δ-function Bose-gas picture of S = 1 antiferromagnetic quantum spin chains near critical fields

摘要: We study the zero-temperature magnetization curve $(M\ensuremath{-}H$ curve) of one-dimensional quantum antiferromagnet spin one. The Hamiltonian H we consider is bilinear-biquadratic form: $H={\ensuremath{\sum}}_{i}f({s}_{i}\ensuremath{\cdot}{s}_{i+1})$ (+Zeeman term) where ${s}_{i}$ operator at site i and $f(X)=X+\ensuremath{\beta}{X}^{2}$ with $0l~\ensuremath{\beta}l1.$ focus on validity $\ensuremath{\delta}$-function Bose-gas picture near two critical fields: upper-critical field ${H}_{s}$ above which saturates lower-critical ${H}_{c}$ associated Haldane gap. As for behavior ${H}_{s},$ take ``low-energy effective S matrix'' approach, correct coupling constant c extracted from down-spin matrix in its low-energy limit. find that resulting value differs spin-wave value. draw $M\ensuremath{-}H$ by using resultant Bose gas, compare it numerical calculation product-wave-function renormalization-group (PWFRG) method, a variant White's density-matrix renormalization group employed. Excellent agreement seen between PWFRG correctly mapped calculation. also test ${H}_{c}.$ Comparing PWFRG-calculated curves prediction, there are distinct regions, I II, $\ensuremath{\beta}$ separated ${\ensuremath{\beta}}_{c}(\ensuremath{\approx}0.41).$ In region I, $0l\ensuremath{\beta}l{\ensuremath{\beta}}_{c},$ positive but rather small. small makes ``critical region'' square-root $M\ensuremath{\sim}\sqrt{H\ensuremath{-}{H}_{c}}$ very narrow. Further, $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}{\ensuremath{\beta}}_{c}\ensuremath{-}0,$ transmutes to different one, $M\ensuremath{\sim}(H\ensuremath{-}{H}_{c}{)}^{\ensuremath{\theta}}$ $\ensuremath{\theta}\ensuremath{\approx}1/4.$ ${\ensuremath{\beta}}_{c}l\ensuremath{\beta}l1,$ more pronounced as compared becomes negative.