δ-function Bose-gas picture of $S=1\mathrm{}$ antiferromagnetic quantum spin chains near critical fields

We study the zero-temperature magnetization curve $(M-H$ curve) of the one-dimensional quantum antiferromagnet of spin one. The Hamiltonian H we consider is of the bilinear-biquadratic form: $H={\sum }_{i}f\left(s_i\cdot s_{i+1\mathrm{}}\right)$ (+Zeeman term) where $s_i$ is the spin operator at site i and $f\left(X\right)=X+\beta X^2$ with $0<~\beta <1.\mathrm{}$ We focus on validity of the $\delta$-function Bose-gas picture near the two critical fields: upper-critical field $H_s$ above which the magnetization saturates and the lower-critical field $H_c$ associated with the Haldane gap. As for the behavior near $H_s,$ we take “low-energy effective S matrix” approach, where the correct effective Bose-gas coupling constant c is extracted from the two down-spin S matrix in its low-energy limit. We find that the resulting value of c differs from the spin-wave value. We draw the $M-H$ curve by using the resultant Bose gas, and compare it with numerical calculation where the product-wave-function renormalization-group (PWFRG) method, a variant of White’s density-matrix renormalization group method, is employed. Excellent agreement is seen between the PWFRG calculation and the correctly mapped Bose-gas calculation. We also test the validity of the Bose-gas picture near the lower-critical field $H_c.$ Comparing the PWFRG-calculated $M-H$ curves with the Bose-gas prediction, we find that there are two distinct regions, I and II, of $\beta$ separated by a critical value ${\beta }_c(\approx 0.41).$ In region I, $0<\mathrm{}\beta <{\beta }_c,$ the effective Bose 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 $\overrightarrow{\beta }{\beta }_c-0,$ the square-root behavior transmutes to a different one, $M\sim (H-H_c{)}^{\theta }$ with $\theta \approx 1/4.$ In region II, ${\beta }_c<\beta <1,$ the square-root behavior is more pronounced as compared with region I, but the effective coupling c becomes negative.