Spin chain description of rotating bosons at $\nu=1$

We consider bosons at Landau level filling $\nu=1$ on a thin torus. In analogy with previous work on fermions at filling $\nu =1/2$, we map the low-energy sector onto a spin-1/2 chain. While the fermionic system may realize the gapless XY-phase, we show that typically this does not happen for the bosonic system. Instead, both delta function and Coulomb interaction lead to gapped phases in the bosonic system, and in particular we identify a phase corresponding to the non-abelian Moore-Read state. In the spin language, the hamiltonian is dominated by a ferromagnetic next-nearest neighbor interaction, which leads to a description consistent with the non-trivial degeneracies of the ground and excited states of this phase of matter. In addition we comment on the similarities and differences of the two systems mentioned above and fermions at $\nu=5/2$.