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In this paper, we study the surface energy of the one-dimensionalsupersymmetric $t-J$ model with unparallel boundary magnetic fields, which is atypical $U(1)$-symmetry broken quantum integrable strongly correlated electronsystem. It is shown that at the ground state, the contribution of inhomogeneousterm in the Bethe ansatz solution of eigenvalues of transfer matrix satisfiesthe finite size scaling law $L^{\beta}$ where $\beta<0$. Based on it, thephysical quantities of the system in the thermodynamic limit are calculated. Weobtain the patterns of Bethe roots and the analytical expressions of density ofstates, ground state energy and surface energy. We also find that there existthe stable boundary bound states if the boundary fields satisfy someconstraints.

Surface energy of the one-dimensional supersymmetric $t-J$ model with general integrable boundary terms in the antiferromagnetic sector