Non-Abelian quantum Hall states of fermions and bosons

In a non-Abelian quantum Hall state, there are types of elementary excitations or quasiparticles that obey non-Abelian statistics. This is an extension of the idea of fractional statistics that means that when several of these quasiparticles are present in the system and are well-separated at well-defined positions, there is a degenerate space of lowest-energy states. When the quasiparticles are then exchanged adiabatically, the result is a matrix operation on this space of states. Greg Moore and the author$^1$ introduced this idea to condensed matter physics in 1991. They proposed a basic example called the Moore-Read Pfaffian state. The physics of the existence of the degenerate states for the quasiparticles in this system can be understood by viewing it as a $p_x-ip_y$ paired state of composite fermions, in which quasiparticles are $hc/2e$ vortices that carry Majorana fermion zero modes. This state is expected to be realized in the filling factor $\nu=5/2$ fractional quantum Hall (FQH) state. In later work, a series (labeled by an integer $k$) of ``parafermion'' states was proposed$^2$. This talk will review these ideas, and describe recent numerical work that strongly supports the idea that the $k=3$ member of the sequence occurs in the filling factor $12/5$ FQH state for electrons$^3$, and also$^4$ in a system of bosons, such as rotating cold atoms, at filling factor $3/2$. It will also describe recent analytical results$^5$ on the explicit quasihole trial wavefunctions of the parafermion states. \newline 1. G. Moore and N. Read, Nucl. Phys. {\bf B 360}, 362 (1991). \newline 2. N. Read and E. Rezayi, Phys. Rev. B {\bf 59}, 8084 (1999). \newline 3. E.H. Rezayi and N. Read, cond-mat/0608346. \newline 4. E.H. Rezayi, N. Read, and N.R. Cooper, Phys. Rev. Lett.{\bf 95}, 160404 (2005). \newline 5. N. Read, Phys. Rev. B {\bf 73}, 245334 (2006).