An Exactly Solvable Model of Generalized Spin Ladder

A detailed study of an S = 1 2 spin ladder model is given. The ladder consists of plaquettes formed by nearest neighbor rungs with all possible SU(2)-invariant interactions. For properly chosen coupling constants, the model is shown to be integrable in the sense that the quantum YangBaxter equation holds and one has an infinite number of conserved quantities. The R-matrix and L-operator associated with the model Hamiltonian are given in a limiting case. It is shown that after a simple transformation, the model can be solved via a Bethe ansatz. The phase diagram of the ground state is exactly derived using the Bethe ansatz equation. PACS numbers: 75.10.Jm SFB 256; SFB 237; BiBoS; CERFIM (Locarno); Acc.Arch., USI (Mendrisio) Institute of Physics, Chinese Academy of Science, Beijing. AvH fellow. 1 Heisenberg spin ladders and generalized spin ladders have attracted considerable attention in recent years, due to the developing experimental results on ladder materials and the hope to get some insight into the physics of metal-oxide superconductors [1]. Especially generalized ladders including other couplings beyond the simplest case of rung and leg exchange interpolate among a variety of systems and exhibit a remarkably rich behavior [2-9]. In particular, it has been shown that the diagonal interactions may cause frustration and change the structure of the ground state [2, 3], while the biquadratic interactions, which can arise due to effective spin-spin interaction mediated by phonons in real magnetic systems [4], tend to produce dimerization and may lead to a phase transition into a “non-Haldane” spin liquid state with absence of magnon excitations [4, 5]. As spin ladders are generally not equivalent to spin chains with nearest neighbor interactions, till now little is known about integrable spin ladder models. In this letter we study a generalized S = 1 2 spin ladder system with both isotropic exchange interactions and biquadratic interactions. Using ideas related to the quantum Yang-Baxter equations [10] we found in our systems some cases of integrable ladder systems, in the sense of models having an infinite number of conserved quantities with explicit R matrices satisfying the Yang-Baxter equation. Properly choosing the spectral parameter, we get a Hamiltonian consisting of only nearest-neighbor and next-nearest-neighbor interactions. This model can be solved via an ordinary Bethe ansatz. The present work was submitted for publication in Euro. Phys. Lett.. During the preparation of a revised version of this paper, another integrable ladder model without diagonal interactions was presented in [11].