Off-diagonal Bethe Ansatz Method Research Articles

Thermodynamic limit of the XXZ central spin model with an arbitrary central magnetic field

The U(1) symmetry of the XXZ central spin model with an arbitrary central magnetic field B is broken, since its total spin in the z-direction is not conserved. We obtain the exact solutions of the system by using the off-diagonal Bethe ansatz method. The thermodynamic limit is investigated based on the solutions. We find that the contribution of the inhomogeneous term in the associated T–Q relation to the ground state energy satisfies an N −1 scaling law, where N is the total number of spins. This result makes it possible to investigate the properties of the system in the thermodynamic limit. By assuming the structural form of the Bethe roots in the thermodynamic limit, we obtain the contribution of the direction of B to the ground state energy. It is shown that the contribution of the direction of the central magnetic field is a finite value in the thermodynamic limit. This is the phenomenon caused by the U(1) symmetry breaking of the system.

Exact solution of the quantum integrable model associated with the twisted {mathrm{D}}_3^{(2)} algebra

We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted {D}_3^{(2)} algebra (or the {D}_3^{(2)} model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the {D}_{n+1}^{(2)} model and other high rank integrable models.

Exact solutions of the Cn quantum spin chain

We study the exact solutions of quantum integrable model associated with the Cn Lie algebra, with either a periodic or an open one with off-diagonal boundary reflections, by generalizing the nested off-diagonal Bethe ansatz method. Taking the C3 as an example we demonstrate how the generalized method works. We give the fusion structures of the model and provide a way to close fusion processes. Based on the resulted operator product identities among fused transfer matrices and some necessary additional constraints such as asymptotic behaviors and relations at some special points, we obtain the eigenvalues of transfer matrices and parameterize them as homogeneous T−Q relations in the periodic case or inhomogeneous ones in the open case. We also give the exact solutions of the Cn model with an off-diagonal open boundary condition. The method and results in this paper can be generalized to other high rank integrable models associated with other Lie algebras.

Exact solution of the Gaudin model with Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11847245 and 11874393).

We study the exact solution of the Gaudin model with Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions. The energy and Bethe ansatz equations of the Gaudin model can be obtained via the off-diagonal Bethe ansatz method. Based on the off-diagonal Bethe ansatz solutions, we construct the Bethe states of the inhomogeneous XXX Heisenberg spin chain with the generic open boundaries. By taking a quasi-classical limit, we give explicit closed-form expression of the Bethe states of the Gaudin model. From the numerical simulations for the small-size system, it is shown that some Bethe roots go to infinity when the Gaudin model recovers the U(1) symmetry. Furthermore, it is found that the contribution of those Bethe roots to the Bethe states is a nonzero constant. This fact enables us to recover the Bethe states of the Gaudin model with the U(1) symmetry. These results provide a basis for the further study of the thermodynamic limit, correlation functions, and quantum dynamics of the Gaudin model.

Graded off-diagonal Bethe ansatz solution of the SU(2|2) spin chain model with generic integrable boundaries

The graded off-diagonal Bethe ansatz method is proposed to study supersymmetric quantum integrable models (i.e., quantum integrable models associated with superalgebras). As an example, the exact solutions of the SU(2|2) vertex model with both periodic and generic open boundary conditions are constructed. By generalizing the fusion techniques to the supersymmetric case, a closed set of operator product identities about the transfer matrices are derived, which allows us to give the eigenvalues in terms of homogeneous or inhomogeneous T−Q relations. The method and results provided in this paper can be generalized to other high rank supersymmetric quantum integrable models.

Exact solution of a topological spin ring with an impurity**Project supported by the National Natural Science Foundation of China (Grant No. 11664001).

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. By combining 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 the 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.

Exact solution of the [formula omitted] model with non-diagonal boundary reflections

The exact solution of the A2(2) model (or the Izergin–Korepin model) with non-diagonal boundary reflections of the type I is studied by using the off-diagonal Bethe ansatz method. The complete operator product identities of the transfer matrices are obtained with the help of fusion. Based on them and the asymptotic behaviors as well as the values at special points, the inhomogeneous T−Q relations of the system are constructed. The method and the results in this paper can be generalized to the high rank An(2) case directly.

