Phase transitions in antiferromagnetic quantum spin chains with bond alternation

We study the phase transitions in antiferromagnetic quantum spin chains with bond alternation, H=J${\sum }_{\mathrm{i}}$[1-(-1${)}^{\mathrm{i}}$δ]${\mathbf{S}}_{\mathrm{i}}$⋅${\mathbf{S}}_{\mathrm{i}+1\mathrm{}}$, of spin S=1, 3/2, and 2. On the one hand, using a transfer matrix technique, we make a simple variational approach, which results in the better estimates of the transition points than by the O(3) nonlinear-σ-model quantum field theory. On the other hand, employing a quantum Monte Carlo method, we calculate the generalized string order parameter, ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(θ)=${\lim }_{\mathrm{L}\to \mathrm{\infty }}$ ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(θ;L) with ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(θ;L)=〈${\mathrm{S}}_{\mathrm{L}\mathrm{/}4}^{\mathrm{z}}$ ${\prod }_{\mathrm{j}\mathrm{=}\mathrm{L}\mathrm{/}4}^{3\mathrm{L}\mathrm{/}4\mathrm{-}1}$exp[iθ${\mathrm{S}}_{\mathrm{j}}^{\mathrm{z}}$]${\mathrm{S}}_{3\mathrm{L}\mathrm{/}4}^{\mathrm{z}}$〉. It turns out that the transition points are successfully detected by observing the overall behavior of ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(θ) at various values of δ. Investigating the dependences of ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(θ;L) on θ, L, and S, we discuss the applicability of the valence-bond-solid picture to the ground states of the present Hamiltonian.