Fermionic R-Operator and Integrability of the One-Dimensional Hubbard Model.pdf

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Fermionic R-Operator and Integrability of the One-Dimensional Hubbard Model.pdf

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Fermionic R-Operator and Integrability of the One-Dimensional Hubbard Model

a r X i v : c o n d - m a t / 9 8 0 6 1 4 4 v 2 [ c o n d - m a t .s t r - e l ] 1 6 J u n 1 9 9 8 Fermionic R-Operator and Integrability of the One-Dimensional Hubbard Model February 1, 2008 Yukiko Umeno ?, Masahiro Shiroishi ?? and Miki Wadati Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan We propose a new type of the Yang-Baxter equation (YBE) and the decorated Yang- Baxter equation (DYBE). Those relations for the fermionic R-operator were introduced recently as a tool to treat the integrability of the fermion models. Using the YBE and the DYBE for the XX fermion model, we construct the fermionic R-operator for the one- dimensional (1D) Hubbard model. It gives another proof of the integrability of the 1D Hubbard model. Furthermore a new approach to the SO(4) symmetry of the 1D Hubbard model is discussed. KEYWORDS fermion XY Z model, one-dimensional Hubbard model, fermionic R-operator, Yang-Baxter equation, SO(4) symmetry 1 Introduction There have been much interests in strongly correlated electron systems. Among the exactly solvable ones, the one-dimensional (1D) Hubbard model, H = ? N∑ j=1 ∑ σ=↑↓ (c?j+1σcjσ + c ? jσcj+1σ) + U N∑ j=1 (nj↑ ? 1 2 )(nj↓ ? 1 2 ). (1.1) ? E-mail: umeno@monet.phys.s.u-tokyo.ac.jp ?? E-mail: siroisi@monet.phys.s.u-tokyo.ac.jp 1 is the most interesting model. Here c?jσ and cjσ are the fermion creation and annihilation operators and njσ is the fermion number operator, njσ = c ? jσcjσ. (1.2) The parameter U is the coupling constant describing the Coulomb interaction between electrons with opposite spins on the same site. Lieb and Wu [1] diagonalized the Hamilto- nian (1.1) by means of the coordinate Bethe ansatz method under the periodic boundary condition (PBC), c?N+1,σ = c ? 1σ, cN+1,σ = c1σ, σ =↑↓ . (1.3) The bulk properties of the 1D Hubbard model have been investigated by analyzing the associated Bethe ansatz equations. [2] The integrability of the 1D Hubba