Magnetization process of bilinear-biquadratic spin chains at finite temperature

We investigate magnetization process $(M-H\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e})$ of $S=1\mathrm{}$ 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 $S=1/2\mathrm{}\hspace{0.5em}\mathrm{XY}$ 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 $\delta$-function Bose-gas approach and “Bethe-ansatz approximation” approach. Near the saturation field, we show that the $\delta$-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 $\delta$-function Bose gas is a good effective model at the Heisenberg point. We further find that the $\delta$-function Bose gas cannot describe the $M-H$ curve at finite temperatures, near the special point where the $M-H$ curve changes qualitatively.