Non-Abelian fractional quantum Hall states for hard-core bosons in one dimension

Abstract

I present a family of one-dimensional bosonic liquids analogous to non-Abelian fractional quantum Hall states. A new quantum number is introduced to characterize these liquids, the chiral momentum, which differs from the usual angular or linear momentum in one dimension. As their two-dimensional counterparts, these liquids minimize a $k$-body hard-core interaction with the minimum total chiral momentum. They exhibit global order, with a hidden organization of the particles in $k$ identical copies of a one-dimensional Laughlin state. For $k=2$ the state is a $p$-wave paired phase corresponding to the Pfaffian quantum Hall state. By imposing conservation of the total chiral momentum, an exact parent Hamiltonian is derived which involves long-range tunneling and interaction processes with an amplitude decaying with the chord distance. This family of non-Abelian liquids is shown to be in formal correspondence with a family of spin-$\frac{k}{2}$ liquids which are total singlets made out of $k$ indistinguishable resonating valence bond states. The corresponding spin Hamiltonians are obtained.