Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states*

The fractional quantum Hall (FQH) effect encompasses a wide range of quantum many-body phases which are characterized by exotic topological orders. Among these, the Read-Rezayi (RR) states harbor parafermionic excitations, whose non-Abelian braiding properties could potentially be used to carry out fault-tolerant quantum computation. Recent intriguing numerical results indicate that the 12/5 FQH state realized in GaAs could be described by the particle-hole conjugate of an RR state. However, numerically constructing the RR states for large systems, which is necessary for their characterization and for the demonstration of the exotic properties of their excitations, is computationally prohibitive. In this work, we use the parton framework to construct states that lie in the same phases as the particle-hole conjugates of the RR states. A nice feature of our parton states is that their wave functions can be evaluated for very large system sizes, thus, paving the way for the numerical investigation of parafermions.