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.

Figure 1

The staggered field dependence of the transverse and longitudinal gaps (${\Delta }_T$, solid circles; ${\Delta }_L$, solid triangles) and the midgap-state energies (${\Delta }_T^m$, open circles; ${\Delta }_L^m$, open squares) for the spin-$1/2$ chain. The lines are fittings to the $h_s^{2/3}|\log h_s{|}^{1/6}$ scaling behavior anticipated from the conformal field theory [4]. The slopes of the plots which determine, up to log corrections, the scaling exponents of the leading order term are shown in the inset with the same axis labels, in the logarithmic scale. Data show a common scaling exponent of $\gamma =2/3$ when the field is not too high.

Figure 2

The excitation energies versus $1/L$ for several lowest states of the spin-$1/2$ chain labeled by $S_{\alpha }^{S_{\mathrm{tot}}}$ (upper panel) and $S_{\alpha }^{S_{\mathrm{tot}}}(|S_{\mathrm{tot}}^z|)$ (lower panel) $[\alpha =A,B,C]$. They all scale to zero as $L\to \infty$ at $h_s=0$. At finite $h_s$ ($h_s=0.05$ shown), they split into a longitudinal (lines) and a transverse (symbols) branch with distinct gap and midgap states [$S_A^1(1)\to {\Delta }_T^m$, $S_B^1(1),S_C^1(1)\to {\Delta }_T$, $S_A^1(0),S_B^1(0)\to {\Delta }_L^m$, and $S_C^1(0)\to {\Delta }_L$; see text for details].

Figure 3

The staggered field dependence of the transverse and longitudinal gaps (${\Delta }_T$, solid circles; ${\Delta }_L$, solid squares) and the midgaps (${\Delta }_T^m$, open circles, $|S_{\mathrm{tot}}^z|=1$; ${\Delta }_L^m$, open squares, $S_{\mathrm{tot}}^z=0$; down-pointing triangles, $S_{\mathrm{tot}}^z=0$; diamonds, $S_{\mathrm{tot}}^z=0$; up-pointing triangles, $|S_{\mathrm{tot}}^z|=2$) in the spin-1 chain. The lines are a guide to the eyes only. The midgap states falling off from the longitudinal continuum are not connected by lines for clarity.

Figure 4

The staggered field dependence of the transverse and longitudinal gaps (${\Delta }_T$, solid circles; ${\Delta }_L$, solid squares) and the corresponding lower midgap-state energies (${\Delta }_T^m$, open circles; ${\Delta }_L^m$, open triangles) in the spin-$3/2$ chain. The lines are fittings to $h_s^{2/3}|\log h_s{|}^{1/6}$ for these low-energy excitations.