Numerical Study of the String Order in S=1 Antiferromagnetic Heisenberg Chain

Abstract

We study the spin-1 antiferromagnetic Heisenberg chain under the three boundary conditions, which are the free, periodic and antiperiodic boundary conditions, by numerical diagonalization up to 16 sites. We checked that the lowest excitation caused by the domain wall, which is imposed by the boundary condition and breaks the long-range string order, yields the same energy gap under all the boundary conditions in the infinite length limit. It is also found that the ground state and the first excited state have the same density (about 3%) of domain walls due to the quantum fluctuation which reduces the string order parameter about 10% smaller than that of the VBS model in the ground state. The analysis of the mean distance between two domain walls suggests that domain walls due to the quantum fluctuation form a bound state in pairs and does not break the long-range string order.