Recently, considerable attention has been drawn by the theory of impurities in both the Kondo problem and the topological state, and many new developments have been reported.^[1–6] We know that the impurities play an important role in the strongly correlated electron systems, and even a small amount of defects may change the properties of the electron systems.^[7–10] Then it is very important to construct the integrable systems including the impurities. Many impurity models with free end boundary conditions have been considered previously. Frahm and Zvyagin studied the effects of a boundary magnetic field and an impurity spin coupled to an open spin chain by means of Bethe ansatz method.^[11] Wang proposed an open spin-1/2 Heisenberg chain coupled with two impurity spins sited at the ends, and exactly solved this model for arbitrary impurity spin and arbitrary exchange constants between the bulk and the impurities using Bethe ansatz method.^[12] The integrable impurity problem of the Heisenberg chain with periodic boundary condition was first considered by Andrei and Johannesson.^[13] They studied the integrable case of a spin S > 1/2 embedded in a spin-1/2 Heisenberg chain. Subsequently, the problem was generalized to arbitrary spins by Lee and Schlottmann.^[14] In contrast to the periodic boundary conditions in the integrable impurity models, the anti-periodic boundary conditions are especially interesting, which are tightly related to the recent study on the topological states of matter.^[15,16]

Since the Yang and Baxter’s pioneering works,^[17–19] the quantum Yang–Baxter equation, which defines the underlying algebraic structure, has become a cornerstone for constructing and solving the integrable models. The T–Q relation method and the algebraic Bethe ansatz method developed from the Yang–Baxter equation have become two very popular methods for dealing with the exact solutions of the known integrable models.^[20] Although the algebraic Bethe ansatz method is a powerful tool exhibiting a rich mathematical structure, its implementation for models without U(1) symmetry still has an obstacle. The main obstacle applying the algebraic Bethe ansatz to these models lies in the absence of a trivial reference state. Significant progress has been made recently for the integrable models without U(1) symmetry. For example, a promising method for approaching such kind of problems is Sklyanin’s separation variables method which has been applied to some integrable models.^[21,22] Another important advance was made by Cao et al.,^[23–25] who proposed the off-diagonal Bethe ansatz method to approach the exact solutions of generic integrable models either with or without U(1) symmetry. The central step of this method lies in the construction of the T–Q ansatz with an extra off-diagonal term. For the integrable models with impurities, on one hand the twisted boundary breaks the U(1) symmetry of the system, on the other hand the introduction of the impurity adds the freedom degree of the system, which make it difficult to resolve the impurity model without U(1) symmetry. A systematic method is still absent to derive the Bethe ansatz equations for integrable impurity models without U(1) symmetry.

It is known that the behavior of a single magnetic impurity in the one-dimensional spin-1/2 Heisenberg model as well as the behavior of a single Kondo impurity in a three-dimensional free electron host is described by similar Bethe ansatz theories.^[26] The spin-1/2 magnetic impurity manifests the total Kondo screening with Fermi liquids-like low temperature behavior of the magnetic susceptibility and specific heat.^[27] The effects of impurity scattering on the surface of topological insulators have been investigated extensively.^[28–30] A key issue is to determine the conditions for the stability of the current-carrying states at the edge of the sample,^[31,32] as this is the feature that most directly impacts prospect for future applications in electronics and spintronics.^[33] Therefore, it is worthwhile to investigate the effects of impurity when the impurity locates at the twisted boundary. In this paper, we consider the Heisenberg spin ring with a Möbius-like topological boundary condition coupled to a spin-S impurity. The Hamiltonian of the model is H = H₀ + H_imp, which was first solved by Andrei and Johannesson,^[13]

This paper is organized as follows. Section 2 serves as an introduction of our notations and some basic ingredients. We briefly describe the inhomogeneous topological spin ring with a spin-S simpurity. In Section 3, base on the properties of the transfer matrix and the quantum determinant, we drive the inhomogeneous T–Q relation for the eigenvalue of the transfer matrix and the corresponding Bethe ansatz equations. In Section 4, the impurity effects on the ground state and the thermodynamic limit for different impurity parameter c are discussed. Concluding remarks will be given in Section 5.

