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We numerically investigate the ground-state phase diagram of an S=2 quantumspin chain with the $XXZ$ and on-site anisotropies described by ${\mathcalH}=\sum_j (S_j^x S_{j+1}^x+S_j^y S_{j+1}^y+\Delta S_j^z S_{j+1}^z) + D \sum_j(S_j^z)^2$, where $\Delta$ denotes the XXZ anisotropy parameter of thenearest-neighbor interactions and $D$ the on-site anisotropy parameter. Werestrict ourselves to the $\Delta>0$ and $D>0$ case for simplicity. Our mainpurpose is to obtain the definite conclusion whether there exists or not theintermediate-$D$ (ID) phase, which was proposed by Oshikawa in 1992 and hasbeen believed to be absent since the DMRG studies in the latter half of 1990's.In the phase diagram with $\Delta>0$ and $D>0$ there appear the XY state, theHaldane state, the ID state, the large-$D$ (LD) state and the N\'eel state. Inthe analysis of the numerical data it is important to distinguish three gappedstates; the Haldane state, the ID state and the LD state. We give a physicaland intuitive explanation for our level spectroscopy method how to distinguishthese three phases.

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