Finite-size scaling for low-energy excitations in integer Heisenberg spin chains

In this paper we study the finite-size scaling for low-energy excitations of S=1 and S=2 Heisenberg chains, using the density matrix renormalization-group technique. A crossover from 1/L behavior (with L as the chain length) for medium chain length to 1/${\mathrm{L}}^2$ scaling for long chain length is found for excitations in the continuum band as the length of the open chain increases. Topological spin S=1/2 excitations are shown to give rise to the two lowest energy states for both open and periodic S=1 chains. In periodic chains these two excitations are ``confined'' next to each other, while for open chains they are two free-edge 1/2 spins. The finite-size scaling of the two lowest energy excitations of open S=2 chains is determined by coupling the two free-edge S=1 spins. The gap and correlation length for S=2 open Heisenberg chains are shown to be 0.082 (in units of the exchange J) and 47, respectively.