Thermodynamic limit of the off-diagonal Bethe ansatz solvable models

In two previous papers [26, 27], the exact solutions of the spin-1/2 chains with arbitrary boundary fields were constructed via the off-diagonal Bethe ansatz (ODBA). Here we introduce a method to approach the thermodynamic limit of those models. The key point is that at a sequence of degenerate points of the crossing parameter \eta=\eta_m, the off-diagonal Bethe ansatz equations (BAEs) can be reduced to the conventional ones. This allows us to extrapolate the formulae derived from the reduced BAEs to arbitrary \eta case with O(N^{-2}) corrections in the thermodynamic limit N\to\infty. As an example, the surface energy of the XXZ spin chain model with arbitrary boundary magnetic fields is derived exactly. This approach can be generalized to all the ODBA solvable models.