Transition from Quantum Hall to Compressible States in the Second Landau Level: New Light on the $\nu =5\mathrm{}/2$ Enigma

Quantum Hall states at filling fraction $\nu =5/2\mathrm{}$ are examined by numerical diagonalization. Spin-polarized and spin-unpolarized states of systems with $N\le 18$ electrons are studied, neglecting effects of Landau level mixing. We find that the ground state is spin polarized. It is incompressible and has a large overlap with paired states like the Pfaffian. For a given sample, the energy gap is about 11 times smaller than at $\nu =1/3\mathrm{}$. Evidence is presented of phase transitions to compressible states, driven by the interaction strength at short distance. A reinterpretation of experiments is suggested.