From frustration-free parent Hamiltonians to off-diagonal long-range order: Moore-Read and related states in second quantization

We construct a recursive second-quantized formula for Moore-Read Pfaffian states. We demonstrate the utility of such second-quantized presentations by directly proving the existence of frustration-free parent Hamiltonians, without appealing to polynomial clustering properties. Furthermore, we show how this formalism is connected to the existence of a nonlocal order parameter for Moore-Read states, and we provide proof that the latter exhibit off-diagonal long-range order (ODLRO) in these quantities. We also develop a similar second-quantized presentation for the fermionic anti- and PH-Pfaffian states, as well as f- and higher wave paired composite fermion states, and we discuss ODLRO in most cases.