A comparative study of the phase diagrams of spin-$\frac{1}{2}$ and spin-$1$ antiferromagnetic chains with dimerization and frustration

Abstract

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

16 pages in latex; 9 figures available on request