Off-diagonal Bethe Ansatz on the so(5) spin chain

The so(5) (i.e., B2) quantum integrable spin chains with both periodic and non-diagonal boundaries are studied via the off-diagonal Bethe Ansatz method. By using the fusion technique, sufficient operator product identities (comparing to those in [1]) to determine the spectrum of the transfer matrices are derived. For the periodic case, we recover the results obtained in [1], while for the non-diagonal boundary case, a new inhomogeneous T−Q relation is constructed. The present method can be directly generalized to deal with the so(2n+1) (i.e., Bn) quantum integrable spin chains with general boundaries.

Exact solution of the sp(4) integrable spin chain with generic boundaries

The off-diagonal Bethe ansatz method is generalized to the integrable model associated with the sp(4) (or C2) Lie algebra. By using the fusion technique, we obtain the complete operator product identities among the fused transfer matrices. These relations, together with some asymptotic behaviors and values of the transfer matrices at certain points, enable us to determine the eigenvalues of the transfer matrices completely. For the periodic boundary condition case, we recover the same T − Q relations obtained via conventional Bethe ansatz methods previously, while for the off-diagonal boundary condition case, the eigenvalues are given in terms of inhomogeneous T − Q relations, which could not be obtained by the conventional Bethe ansatz methods. The method developed in this paper can be directly generalized to generic sp(2n) (i.e., Cn) integrable model.

Exact solution of the B1 model

The B1 vertex model is studied by the off-diagonal Bethe ansatz method. New closed operator product identities of the fused transfer matrices are obtained by using the fusion technique. Based on them and the asymptotic behaviors as well as the values of the fused transfer matrices at certain points, the exact solutions of the B1 model with periodic and with integrable off-diagonal open boundary conditions are obtained. We find that the degree of the polynomial of the inhomogeneous T − Q relation is lower. The B1 model is equivalent to the integrable spin-1 Heisenberg chain, thus the method and the results in this paper can be generalized to the high spin Heisenberg model and the integrable models associated with Bn Lie algebra.

An anisotropic four-component spin chain with integrable boundary terms

The anisotropic four-component spin chain with integrable off-diagonal boundary terms is studied by the nested off-diagonal Bethe ansatz method. Based on the intrinsic properties of the associated R-matrix, the recursive operator product identities of the fused transfer matrices are obtained by using the fusion. The asymptotic behaviors and the values of fused transfer matrices at certain special points are derived explicitly. The nested inhomogeneous T − Q relations of the system are constructed, and the self-consistency of these relations is demonstrated. The method and the results can be applied to study the eigenvalue problems of multicomponent integrable models with open boundaries.

T-Q relations for the integrable two-species asymmetric simple exclusion process with open boundaries

We study the integrable two-species asymmetric simple exclusion process (ASEP) for two inequivalent types of open, non particle conserving boundary conditions. Employing the nested off-diagonal Bethe ansatz method, we construct for each case the corresponding homogeneous - relations and obtain the Bethe ansatz equations. Numerical checks for small system sizes show completeness for some Bethe ansatz equations, and partial completeness for others.

On the solutions of the Zn-Belavin model with arbitrary number of sites

The periodic Zn-Belavin model on a lattice with an arbitrary number of sites N is studied via the off-diagonal Bethe Ansatz method (ODBA). The eigenvalues of the corresponding transfer matrix are given in terms of an unified inhomogeneous T−Q relation. In the special case of N=nl with l being also a positive integer, the resulting T−Q relation recovers the homogeneous one previously obtained via algebraic Bethe Ansatz.

Bethe ansatz solutions of the τ 2-model with arbitrary boundary fields

The quantum $\tau_2$-model with generic site-dependent inhomogeneity and arbitrary boundary fields is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix are given in terms of an inhomogeneous T-Q relation, which is based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices. Moreover, the associated Bethe Ansatz equations are also obtained.

