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

Kouichi Okunishi; Yasuhiro Hieida; Yasuhiro Akutsu

doi:10.1103/PhysRevB.59.6806

paper

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

arXiv:cond-mat/9807266 · doi:10.1103/PhysRevB.59.6806

Abstract

We study the zero-temperature magnetization curve (M-H curve) of the S=1 bilinear-biquadratic spin chain, whose Hamiltonian is given by $H = \sum_{i} S_{i} S_{i + 1} + β (S_{i} S_{i + 1})^{2} w i t h$ 0 \leq β<1 $. W e f oc u so n v a l i d i t y o f t h e d e l t a - f u n c t i o nb ose - g a s p i c t u r e n e a r t h e tw ocr i t i c a l f i e l d s : t h es a t u r a t i o n f i e l d$ H_s $an d t h e l o w er cr i t i c a l f i e l d$ H_c $a ssoc ia t e d w i t h t h eH a l d an e g a p . N e a r$ H_s $, w e t ak e ‘‘ l o w - e n er g y e f f ec t i v e S - ma t r i x^{''} a pp r o a c h, w hi c h g i v escor r ec t e f f ec t i v e b ose - g a sco u pl in g co n s t an t$ c $, d i f f er e n t f r o m t h es p in - w a v e v a l u e . C o m p a r in g t h e M - H c u r v eo f t h e b ose g a s w i t h t h e p r o d u c t - w a v e f u n c t i o n r e n or ma l i z a t i o n g r o u p (P W F R G) c a l c u l a t i o n, e x ce l l e n t a g r ee m e n t i ssee n . N e a r$ H_c $, co m p a r in g t h e P W F R G r es u l tw i t h t h e b ose - g a s p r e d i c t i o n, w e f in d t ha tt h er e a r e tw o d i s t in c t r e g i o n so f$ β $se p a r a t e d b y a cr i t i c a l v a l u e$ β_c(\approx 0.41) $. I n t h er e g i o n$ 0<β<β_c $, t h ee f f ec t i v eco u pl in g$ c $i s p os i t i v e b u t r a t h er s ma l l . T h es ma l l v a l u eo f$ c $mak es t h e ‘‘ cr i t i c a l r e g i o n^{''} o f t h es q u a r e - r oo t b e ha v i or$ M\sim \sqrt{H-H_c} $v er y na r r o w . F u r t h er, w e f in d t ha t in t h e$ β\to β_c-0 $, t h es q u a r e - r oo t b e ha v i or t r an s m u t es t o a d i f f er e n t o n e,$ M\sim (H-H_c)^{1/4} $. I n t h er e g i o n$ β_c<β<1 $, t h es q u a r e - r oo t b e ha v i or i sr a t h er d i s t in c t, b u tt h ee f f ec t i v eco u pl in g$ c$ becomes negative.

6 pages, RevTeX, 8 ps figures