Discussion on the convergence in the orbital number

The convergence in the orbital number is an important issue in MCTDH-X. The choice of the orbital number is a trade-off between computation speed and simulation accuracy. Without enough orbitals, we will not only obtain incorrect particle correlations, but sometimes even an incorrect density distribution. The simplest method to determine the convergence is by comparing the occupations. A higher-order orbital is considered insignificant, when its occupation is significantly lower than the lower-order orbitals. This criterion sometimes fails when the seemingly insignificant orbitals contain important information about particle correlations and fluctuations of the simulated many-body state.

An intuitive example is strongly-interacting bosons. Fig. S1(a) shows the real-space density distribution of bosons with extremely strong interaction simulated with different orbital number. When orbital is used, the interaction can only be captured in the Gross-Pitaevskii mean-field manner, and the many-body state simply reduces to a Thomas-Fermi cloud. As soon as the second orbital is used, the double-peak density distribution can be captured, though an obvious discrepancy with the fermionic distribution can be observed. Such a discrepancy diminishes as more orbitals are used.

A more intuitive discussion on the convergence of MCTDH-X results with the orbital number can be illustrated by the comparison between the standard single-band Bose-Hubbard model and the simulation results for bosons in a Mott insulator state.