How to distinguish the Haldane/Large-D state and the intermediate-D state in an S = 2 quantum spin chain with the XXZ and on-site anisotropies

We numerically investigate the ground-state phase diagram of an S = 2 quantum spin chain with the XXZ and on-site anisotropies described by = Σ_j(S_j^xS_{j+ 1}^x +S_j^yS_{j + 1}^y + Δ S_j^z S_j+1^z) + D Σ_j(S_j^z)², where Δ denotes the XXZ anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. We restrict ourselves to the Δ > 0 and D > 0 case for simplicity. Our main purpose is to obtain the definite conclusion whether there exists or not the intermediate-D (ID) phase, which was proposed by Oshikawa in 1992 and has been believed to be absent since the DMRG studies in the latter half of 1990's. In the phase diagram with Δ > 0 and D > 0 there appear the XY state, the Haldane state, the ID state, the large-D (LD) state and the Néel state. In the analysis of the numerical data it is important to distinguish three gapped states; the Haldane state, the ID state and the LD state. We give a physical and intuitive explanation for our level spectroscopy method how to distinguish these three phases.