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 ν = p/(2p + 1) (and ν = 1 − p/(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 e^−2π2 / L²₁ which serves as an expansion parameter away from the TT limit (L₁ → 0). We also corroborate earlier results on the smooth Fermi surface deformation of the gapless state at ν = 1/2, interpolating between a state described by a critical S = 1/2 chain and the bulk.