Erratum: “Spin and Chiral Orderings of Frustrated Quantum Spin Chains” [J. Phys. Soc. Jpn. 68, 3185 (1999)

Abstract

In p. 3186, below Eq. (3), it was incorrectly stated that by the exact-diagonalization (ED) calculation for the open chains with even N [8 N 20 (S = 1/2) and 8 N 16 (S = 1)], the ground state was found to belong to the subspace of S total 1⁄4 0 with even parity.1) In fact, for the S = 1/2 XY and Heisenberg open chains, the ground state belongs to the subspace of S total 1⁄4 0 with even (odd) parity when N = 4n (4n + 2), where n is an integer. For the S = 1 case, the ground state for a wide parameter region belongs to the subspace of S total 1⁄4 0 with even parity, while the ground state with odd parity appears in some parameter regions, whose position is confined within a range 0:3 . j . 0:6 and changes depending on the system size N and on whether XY or Heisenberg. This correction has no effect on the other parts of the paper. The data of the Binder parameter shown in Fig. 1 were obtained by the ED calculation for the whole subspace of S total 1⁄4 0 including both the even and odd parity states, and therefore, they are not affected by the correction. As for the density-matrix renormalization-group (DMRG) calculation for the S = 1 XY chains, we note that the appearance of the odd-parity ground state found by the ED is a boundary effect in small systems with frustration, and the infinite-system DMRG for the subspace of S total 1⁄4 0 with even parity should capture the ground-state properties in thermodynamic limit correctly. Indeed, we have newly performed the DMRG calculation for the whole subspace of S total 1⁄4 0 and obtained the same results as the previous ones within a numerical accuracy. As to the data shown in Fig. 2, for example, the differences between the previous and newly-obtained results averaged on the distance r are at most 1:2 10 3 (chiral correlation), 1:5 10 4 (string correlation), and 4:8 10 4 (spin correlation), except for the chiral and string correlations at the transition point j 1⁄4 jc1 1⁄4 0:473 for which the averaged differences are 1:2 10 2 or less (chiral) and 1:1 10 3 (string). These differences have no effect on our argument on the behaviors of the correlation functions, and therefore, our conclusions in the paper remain the same. We also correct a misprint in Fig. 2(a): The distance r of the chiral correlation C ðrÞ was shifted by one due to a simple error in the data processing. Therefore, the label for the x-axis should read r + 1 instead of r. Finally, we note that, although the ED data of the Binder parameter in Fig. 1 suggested the absence of the chiral order in the S = 1/2 XY chains, it has been revealed by the DMRG that the chiral order indeed appears in the S = 1/2 chains with easy-plane anisotropy. The DMRG results and the reason why the ED analysis missed the chiral order in the S = 1/2 chains have already been reported in Ref. 2.