Dynamical Transition from Triplets to Spinon Excitations: A Series Expansion Study of the $J_1-J_2-\delta$ spin-half chain

We study the spin-half Heisenberg chain with alternating nearest neighbor interactions $J_1(1+\delta )$ and $J_1(1-\delta )$ and a uniform second neighbor interaction $J_2=y(1-\delta )$ by series expansions around the limit of decoupled dimers ($\delta =1$). By extrapolating to $\delta =0$ and tuning $y$, we study the critical point separating the power-law and spontaneously dimerized phases of the spin-half antiferromagnet. We then focus on the disorder line $y=0.5$, $0\le \delta \le 1$, where the ground states are known exactly. We calculate the triplet excitation spectrum, their spectral weights and wavevector dependent static susceptibility along this line. It is well known that as $\delta \to 0$, the spin-gap is still non-zero but the triplets are replaced by spinons as the elementary excitations. We study this dynamical transition by analyzing the series for the spectral weight and the static susceptibility. In particular, we show that the spectral weight for the triplets vanishes and the static spin-susceptibility changes from a simple pole at imaginary wavevectors to a branch cut at the transition.