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Excitation spectra of S=1/2 and S=1 frustrated Heisenberg antiferromagneticchains with bond alternation (explicit dimerization) are studied using acombination of analytical and numerical methods. The system undergoes adimerization transition at a critical bond alternation parameter$\delta=\delta_{\rm c}$, where $\delta_{\rm c} = 0$ for the S=1/2 chain. TheSU(2)-symmetric sine-Gordon theory is known to be an effective field theory ofthe system except at the transition point. The sine-Gordon theory has aSU(2)-triplet and a SU(2)-singlet of elementary excitation, and the mass ratio$r$ of the singlet to the triplet is $\sqrt{3}$. However, our numericalcalculation with the infinite time-evolving block decimation method shows that$r$ depends on the frustration (next-nearest-neighbor coupling) and isgenerally different from $\sqrt{3}$. This can be understood as an effect ofmarginal perturbation to the sine-Gordon theory. In fact, at the criticalfrustration separating the second-order and first-order dimerizationtransitions, the marginal operator vanishes and $r=\sqrt{3}$ holds. We derivethe mass ratio $r$ analytically using form-factor perturbation theory combinedwith a renormalization-group analysis. Our formula agrees well with thenumerical results, confirming the theoretical picture. The present theory alsoimplies that, even in the presence of a marginally irrelevant operator, themass ratio approaches $\sqrt{3}$ in the very vicinity of the second-orderdimerization critical point $\delta \sim \delta_c$. However, such a region isextremely small and would be difficult to observe numerically.

Mass ratio of elementary excitations in frustrated antiferromagnetic chains with dimerization