  The relation between quantum phase transitions, entanglement, and geometric
phases is investigated with a system of two qubits with XY type interaction. A
seam of level crossings of the system is a circle in parameter space of the
anisotropic coupling and the transverse magnetic field, which is identical to
the disorder line of an one-dimensional XY model. The entanglement of the
ground state changes abruptly as the parameters vary across the circle except
specific points crossing to the straight line of the zero magnetic field. At
those points the entanglement does not change but the ground state changes
suddenly. This is an counter example that the entanglement is not alway a good
indicator to quantum phase transitions. The rotation of the circle about an
axis of the parameter space produces the magnetic monopole sphere like a
conducting sphere of electrical charges. The ground state evolving
adiabatically outside the sphere acquires a geometric phase, whereas the ground
state traveling inside the sphere gets no geometric phase. The system also has
the Renner-Teller intersection which gives no geometric phases.
