Boundary K-supermatrices for the one-dimensional small-polaron open chain

Huan-Qiang Zhougif
 
 

CCAST (World Laboratory), PO Box 8730, Beijing 100080, People's Republic of China, and

Department of Physics, Chongqing University Chongqing, Sichuan 630044, People's Republic of Chinagif

Received 21 Sept. 1998

Abstract:

The Lax pair for the one-dimensional small-polaron open chain is explicitly constructed. From this the general boundary K-supermatrices are found. Our construction provides a direct demonstration for the integrability of the system.

In the last decade, much attention has been paid to the study of completely integrable lattice spin open chains [1] - [8]. As was shown by Sklynanin [1], there is a variant of the usual formalism of the quantum inverse scattering method (QISM) [9] - [11], which may be used to describe systems on a finite interval with independent boundary conditions on each end. Central to his approach is the introduction of an algebraic structure called the reflection equations (RE) [12]. Although Sklyanin's argument was carried out only for the P and T invariant R-matrices, it is now known that the formalism may be extended to apply to any systems integrable by the quantum R-matrix approach [8]. Much attention has been paid to the solutions of RE which present the boundary K-matrices compatible with the integrability. Recently, the boundary K-matrices have been constructed by several groups [6] for the Heisenberg spin- open chain and by the present author [7,8] for the one-dimensional (1D) Hubbard open chain and for the 1D Bariev open chain.

On the other hand, the traditional basis for applying QSIM to a completely integrable system is to represent the equations of motion of the system into Lax form. Following Korepin et al [9,10], one may show that, for systems with periodic boundary conditions, the existence of the quantum R-matrix allows one to express the original equations of motion in Lax form. In particular, the Lax pairs for a variety of physically interesting models were given in [13] - [16]. Thus, one may expect that there is a variant of the usual Lax pair formulation to describe quantum integrable lattice open chains. Recently, we have shown that such a formulation does exist [17].

The aim of this letter is to present the Lax pair for the 1D small-polaron open chain in explicit form. From this the general boundary K-supermatrices are determined. Our construction provides a direct description for the quantum integrability of the system.

Let us first recall the Lax pair formulation for completely integrable lattice fermion open chains described in [17]. Instead of directly considering the equations of motion, let us study an operator version of an auxiliary linear problem:

 Here , and  are some supermatrices depending on the spectral parameter  and the dynamical variables. The consistency conditions for equation (1) yield the Lax equations

 A lattice fermion open chain is said to be completely integrable if we can express the equations of motion in the Lax form (2), provided the boundary K-supermatrices exist as the solutions of equations (4) and (5) below. In fact, it is readily shown that a transfer matrix

 does not depend on time t, provided the constrains hold:

 and

 with

 Here the supertrace `str' is taken over the auxiliary superspace. This implies that the system under study possesses an infinite number of conserved quantities.

Now, let us study the 1D small-polaron open chain with Hamiltonian

 Here  and  are, respectively, the creation and annihilation operators at lattice site j, and satisfy the usual anti-commutation relations

 and  is the density operator, . Furthermore, is a coupling parameter and  and are some members of Grassmann algebra with  even and  odd, satisfying .

It is not difficult to check that the equations of motion derived from the Hamiltonian (7) may be cast into the Lax form (2). Indeed, in our case, the L and M matrices take the form [16]

 and

 with

 Here we emphasize that the L and M matrices are supermatrices with parities . From equation (2), it follows that

 with

 and

 with

 where .

We now proceed to study the constraint conditions (4) and (5). Let us assume the boundary K-supermatrix  to take the form

 and, substituting into equation (4), one may check that, out of the 16 homogeneous linear equations about  and, only three are independent. After some algebraic calculations, we find

 (up to an unimportant scalar factor). In order to determine the boundary supermatrix , let us first note that

 Obviously, the matrix elements of  (anti)-commute with those of . Keeping this fact in mind, and noting that the matrix elements of  are independent, we immediately obtain

 (up to an unimportant scalar factor).

Now let us show that the Hamiltonian (7) may be related with the transfer matrix (3). Indeed, expanding the transfer matrix (3) in powers of , we have

 Thus, we have shown that the model under consideration admits the Lax pair formulation.

In conclusion, we have presented the Lax pair for the 1D small-polaron open chain. The boundary K-supermatrices thus constructed should be the solutions of the graded version of the reflection equations. Thus our construction provides a basis for establishing the graded version of Sklyanin's formalism to describe integrable systems with boson and fermion fields in the finite interval. Here we emphasize that in contrast with the periodic case [16,18], the Hamiltonian (7) may not be mapped into the 1D Heisenberg XXZ open chain via the Jordan - Wigner transformation. This implies that the system under consideration is essentially new. The extension of our construction to other open fermion chains, such as the 1D Hubbard open chain [7] and the 1D Bariev open chain [8], is also interesting. We hope to return to these questions in the near future.

This work was supported in part by the National Natural Science Foundation of China under grant No 19505009. I am grateful to Professor Xing-Chang Song and Professor Chong-Sheng Li for their support and encouragement.

Please note: this document was submitted by a third party for peer review and has been included in the online working set due to the extremely favorable review it received. -- Articulum Editor

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Footnotes

...Zhou
E-mail address: cul@cbistic.sti.ac.cn

 

 

...China
Mailing address.