We study the low-energy excitations of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain and $N$-leg $(N=2,3,4)$ ladders in a staggered magnetic field $h_s$. We show that $h_s$ induces gap and midgap states in all the cases and we will examine their field scaling behavior. A modified boundary scheme is devised to extract accurate bulk excitation behavior. The gap values converge rapidly as $N$ increases, leading to a field scaling exponent $\gamma =\frac{1}{2}$ for both the longitudinal and transverse gaps of the square lattice $(N\to \infty )$. The midgap states induced by the boundary edge effects share the bulk gap scaling exponents but their overall scaling behavior in the large-$N$ limit needs further investigation.

• Received 25 July 2005

DOI:https://doi.org/10.1103/PhysRevB.73.012411