  A new numerical algorithm for interacting fermion systems to treat the
grand-canonical ensemble is proposed and examined by extending the
path-integral renormalization group method. To treat the grand-canonical
ensemble, the particle-hole transformation is applied to the Hamiltonian and
basis states. In the interaction-term projection, the Stratonovich-Hubbard
transformation which hybridizes up and down spin electrons is introduced. By
using this method, the phase diagram of the two-dimensional Hubbard model with
next-nearest-neighbor transfer is accurately determined by treating the
filling-control (FC) and bandwidth-control (BC) Mott transitions on the same
ground. A V-shaped Mott insulating phase is obtained in the plane of the
chemical potential and the Coulomb interaction, where the transitions at the
corner (BC) and the edges (FC) show contrasted characters with large critical
fluctuations near the edges coexisting with the first-order transition at the
corner. This contrasted behavior is shown to be consistent with the V-shape
structure of the phase boundary because of a general relation, in which the
slope of the metal-insulator transition line in the phase diagram is expressed
by thermodynamic quantities. The V-shaped opening of the Mott gap is favorably
compared with the experimental results of the transition metal oxides.
