Magnetization process of bilinear-biquadratic spin chains at finite temperature

We investigate magnetization process (M-H0ex0excurve) of $S=1$ bilinear-biquadratic (BLBQ) spin chain near critical fields at finite temperatures. We use the density matrix renormalization group (DMRG) for the two-dimensional classical lattice model mapped by the Trotter decomposition, with a help of the Baxter's variational principle. By comparing the DMRG result of S=1/2 XY chain with the exact solution, we show that the DMRG is an efficient tool to calculate the M-H curve at finite temperatures. Further, we compute the M-H curve of the BLBQ chain. We compare the DMRG curve of the magnetization process of the BLBQ chain with those obtained by analytic approaches: correctly mapped δ-function Bose-gas approach and ``Bethe-ansatz approximation'' approach. Near the saturation field, we show that the δ-function Bose gas and the Bethe-ansatz approximation describe the M-H curve well in both at zero temperature and finite temperatures. Near the lower critical field, we find that the δ-function Bose gas is a good effective model at the Heisenberg point. We further find that the δ-function Bose gas cannot describe the M-H curve at finite temperatures, near the special point where the M-H curve changes qualitatively.