbard model with nonparallel boundary magnetic fields based on the off-diagonal Bethe ansatz solution. A particular emphasis is placed on the half-filling condition to investigate the distinct patterns of Bethe roots in the reduced Bethe ansatz equations for different boundary parameters. The ground state of the system can be divided into five regions according to the distribution of Bethe roots. By analyzing these patterns, we calculate the densities of states, ground-state energy density, and surface energy. The results reveal the existence of stable- boundary bound states, which are dependent on specific constraints regarding the boundary magnetic fields.