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Exact solution of a topological spin ring with an impurity

The spin-1/2 Heisenberg chain coupled to a spin-S impurity moment with anti-periodic boundary condition is studied via the off-diagonal Bethe ansatz method. The twisted boundary breaks the U(1) symmetry of the system, which leads to that the spin ring with impurity can not be solved by the conventional Bethe ansatz methods. Combing with the properties of the R-matrix, the transfer matrix and the quantum determinant, we derive the T-Q relation and the corresponding Bethe ansatz equations. The residual magnetizations of the ground states and the impurity specific heat are investigated. It is found that the residual magnetizations in this model strongly depend on the constraint of the topological boundary condition, the inhomogeneity of impurity comparing with the hosts could depress the impurity specific heat in the thermodynamic limit. This method can be expand to other integrable impurity models without U(1) symmetry.