Numerical study of the hidden antiferromagnetic order in the Haldane phase

We investigate the string correlation functions proposed by den Nijs and Rommelse for S=1 Heisenberg antiferromagnets in one dimension. The Hamiltonian with D${\mathrm{\Sigma }}_{\mathit{i}}$(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ ${)}^2$ term is diagonalized by the Lanczos method to obtain the ground state. We calculate both the usual spin-spin correlation functions and the string correlation functions not only in the z direction (quantized direction) but also in the x direction to investigate the ${\mathit{Z}}_2$×${\mathit{Z}}_2$ symmetry breaking recently proposed by Kennedy and Tasaki. We find that the long-range string correlation, which is argued to exist in the Haldane disordered phase, in fact, exists at the Heisenberg point D=0 by a finite-size analysis. We can show explicitly that the string correlation in the x direction signifies the difference between the Haldane phase and the Néel phase, which appears for the D<0, ‖D‖≳1 case. In the large-D (D≳1) phase, all spin-spin correlations are of short range as expected. There is a significant enhancement in the usual and string correlations in the x direction at the boundary between the Haldane phase and the large-D phase.