Comparison of the $\pi$ and $\mathrm{\Delta }$ operator approximations for the two-dimensional t-J model

We compare the spectra of the new $\pi$ operator of the SO(5) theory and the conventional $\mathrm{\Delta }$ operator for the two-dimensional t-J model. We also calculate the weight transferred to the two-hole ground state from half-filling by these operators. We find that the spectra of these operators are quite similar and the weight for the $\pi$ operator is smaller than the weight for the $\mathrm{\Delta }$ operator. We argue that the two-dimensional t-J model does not have a good approximate SO(5) symmetry claimed in Ref. 1.