A general method is proposed for constructing the Bethe ansatz equations of integrable models without $U(1)$ symmetry. As an example, the exact spectrum of the $XXZ$ spin ring with a Möbius-like topological boundary condition is derived by constructing a modified $T-Q$ relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological $XX$ spin ring are discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.

• Received 6 June 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.137201