Delta-Function Bose Gas Picture of S=1 Antiferromagnetic Quantum Spin Chains Near Critical Fields

We study the zero-temperature magnetization curve (M-H curve) of the S=1 bilinear-biquadratic spin chain, whose Hamiltonian is given by $H=\sum_{i} S_i S_{i+1}+\beta (S_iS_{i+1})^2 with$0 \leq \beta <1$. We focus on validity of the delta-function bose-gas picture near the two critical fields: the saturation field$H_s$and the lower critical field$H_c$associated with the Haldane gap. Near$H_s$, we take ``low-energy effective S-matrix'' approach, which gives correct effective bose-gas coupling constant$c$, different from the spin-wave value. Comparing the M-H curve of the bose gas with the product-wavefunction renormalization group (PWFRG) calculation, excellent agreement is seen. Near$H_c$, comparing the PWFRG result with the bose-gas prediction, we find that there are two distinct regions of$\beta$separated by a critical value$\beta_c(\approx 0.41)$. In the region$0<\beta<\beta_c$, the effective coupling$c$is positive but rather small. The small value of$c$makes the ``critical region'' of the square-root behavior$M\sim \sqrt{H-H_c}$very narrow. Further, we find that in the$\beta \to \beta_c-0$, the square-root behavior transmutes to a different one,$M\sim (H-H_c)^{1/4}$. In the region$\beta_c<\beta <1$, the square-root behavior is rather distinct, but the effective coupling$c$ becomes negative.