Lax pair for SU ( n ) Hubbard model

For one dimensional SU(n) Hubbard model, a pair of Lax operators are derived, which give a set of fundamental equations for the quantum inverse scattering method under both periodic and open boundary conditions. This provides another proof of the integrability of the model under periodic boundary condition. PACS Numbers: 46.10.+z, 05.40.+j, 05.60.+w e-mail: [email protected] Integrable strongly correlated electron systems have been an important research subject in condensed matter physics and mathematics. One of the significant models is the 1-D Hubbard. The exact solution was given by Lieb and Wu [1]. However, the integrability was shown twenty years later by Shastry, Olmedilla and Wadati [2, 3]. The integrability and the exact solution of the system under the open boundary condition were discussed by several authors [4, 5]. The Lax pair was first given by Wadati, Olmedilla and Akutsu [6]. Recently, Maassarani and Mathieu have constructed the hamiltonian SU(n) XX model and proved its integrability [7]. Considering two coupled SU(n) XX models, Maassarani succeeded in generalizing Shastry’s method to SU(n) Hubbard model [8]. Further, he solved the YangBaxter equation to prove the integrability of one dimensional SU(n) Hubbard model [9]. ( It is also proved by Martins for n = 3, 4 [10].) In this paper, we apply the quantum inverse scattering method to 1-dimensional SU(n) Hubbard model and derive the explicit form of the Lax pair, which gives another proof of the integrability. It is worthy to point out that one may not apply the reflection equation to study the integrable open boundary due to the fact that the R matrix given in reference [9] does not satisfy the crossing symmetry condition. The Lax pair formalism will give an effective method for such a system. The 1-dimensional SU(n) model in the Schrödinger picture is given by