Midgap States in Antiferromagnetic Heisenberg Chains with A Staggered Field

We study low-energy excitations in antiferromagnetic Heisenberg chains with a staggered field which splits the spectrum into a longitudinal and a transverse branch. Bound states are found to exist inside the field induced gap in both branches. They originate from the edge effects and are inherent to spin-chain materials. The sine-Gordon scaling $h_s^{2/3}|\log h_s|^{1/6}$ ($h_s$: the staggered field) provides an accurate description for the gap and midgap energies in the transverse branch for $S=1/2$ and the midgap energies in both branches for $S=3/2$ over a wide range of magnetic field; however, it can fit other low-energy excitations only at much lower field. Moreover, the integer-spin S=1 chain displays scaling behavior that does not fit this scaling law. These results reveal intriguing features of magnetic excitations in spin-chain materials that deserve further investigation.