  We report on the study of the fractional quantum Hall effect at the filling
factor 5/2 using exact diagonalization method with torus geometry. The
particle-hole symmetry breaking effect is considered using an additional
three-body interaction. Both Pfaffian and anti-Pfaffian states can be the
ground state depending on the sign of the three-body interaction. The results
of the low-energy spectrum, the wave function overlap, and the particle-hole
parity evolution, have shown the clear evidence of a direct sharp transition
(possibly first-order) from the Pfaffian to the anti-Pfaffian state at the
Coulomb point. A quantum phase diagram is established, where one finds further
transitions from the Pfaffian or anti-Pfaffian state to the nearby compressible
phases induced by a change of the pseudopotential.
