arXiv1996
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 $δ$ of the nearest neighbor exchanges. In the spin-$1 \over 2$ chain, the system is gapless for $δ=0$ and $J_2 < J_{2c} =0.241$, and is gapped everywhere else in the $J_2 - δ$ plane. At $J_{2c}$, for small $δ$, the gap increases as $δ^α$, where $α= 0.667 \pm 0.001$. $2J_2 + δ= 1$ is a disorder line. To the left of this line, the structure factor $S(q)$ peaks at $q_{max} = π$ (Neel `phase'), while to the right, $q_{max}$ decreases from $π$ to $π/2$ (spiral `phase') as $J_2$ increases. There is also a `$\uparrow \uparrow \downarrow \downarrow$ phase' for large values of both $J_2$ and $δ$. In the spin-$1$ case, we find a line running from a gapless point at $(J_2 , δ) = (0,0.25 \pm 0.01)$ 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 $δ=0.136$. Similar to the spin-$1 \over 2$ case, to the left of this line, the peak in the structure factor is at $π$ (Neel `phase'), while to the right of the line, it is at less than $π$ (spiral `phase'). For $δ=1$, 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.