Beyond the Tao-Thouless limit of the fractional quantum Hall effect: spin chains and Fermi surface deformation

We discuss the relationship between the fractional quantum Hall effect in the vicinity of the thin-torus, a.k.a. Tao-Thouless (TT), limit and quantum spin chains. We argue that the energetics of fractional quantum Hall states in Jain sequence at filling fraction $\nu =p\mathrm{/}(2p+1)$ (and $\nu =1-p\mathrm{/}(2p+1)$) in the lowest Landau level is captured by S=1 spin chains with $p$ spins in the unit cell. These spin chains naturally arise at sub-leading order in ${\text{e}}^{-2{\pi }^2\mathrm{/}{L}_1^2}$ which serves as an expansion parameter away from the TT limit ($L_1\to 0$). We also corroborate earlier results on the smooth Fermi surface deformation of the gapless state at $\nu =1\mathrm{/}2$, interpolating between a state described by a critical $S=1\mathrm{/}2$ chain and the bulk.