Long-Distance Universality of Laughlin State and Calogero-Sutherland Model

We study the universal long-distance behaviour of the Laughlin state for the fractional quantum Hall effect and the ground state of the Calogero-Sutherland model (one dimensional $1/r^2$ interaction model). In particular, it is shown that these two wave functions coincide exactly when Laughlin state is confined in a narrow cylinder geometry. The seeming difference of dimensionality is merely a difference of representation of wave functions. We also give a recipe to interpret operators acting on states in the lowest Landau level in terms of the usual one dimensional Fermion operators, which is important for extracting the Tomonaga-Luttinger liquid behaviour of the edge states.