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-SIGMA(i)(S(i)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 Z2 X Z2 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 Neel phase, which appears for the D < 0, \D\greater-than-or-similar-to 1) case. In the large-D (D greater-than-or-similar-to 1) phase, all spin-spin correlations are short ranged 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.