Functional representation of the partition function and phase transitions in the Hubbard model

The functional representation of the partition function for the Hubbard model with strong Coulomb repulsion is obtained as integral by auxiliary Grassmann variables.
The approximative study of the phase transitions showes:
1) for the small doping constant $\delta$ the system is an antiferromagnetic isolator;
2) for doping $\delta\ge t/U$ the system is a ferromagnetic metal for $U/t>10$, and a superconductor metal for $U/t\lesssim10$.
Bibliography: 13 titles.