A comparative study of the phase diagrams of spin-1 over 2 and spin-1 antiferromagnetic chains with dimerization and frustration

We use the density matrix renormalization group method to study the ground state `phase' diagram and some low-energy properties of isotropic antiferromagnetic spin-$1 \over 2$ and spin-$1$ chains with a next-nearest neighbor exchange $J_2 ~$ and an alternation $\delta$ of the nearest neighbor exchanges. In the spin-$1 \over 2$ chain, the system is gapless for $\delta=0$ and $J_2 < J_{2c} =0.241$, and is gapped everywhere else in the $J_2 - \delta$ plane. At $J_{2c}$, for small $\delta$, the gap increases as $\delta^{\alpha}$, where $\alpha = 0.667 \pm 0.001$. $2J_2 + \delta = 1$ is a disorder line. To the left of this line, the structure factor $S(q)$ peaks at $q_{max} = \pi$ (Neel `phase'), while to the right, $q_{max}$ decreases from $\pi$ to $\pi/2$ (spiral `phase') as $J_2$ increases. There is also a `$\uparrow \uparrow \downarrow \downarrow$ phase' for large values of both $J_2$ and $\delta$. In the spin-$1$ case, we find a line running from a gapless point at $(J_2 , \delta) = (0,0.25 \pm 0.01)$ upto a `gapless' point at $(0.73 \pm 0.005,0)$ such that the open chain ground state is four-fold degenerate below the line and is unique above it. There is a disorder line in this case also and it has the same equation as in the spin-$1 \over 2$ case, but the line ends at about $\delta =0.136$. Similar to the spin-$1 \over 2$ case, to the left of this line, the peak in the structure factor is at $\pi$ (Neel `phase'), while to the right of the line, it is at less than $\pi$ (spiral `phase'). For $\delta =1$, the system corresponds to a spin ladder and the system is gapped for all values of the interchain coupling for both spin-$1 \over 2$ and spin-$1$ ladders.