Spontaneous Particle-Hole Symmetry Breaking in the $\nu=5/2$ Fractional Quantum Hall Effect

The essence of the $\nu=5/2$ fractional quantum Hall effect is believed to be well captured by the Moore-Read Pfaffian (or anti-Pfaffian) description. However, an important mystery regarding the formation of the Pfaffian state is the role of the three-body interaction Hamiltonian $H_3$ that produces the Pfaffian as an exact ground state and the concomitant particle-hole symmetry breaking. We show that a two-body interaction Hamiltonian $H_2$ constructed via particle-hole symmetrization of $H_3$ produces a ground state nearly exactly approximating the Pfaffian and anti-Pfaffian states, respectively, in the spherical geometry. More importantly, the ground state energy of $H_2$ is shown to exhibit a ``Mexican-hat'' structure as a function of particle number in the vicinity of half filling for a given flux indicating spontaneous particle-hole symmetry breaking. We do not find any such Mexican-hat signature in the second Landau level Coulomb interaction at 5/2.