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Based on its off-diagonal Bethe ansatz solution, we study the thermodynamiclimit of the spin-$\frac{1}{2}$ XYZ spin chain with the antiperiodic boundarycondition. The key point of our method is that there exist some degeneratepoints of the crossing parameter $\eta_{m,l}$, at which the associatedinhomogeneous $T-Q$ relation becomes a homogeneous one. This makesextrapolating the formulae deriving from the homogeneous one to an arbitrary$\eta$ with $O(N^{-2})$ corrections for a large $N$ possible. The ground stateenergy and elementary excitations of the system are obtained. By taking thetrigonometric limit, we also give the results of antiperiodic XXZ spin chainwithin the gapless region in the thermodynamic limit, which does not have anydegenerate points.

Thermodynamic limit of the spin-$\frac{1}{2}$ XYZ spin chain with the antiperiodic boundary condition