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arXiv:cond-mat/9711150

Bethe ansatz for the one-dimensional small-polaron model with open boundary conditions

Heng Fan, Xi‐Wen Guan

Published Nov 16, 1997• 1 citation
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Abstract

The one-dimensional small-polaron model with open boundary conditions is considered in the framework of the quantum inverse scattering method. The spin model which is equivalent to the small-polaron model is the Heisenberg $XXZ$ spin chain in an external magnetic field. For spin model, the solutions of the reflection equation (RE) and the dual RE are obtained. The eigenvalues and eigenvectors problems are solved by using the algebraic Bethe ansatz method. The case of the fermion model is also studied, the commuting of the transfer matrix can be proved, and the eigenvalues and the Bethe ansatz equations for fermion model are obtained.

DOI
10.48550/arxiv.cond-mat/9711150

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The one-dimensional small-polaron model with open boundary conditions is considered in the framework of the quantum inverse scattering method. The spin model which is equivalent to the small-polaron model is the Heisenberg XXZXXZXXZ spin chain in an external magnetic field. For spin model, the solutions of the reflection equation (RE) and the dual RE are obtained. The eigenvalues and eigenvectors problems are solved by using the algebraic Bethe ansatz method. The case of the fermion model is also studied, the commuting of the transfer matrix can be proved, and the eigenvalues and the Bethe ansatz equations for fermion model are obtained.
The one-dimensional small-polaron model with open boundary conditions is considered in the framework of the quantum inverse scattering method. The spin model which is equivalent to the small-polaron model is the Heisenberg X X Z spin chain in an external magnetic field. For spin model, the solutions of the reflection equation (RE) and the dual RE are obtained. The eigenvalues and eigenvectors problems are solved by using the algebraic Bethe ansatz method. The case of the fermion model is also studied, the commuting of the transfer matrix can be proved, and the eigenvalues and the Bethe ansatz equations for fermion model are obtained.
Exact integrability and algebraic Bethe ansatz of the small-polaron model with the open boundary condition are discussed in the framework of the quantum inverse scattering method (QISM). We employ a new approach where the fermionic R-operator which consists of fermion operators is a key object. It satisfies the Yang-Baxter equation and the reflection equation with its corresponding K-operator. Two kinds of 'super-transposition' for the fermion operators are defined and the dual reflection equation is obtained. These equations prove the integrability and the Bethe ansatz equation which agrees with the one obtained from the graded Yang-Baxter equation and the graded reflection equations.
The nested Bethe ansatz (BA) method is applied to find the eigenvalues and the eigenvectors of the transfer matrix for spin-ladder model with open boundary conditions. Based on the reflection equation, we find the general diagonal solution, which determines the general boundary interaction in the Hamiltonian. We introduce the spin-ladder model with open boundary conditions. By finding the solution K± of the reflection equation which determines the nontrivial boundary terms in the Hamiltonian, we diagonalize the transfer matrix of the spin-ladder model with open boundary conditions in the framework of nested BA.
The small polaron with generic, nondiagonal boundary terms is investigated within the framework of quantum integrability. The fusion hierarchy of the transfer matrices and its truncation for particular values of the anisotropy parameter are both employed, so that the spectral problem is formulated in terms of a TQ equation. The solution of this equation for generic boundary conditions is based on a deformation of the diagonal case. The eigenvalues of the model are extracted and the corresponding Bethe ansatz equations are presented. Finally, we comment on the eigenvectors of the model and explicitly compute the eigenstate of the model which evolves into the Fock vacuum when the off-diagonal boundary terms are switched off.
Employing the graded versions of the Yang-Baxter equation and the reflection equations, we construct two kinds of integrable impurities for a small-polaron model with general open boundary conditions: (a) we shift the spectral parameter of the local Lax operator at arbitrary sites in the bulk and (b) we embed the impurity fermion vertex at each boundary of the chain. The Hamiltonians with different types of impurity terms are given explicitly. The Bethe ansatz equations, as well as the eigenvalues of the Hamiltonians, are constructed by means of the quantum inverse scattering method. In addition, we discuss the ground-state properties in the thermodynamic limit.