Thermodynamics of integrable chains with alternating spins.

We consider a two-parameter (c̄ , c̃) family of quantum integrable Hamiltonians for a chain of alternating spins of spin s = 1/2 and s = 1. We determine the thermodynamics for low-temperature T and small external magnetic field H, with T << H. In the antiferromagnetic (c̄ > 0 , c̃ > 0) case, the model has two gapless excitations. In particular, for c̄ = c̃, the model is conformally invariant and has central charge cvir = 2. When one of these parameters is zero, the Bethe Ansatz equations admit an infinite number of solutions with lowest energy. † Laboratoire de Physique Théorique et Hautes Energies, Tour 16 1er. étage, Université Paris VI, 4, place Jussieu, 75252 Paris Cedex 05, FRANCE * Department of Physics, University of Miami, Coral Gables, FL 33124, USA 1 The one-dimensional Heisenberg model, like the hydrogen atom, has served as the guiding example for a very large body of both experimental and theoretical work. Progress has recently been made on closely related models, consisting of chains with alternating spins, such as spin 1/2 and spin 1. On the experimental side, materials (e.g., [MnCp2] [TCNE]) have been synthesized which, at temperatures above a certain transition temperature Tc, behave as one-dimensional ferromagnets of alternating spins. On the theoretical side, quantum integrable models of chains with alternating spins have recently been constructed. In this Letter we investigate the thermodynamics of a two-parameter family of such integrable models. Depending on the values of the parameters, we find both antiferromagnetic and ferromagnetic behavior. When one of these parameters is zero, the Bethe Ansatz equations admit an infinite number of solutions with lowest energy. We consider a system of N spins 1 2~σ2 , 1 2~σ4 , · · · , 1 2~σ2N of spin 1/2 and N spins ~s1 , ~s3 , · · · , ~s2N−1 of spin 1 in an external magnetic field H(≥ 0) with the Hamiltonian H given by H = c̄H̄+ c̃H̃ −HS , (1) where S = ∑N n=1 1 2 σ 2n + ∑N n=1 s z 2n−1, H̄ = − 1 9 N ∑ n=1 (2~σ2n · ~s2n+1 + 1) (2~σ2n+2 · ~s2n+1 + 3) , (2)