Exact Solution of Two-Site Bose–Hubbard Model with Generic Open Boundaries**Supported by the National Natural Science Foundation of China under Grant Nos. 11375141, 11425522, 11434013, 11347025, 11447239, and Beijing Center for Mathematics and Information Interdisciplinary Sciences are gratefully acknowledged.

The Bose–Hubbard model is a paradigm for the study of strongly correlated bosonic systems. We study the two-site Bose–Hubbard model with generic integrable open boundaries specified by the most general non-diagonal reflecting matrices. Besides the inhomogeneous parameters, the model itself has three free boundary parameters, which break the U(1)-symmetry, in other words, break the particle number conservation. The Hamiltonian H under these circumstances is constructed. With the help of the off-diagonal Bethe Ansatz method, we successfully obtain the corresponding Bethe Ansatz equations as well as the eigenvalues.

Off-diagonal Bethe Ansatz solution of the τ 2-model

The generic quantum τ 2-model (also known as Baxter-Bazhanov-Stroganov (BBS) model) with periodic boundary condition is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix (solutions of the recursive functional relations in τ j -hierarchy) with generic site-dependent inhomogeneity parameters are given in terms of an inhomogeneous T − Q relation with polynomial Q-functions. The associated Bethe Ansatz equations are obtained. Numerical solutions of the Bethe Ansatz equations for small number of sites indicate that the inhomogeneous T − Q relation does indeed give the complete spectrum.

Small polaron with generic open boundary conditions: Exact solution via the off-diagonal Bethe ansatz

The small polaron, a one-dimensional lattice model of interacting spinless fermions, with generic non-diagonal boundary terms is studied by the off-diagonal Bethe ansatz method. The presence of the Grassmann valued non-diagonal boundary fields gives rise to a typical U(1)-symmetry-broken fermionic model. The exact spectra of the Hamiltonian and the associated Bethe ansatz equations are derived by constructing an inhomogeneous T–Q relation.

Exact solution of the alternating XXZ spin chain with generic non-diagonal boundaries

The integrable XXZ alternating spin chain with generic non-diagonal boundary terms specified by the most general non-diagonal K-matrices is studied via the off-diagonal Bethe Ansatz method. Based on the intrinsic properties of the fused R-matrices and K-matrices, we obtain certain closed operator identities and conditions, which allow us to construct an inhomogeneous T−Q relation and the associated Bethe Ansatz equations accounting for the eigenvalues of the transfer matrix.

Spectrum of the Open Asymmetric Simple Exclusion Process with Arbitrary Boundary Parameters**Supported by the National Natural Science Foundation of China under Grant Nos 11375141, 11475135, 11434013 and 11425522, the Ministry of Education Doctoral Program Fund under Grant No 20126101110004, and the Northwest University Graduate Student Innovation Fund under Grant No YZZ14104.

We study the one-dimensional asymmetric simple exclusion process (ASEP) with generic open boundaries (including current-counting deformation), and obtain the exact solutions of this ASEP via the off-diagonal Bethe ansatz method. In particular, numerical results for the small size asymmetric simple exclusion process indicate that the spectrum obtained by the Bethe ansatz equations is complete. Moreover, we present the eigenvalue of the totally asymmetric exclusion process and the corresponding Bethe ansatz equations.

Exact spectrum of the spin-s Heisenberg chain with generic non-diagonal boundaries

The off-diagonal Bethe ansatz method is generalized to the high spin integrable systems associated with the su(2) algebra by employing the spin-s isotropic Heisenberg chain model with generic integrable boundaries as an example. With the fusion techniques, certain closed operator identities for constructing the functional T-Q relations and the Bethe ansatz equations are derived. It is found that a variety of inhomogeneous T-Q relations obeying the operator product identities can be constructed. Numerical results for two-site s=1 case indicate that an arbitrary choice of the derived T-Q relations is enough to give the complete spectrum of the transfer matrix.