Let V₀, V₁, …, V_N be 2D vector spaces over complex numbers C. The V₀ indicates the auxiliary space and the tensor product V₁ ⊗ ⋯ ⊗ V_N indicates the quantum space, where V_j corresponds to the j-th site of the spin chain with j = 1,2,…,N. On the tensor product V₀⊗ V_j, we define the R-matrix

We suppose that |Ψ〉 is an eigenstate of t(u) and independent of u, and has t(u)|Ψ〉 = Λ(u)|Ψ〉. Base on the commutation relations Eqs. (18)–(24), the following useful formulas can be derived:^[35]

The interaction between the impurity and the spin ring depends on the parameter c, and c could be real or imaginary values. Therefore, we discuss the ground state properties for different c values with the standard method proposed by Yang and Yang.^[17] Note that the distributions of the roots { μ_j } and { ν_j } are not independent, which essentially are decided by the selection rules of elementary excitation at the anti-periodic boundary condition of the system. In the homogeneous limits θ_j → 0, with the help of Eqs. (34) and (35), we obtain the following Betha ansatz equations:

First, we consider the case of imaginary c = iζ. The exchange interaction between the impurity and the spin ring is always positive corresponding to antiferromagnetic coupling. All the modes of { μ_j} take imaginary values in the ground state to minimize the energy. We set μ_j = λ_j + iγ_j, and introduce the following function:

For the case of real value c, the coupling between the impurity and the spin ring can be ferromagnetic or antiferromagnetic for |c| < S + 1/2 and S+1/2 < |c|, respectively. Obviously, there exists a critical point |c| = S+1/2, which corresponds to a quantum phase transition to a non-Fermi liquid quantum critical region. When S + 1/2 < |c|, the real modes of μ at Re {μ_j} = c – (S + 1/2) can exist in the ground state. With a similar discuss, the self magnetization of the ground state is M_g = S, which means that the impurity can not be screened. When S + 1/2 > |c|, the exchange interaction between the impurity and the bulk is in the antiferromagnetic region, and no bound state can exist in the ground state. In this case, the residual magnetization of the ground state M_g = S–1/2, which means that the impurity moment is partially screened (for S > 1/2).

In order to investigate the impurity effects in the thermodynamic limit, we consider a special case of c = iζ and μ_j = iγ_j to investigate the thermodynamic of the present model. In the thermodynamic limit, in the leading order of O(N^–1), equation (40) takes the form

Fig. 1.

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	Fig. 1. The impurity specific heat Cs as a function of temperature T for the system with the impurity spin-1/2 (a) and the impurity spin-1 (b), where the impurity parameter ζ = 0.5, 1, 1.5.

From the above discussion, we conclude that the impurity can not be screened for the antiferromagnetic cases. The impurity moments only can be partially screened for the ferromagnetic case. The coupling interaction of the impurity is different to that in the one-dimensional quantum system with U(1) system.^[12] The topological boundary condition will produce a finite boundary energy described by a phase factor, which constraints the interaction between the impurity and the spin kink. The screen behaviors of the impurity strongly depend on the competition between the quantum fluctuation and the constraint of the topological boundary condition. When the impurity parameter c is a pure imaginary value, the contribution of the impurity on the specific heat mainly depends on the impurity spin, and the effects of the anti-periodic boundary are depressed in the thermodynamic limit. From Fig. 1, we note that the peak of specific heat moves down gradually with the increase of ζ, which means that the inhomogeneity of the impurity comparing with the hosts could depress the contribution of the impurity on the specific heat. These results are similar to the situation for the Kondo problem in a Luttinger liquid predicted by Furysaki and Nagaosa.^[39] The Kondo coupling in this model indicates the topological nature of the system.

The spin-1/2 Heisenberg chain described by the Hamiltonian (1) with anti-periodic boundary condition is studied via the off-diagonal Bethe ansatz method. The crucial ideal of the off-diagonal Bethe ansatz method is to construct the function relations between the eigenvalues of the transfer matrix and the elements of the monodromy matrix. The eigenvalues of the transfer matrix are given in terms of the T–Q relations, and the Bethe ansatz equations are given by the regularity of the eigenvalue. Based on the Bethe ansatz equations, we investigate the distribution of roots and the residual magnetization in the ground states by the usual method.^[35] It is found that the screen behavior of the impurity strongly depends on the topological boundary condition, the anti-periodic boundary condition contributes a finite boundary energy described by a phase factor. The selection rules of Bethe roots decide the properties of the ground states and the elementary excitation of the topological spin ring, which indicates the topological nature of the system. In the thermodynamic limit, the impurity contribution to the specific heat is also investigated, and the results indicate that the thermodynamic properties can be adjusted by the impurity parameters. When the impurity does not interact with the boundary fields, the function relation between the eigenvalue and the quantum determinant still holds via some properties of the R-matrix and K-matrices. In this case, this method can also deal with the impurity integrable models with arbitrary boundary conditions, which will be investigated in our next work.