Mass ratio of elementary excitations in frustrated antiferromagnetic chains with dimerization

Abstract

Excitation spectra of $S=1/2$ and 1 frustrated Heisenberg antiferromagnetic chains with bond alternation (explicit dimerization) are studied using a combination of analytical and numerical methods. The system undergoes a dimerization transition at a critical bond alternation parameter $\ensuremath{\delta}={\ensuremath{\delta}}_{\mathrm{c}}$, where ${\ensuremath{\delta}}_{\mathrm{c}}=0$ for the $S=1/2$ chain. The $\text{SU}(2)$-symmetric sine-Gordon theory is known to be an effective field theory of the system except at the transition point. The sine-Gordon theory has a $\text{SU}(2)$-triplet and a $\text{SU}(2)$-singlet of elementary excitation, and the mass ratio $r$ of the singlet to the triplet is $\sqrt{3}$. However, our numerical calculation with the infinite time-evolving block decimation method shows that $r$ depends on the frustration (next-nearest-neighbor coupling) and is generally different from $\sqrt{3}$. This can be understood as an effect of marginal perturbation to the sine-Gordon theory. In fact, at the critical frustration separating the second-order and first-order dimerization transitions, the marginal operator vanishes and $r=\sqrt{3}$ holds. We derive the mass ratio $r$ analytically using form-factor perturbation theory combined with a renormalization-group analysis. Our formula agrees well with the numerical results, confirming the theoretical picture. The present theory also implies that, even in the presence of a marginally irrelevant operator, the mass ratio approaches $\sqrt{3}$ in the very vicinity of the second-order dimerization critical point $\ensuremath{\delta}\ensuremath{\sim}{\ensuremath{\delta}}_{c}$. However, such a region is extremely small and would be difficult to observe numerically.