  The high Tc cuprates have a complex phase diagram with many competing phases.
We propose a bosonic effective quantum Hamiltonian based on the projected SO(5)
model with extended interactions, which can be derived from the microscopic
models of the cuprates. The global phase diagram of this model is obtained
using mean-field theory and the Quantum Monte Carlo simulation, which is
possible because of the absence of the minus sign problem. We show that this
single quantum model can account for most salient features observed in the high
Tc cuprates, with different families of the cuprates attributed to different
traces in the global phase diagram. Experimental consequences are discussed and
new theoretical predictions are presented.
