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 $0\le \beta <1$. 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, $M\sim (H-H_c{)}^{1\mathrm{/}4}$. In the region ${\beta }_c<\beta <1$, the square-root behavior is rather distinct, but the effective coupling $c$ becomes negative.